Basic Derivatives of Functions
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Derivative of Exponents and Logarithmic Functions
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Trigonometric Derivatives
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Derivative of Power Functions
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Implicit Differentiation
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Calculations Using The Definition of Derivative
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The Derivative of an Inverse of a Function
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Logarithmic Differentiation
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[{"Name":"Basic Derivatives of Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Polynomials","Duration":"13m 42s","ChapterTopicVideoID":10128,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.580","Text":"In this clip, I\u0027m going to discuss the concept of the derivative of a function,"},{"Start":"00:05.580 ","End":"00:08.055","Text":"but only from the technical aspect,"},{"Start":"00:08.055 ","End":"00:12.945","Text":"not what it means or where it came from or what to do with it,"},{"Start":"00:12.945 ","End":"00:15.090","Text":"just the technical aspects."},{"Start":"00:15.090 ","End":"00:19.815","Text":"I\u0027ll put something in writing to each function, f of x,"},{"Start":"00:19.815 ","End":"00:24.450","Text":"we associate a new function called the derivative of f"},{"Start":"00:24.450 ","End":"00:28.860","Text":"in which we denote f prime of x."},{"Start":"00:28.860 ","End":"00:31.575","Text":"Important words is derivative,"},{"Start":"00:31.575 ","End":"00:36.760","Text":"and the notation is by adding a little prime sign here."},{"Start":"00:36.760 ","End":"00:39.725","Text":"Let me give a couple of examples now,"},{"Start":"00:39.725 ","End":"00:42.935","Text":"of a function with its derivative."},{"Start":"00:42.935 ","End":"00:52.250","Text":"If the function is f of x equals x squared plus 1 over x plus 8,"},{"Start":"00:52.250 ","End":"01:00.890","Text":"then the derivative turns out to be f prime of x equals 2x minus 1 over x squared."},{"Start":"01:00.890 ","End":"01:04.745","Text":"Another example, this time we can use the y notation."},{"Start":"01:04.745 ","End":"01:09.140","Text":"If y equals x natural log of x,"},{"Start":"01:09.140 ","End":"01:13.265","Text":"then the derivative this time we denote it as y prime,"},{"Start":"01:13.265 ","End":"01:18.710","Text":"happens to be natural log of x plus 1."},{"Start":"01:18.710 ","End":"01:21.320","Text":"In this clip and in some of the following clips,"},{"Start":"01:21.320 ","End":"01:26.150","Text":"we\u0027re going to learn how to compute this function from this function."},{"Start":"01:26.150 ","End":"01:30.740","Text":"In other words, how do we get from one of these to one of these?"},{"Start":"01:30.740 ","End":"01:32.120","Text":"What is the process?"},{"Start":"01:32.120 ","End":"01:37.325","Text":"It turns out there are various rules and this is what we\u0027re going to learn,"},{"Start":"01:37.325 ","End":"01:40.805","Text":"but the process of converting this one to this one"},{"Start":"01:40.805 ","End":"01:44.690","Text":"from the function to the derivative is called differentiation."},{"Start":"01:44.690 ","End":"01:49.400","Text":"But in some books it\u0027s also called derivation."},{"Start":"01:49.400 ","End":"01:51.410","Text":"If you differentiate this function,"},{"Start":"01:51.410 ","End":"01:52.535","Text":"you get this one."},{"Start":"01:52.535 ","End":"02:00.485","Text":"This function is derived from this one so differentiation, derivation, synonymous."},{"Start":"02:00.485 ","End":"02:03.560","Text":"To learn the art of differentiation,"},{"Start":"02:03.560 ","End":"02:07.295","Text":"i.e getting from here to here involves"},{"Start":"02:07.295 ","End":"02:11.795","Text":"lots of rules which are called differentiation rules."},{"Start":"02:11.795 ","End":"02:13.220","Text":"There are loads of these,"},{"Start":"02:13.220 ","End":"02:17.525","Text":"I mean dozens now are probably hundreds of differentiation rules."},{"Start":"02:17.525 ","End":"02:20.615","Text":"We\u0027ll learn some of them, and for the rest,"},{"Start":"02:20.615 ","End":"02:25.490","Text":"there are reference books and all the calculus books and on the Internet,"},{"Start":"02:25.490 ","End":"02:27.380","Text":"in the Wikipedia and so on."},{"Start":"02:27.380 ","End":"02:29.045","Text":"There will be a lot of others."},{"Start":"02:29.045 ","End":"02:31.625","Text":"We\u0027ll just learn a few of them and how to use them."},{"Start":"02:31.625 ","End":"02:36.485","Text":"There are rules for differentiating all the elementary functions,"},{"Start":"02:36.485 ","End":"02:40.670","Text":"constant functions, polynomials, sine, cosine,"},{"Start":"02:40.670 ","End":"02:43.835","Text":"and other trigonometric rules, logarithms,"},{"Start":"02:43.835 ","End":"02:47.570","Text":"exponents and there are also rules for how to combine things,"},{"Start":"02:47.570 ","End":"02:52.040","Text":"how you get the sum of functions or the product of"},{"Start":"02:52.040 ","End":"02:57.245","Text":"functions or quotients and differences and all this thing."},{"Start":"02:57.245 ","End":"03:04.075","Text":"Let\u0027s begin on the next page and we\u0027ll learn dozen so rules. Here we go."},{"Start":"03:04.075 ","End":"03:07.595","Text":"By the way, differentiation rules,"},{"Start":"03:07.595 ","End":"03:11.060","Text":"sometimes also called derivative rules."},{"Start":"03:11.060 ","End":"03:12.500","Text":"Here\u0027s the first one,"},{"Start":"03:12.500 ","End":"03:15.445","Text":"but let me start the counting at 0."},{"Start":"03:15.445 ","End":"03:18.920","Text":"We\u0027ll use the y prime notation."},{"Start":"03:18.920 ","End":"03:23.615","Text":"The first rule says that if y equals a,"},{"Start":"03:23.615 ","End":"03:26.015","Text":"by which I mean a constant, a number,"},{"Start":"03:26.015 ","End":"03:31.580","Text":"then the derivative y prime is 0."},{"Start":"03:31.580 ","End":"03:38.525","Text":"For example, if y equals 4 is the function,"},{"Start":"03:38.525 ","End":"03:43.505","Text":"then its derivative y prime is 0."},{"Start":"03:43.505 ","End":"03:45.950","Text":"If I take y equals,"},{"Start":"03:45.950 ","End":"03:47.810","Text":"let me think a bit more complicated,"},{"Start":"03:47.810 ","End":"03:51.830","Text":"e plus the square root of 2 over Pi."},{"Start":"03:51.830 ","End":"03:58.280","Text":"Still, the derivative is 0 because this doesn\u0027t have any x\u0027s in it."},{"Start":"03:58.280 ","End":"03:59.975","Text":"It\u0027s still just a number."},{"Start":"03:59.975 ","End":"04:01.880","Text":"The y notation is just shorter,"},{"Start":"04:01.880 ","End":"04:05.105","Text":"but I could write the same thing with f notation."},{"Start":"04:05.105 ","End":"04:09.305","Text":"That if f of x is equal to a,"},{"Start":"04:09.305 ","End":"04:13.160","Text":"then f prime of x is equal to 0."},{"Start":"04:13.160 ","End":"04:15.950","Text":"But generally, we\u0027ll use the y notation."},{"Start":"04:15.950 ","End":"04:17.755","Text":"Now on to the next one,"},{"Start":"04:17.755 ","End":"04:22.120","Text":"that if y equals x to the power of n,"},{"Start":"04:22.120 ","End":"04:24.430","Text":"where n is some number, a whole number."},{"Start":"04:24.430 ","End":"04:26.320","Text":"Then the derivative,"},{"Start":"04:26.320 ","End":"04:31.260","Text":"y prime is nx to the n minus 1."},{"Start":"04:31.260 ","End":"04:33.145","Text":"We\u0027ll give some examples in a moment."},{"Start":"04:33.145 ","End":"04:36.005","Text":"I would just also like to say that in general,"},{"Start":"04:36.005 ","End":"04:37.550","Text":"when we write derivative,"},{"Start":"04:37.550 ","End":"04:40.690","Text":"sometimes we don\u0027t write f of x equals or y equals."},{"Start":"04:40.690 ","End":"04:50.135","Text":"We just simply say that x to the n derivative is nx to the n minus 1, just for brevity."},{"Start":"04:50.135 ","End":"04:58.345","Text":"An example of one of these more concrete would be that if y equals x to the 10th, say,"},{"Start":"04:58.345 ","End":"05:07.985","Text":"then y prime would equal 10x to the power of 10 minus 1."},{"Start":"05:07.985 ","End":"05:12.080","Text":"But usually we do the 10 minus 1 in our heads and write"},{"Start":"05:12.080 ","End":"05:16.850","Text":"straightaway 10x to the 9th immediately."},{"Start":"05:16.850 ","End":"05:22.380","Text":"Or if you want to write it as x to the 10th,"},{"Start":"05:22.420 ","End":"05:28.610","Text":"derivative equals 10x to the 9th."},{"Start":"05:29.210 ","End":"05:33.260","Text":"Often in the books It\u0027s appears this way without the y."},{"Start":"05:33.260 ","End":"05:38.415","Text":"Another example, y equals x to the 8th."},{"Start":"05:38.415 ","End":"05:42.585","Text":"Y prime is. 8x,"},{"Start":"05:42.585 ","End":"05:46.395","Text":"and I\u0027ll do the 8 minus 1 in my head and I get 7."},{"Start":"05:46.395 ","End":"05:49.740","Text":"If y equals x cubed,"},{"Start":"05:49.740 ","End":"05:55.185","Text":"then y prime is 3x squared,"},{"Start":"05:55.185 ","End":"05:57.390","Text":"I did the 3 minus 1 in my head."},{"Start":"05:57.390 ","End":"05:59.575","Text":"One final example here."},{"Start":"05:59.575 ","End":"06:03.190","Text":"If y equals just x,"},{"Start":"06:03.190 ","End":"06:08.545","Text":"then what we do is we say y equals x to the power of 1."},{"Start":"06:08.545 ","End":"06:11.275","Text":"If nothing\u0027s written, it means to the power of 1,"},{"Start":"06:11.275 ","End":"06:19.435","Text":"in which case y prime is equal to 1x to the power of 0,"},{"Start":"06:19.435 ","End":"06:22.000","Text":"but x to the 0 is 1,"},{"Start":"06:22.000 ","End":"06:25.520","Text":"which means that this is just equal to 1."},{"Start":"06:25.520 ","End":"06:31.880","Text":"Now, usually this is such a frequent one that you remember the result right away,"},{"Start":"06:31.880 ","End":"06:33.830","Text":"that if y is x,"},{"Start":"06:33.830 ","End":"06:37.410","Text":"then y prime equals 1."},{"Start":"06:37.600 ","End":"06:42.395","Text":"In one step, and I highlighted it because it\u0027s so important,"},{"Start":"06:42.395 ","End":"06:46.595","Text":"just remember that the derivative of the function x is 1."},{"Start":"06:46.595 ","End":"06:48.590","Text":"Also in the shortcut notation,"},{"Start":"06:48.590 ","End":"06:52.395","Text":"we can just write the derivative of x is 1,"},{"Start":"06:52.395 ","End":"06:55.010","Text":"1 meaning the constant function, 1."},{"Start":"06:55.010 ","End":"06:58.040","Text":"Now just let me tidy up a moment."},{"Start":"06:58.040 ","End":"07:00.410","Text":"Erase this stuff here."},{"Start":"07:00.410 ","End":"07:05.645","Text":"It\u0027s in the way, crutch work at the side and continue with the examples."},{"Start":"07:05.645 ","End":"07:09.935","Text":"If y equals x squared,"},{"Start":"07:09.935 ","End":"07:13.580","Text":"then y prime is 2x,"},{"Start":"07:13.580 ","End":"07:16.075","Text":"it\u0027s 2x to the 1."},{"Start":"07:16.075 ","End":"07:21.145","Text":"But 2x to the 1 is just 2x."},{"Start":"07:21.145 ","End":"07:24.050","Text":"Now something more interesting."},{"Start":"07:24.050 ","End":"07:30.770","Text":"What happens if we try something like y equals 1 over x?"},{"Start":"07:30.770 ","End":"07:33.425","Text":"How is this related to this rule?"},{"Start":"07:33.425 ","End":"07:39.530","Text":"Well, actually this does belong here because we can write 1 over x as"},{"Start":"07:39.530 ","End":"07:46.730","Text":"y equals x to the power of minus 1 if you remember your algebra rules of exponents,"},{"Start":"07:46.730 ","End":"07:49.340","Text":"which means that we can use this formula."},{"Start":"07:49.340 ","End":"07:52.870","Text":"Let me highlight this since this is the more general one."},{"Start":"07:52.870 ","End":"07:57.545","Text":"What we get if we let n equals minus 1 here,"},{"Start":"07:57.545 ","End":"08:02.730","Text":"we get that y prime is equal to n,"},{"Start":"08:02.730 ","End":"08:04.740","Text":"which is minus 1,"},{"Start":"08:04.740 ","End":"08:07.725","Text":"x to n minus 1,"},{"Start":"08:07.725 ","End":"08:09.695","Text":"we have to subtract 1 from here,"},{"Start":"08:09.695 ","End":"08:12.860","Text":"so it becomes minus 2."},{"Start":"08:12.860 ","End":"08:15.230","Text":"This is an answer,"},{"Start":"08:15.230 ","End":"08:17.810","Text":"but it\u0027s usual to tidy up."},{"Start":"08:17.810 ","End":"08:22.430","Text":"What I mean is if the original question was given without negative exponents,"},{"Start":"08:22.430 ","End":"08:26.840","Text":"we should convert this back to put the minus 1 in the numerator"},{"Start":"08:26.840 ","End":"08:31.664","Text":"and x to the minus 2 is over x squared."},{"Start":"08:31.664 ","End":"08:35.300","Text":"I can write that if y equals 1 over x,"},{"Start":"08:35.300 ","End":"08:42.320","Text":"then y prime is equal to minus 1 over x squared."},{"Start":"08:42.320 ","End":"08:46.340","Text":"Just like this is a very important one that occurs frequently"},{"Start":"08:46.340 ","End":"08:48.170","Text":"and we should memorize it,"},{"Start":"08:48.170 ","End":"08:51.170","Text":"then this one also should be memorized,"},{"Start":"08:51.170 ","End":"08:52.820","Text":"so I\u0027ll highlight it."},{"Start":"08:52.820 ","End":"08:55.355","Text":"Let\u0027s continue on the next page."},{"Start":"08:55.355 ","End":"09:02.570","Text":"Next exercise is y equals 1 over x to the 4th."},{"Start":"09:02.570 ","End":"09:06.150","Text":"We want to know what y prime equals."},{"Start":"09:06.150 ","End":"09:11.165","Text":"Like before we use the exponents and the rules from algebra"},{"Start":"09:11.165 ","End":"09:16.895","Text":"to write this as y equals x to the minus 4,"},{"Start":"09:16.895 ","End":"09:21.890","Text":"which gives us the y prime equals from this rule,"},{"Start":"09:21.890 ","End":"09:27.020","Text":"minus 4x to the minus 4 minus 1."},{"Start":"09:27.920 ","End":"09:36.650","Text":"At the end, we convert that to minus 4 over x to the 5th."},{"Start":"09:36.650 ","End":"09:37.910","Text":"This is not mandatory,"},{"Start":"09:37.910 ","End":"09:39.259","Text":"but it\u0027s a static."},{"Start":"09:39.259 ","End":"09:45.290","Text":"Meaning if you are given the question in the form of 1 over and no negative exponents,"},{"Start":"09:45.290 ","End":"09:47.750","Text":"then we should really keep to that."},{"Start":"09:47.750 ","End":"09:52.310","Text":"I would write the answer as minus 4 over x to the 5th,"},{"Start":"09:52.310 ","End":"09:54.920","Text":"but this would be acceptable."},{"Start":"09:54.920 ","End":"10:00.940","Text":"Next question is y equals square root of x."},{"Start":"10:00.940 ","End":"10:05.180","Text":"Then we want to know what is y prime so we do a bit of exercise at"},{"Start":"10:05.180 ","End":"10:10.010","Text":"the side where we write it as square root of x,"},{"Start":"10:10.010 ","End":"10:11.960","Text":"x to the power of 1/2."},{"Start":"10:11.960 ","End":"10:14.690","Text":"I hope you remember your exponents from algebra."},{"Start":"10:14.690 ","End":"10:24.825","Text":"Then we get according to the rule that y prime is 1/2x and reduce the power by 1."},{"Start":"10:24.825 ","End":"10:27.505","Text":"We get minus 1/2."},{"Start":"10:27.505 ","End":"10:30.020","Text":"If we rearrange this,"},{"Start":"10:30.020 ","End":"10:34.685","Text":"we get that this is equal to 1 over,"},{"Start":"10:34.685 ","End":"10:41.525","Text":"now 2 goes to the bottom and the 1/2 because it\u0027s minus,"},{"Start":"10:41.525 ","End":"10:43.100","Text":"also goes to the bottom,"},{"Start":"10:43.100 ","End":"10:45.930","Text":"so it\u0027s x to the power of 1/2."},{"Start":"10:45.930 ","End":"10:51.050","Text":"But we can go still further and bring it more to the original format using"},{"Start":"10:51.050 ","End":"10:58.005","Text":"square roots by writing this finally as 1 over 2 square root of x."},{"Start":"10:58.005 ","End":"10:59.715","Text":"That\u0027s how I would leave it."},{"Start":"10:59.715 ","End":"11:05.330","Text":"Y prime is 1 over twice the square root of x."},{"Start":"11:05.330 ","End":"11:10.085","Text":"This is another one of those that we should remember."},{"Start":"11:10.085 ","End":"11:12.230","Text":"I\u0027m going to highlight it."},{"Start":"11:12.230 ","End":"11:21.435","Text":"The last one will be y equals 1 over x square root of x."},{"Start":"11:21.435 ","End":"11:24.675","Text":"We want to know what is y prime."},{"Start":"11:24.675 ","End":"11:30.665","Text":"We do bit of algebra at the side, y equals,"},{"Start":"11:30.665 ","End":"11:34.970","Text":"I\u0027ll leave it to you to show that this is x to the power"},{"Start":"11:34.970 ","End":"11:40.275","Text":"of minus 3 over 2 or minus 1 1/2."},{"Start":"11:40.275 ","End":"11:42.750","Text":"Basically, x to the 1, x to the 1/2,"},{"Start":"11:42.750 ","End":"11:44.175","Text":"x to the 1 1/2,"},{"Start":"11:44.175 ","End":"11:48.380","Text":"1 1/2 is 3 over 2 and the minus because it\u0027s on the denominator."},{"Start":"11:48.380 ","End":"11:51.600","Text":"But I\u0027ll assume you know some algebra."},{"Start":"11:51.600 ","End":"11:59.900","Text":"Then we get from the rule here that y prime is equal to minus 3"},{"Start":"11:59.900 ","End":"12:08.625","Text":"over 2x to the power of this minus 1 brings us down to minus 5 over 2."},{"Start":"12:08.625 ","End":"12:10.925","Text":"This is an acceptable answer."},{"Start":"12:10.925 ","End":"12:16.420","Text":"But if we want to be nice and bring it more to the original format,"},{"Start":"12:16.420 ","End":"12:19.535","Text":"the way it was given, you could,"},{"Start":"12:19.535 ","End":"12:21.650","Text":"with your knowledge of algebra,"},{"Start":"12:21.650 ","End":"12:25.640","Text":"get to the conclusion that this equals"},{"Start":"12:25.640 ","End":"12:33.170","Text":"minus 3 over 2x squared,"},{"Start":"12:33.170 ","End":"12:35.779","Text":"square root of x."},{"Start":"12:35.779 ","End":"12:37.625","Text":"But you don\u0027t have to."},{"Start":"12:37.625 ","End":"12:39.520","Text":"You could leave it like this."},{"Start":"12:39.520 ","End":"12:41.490","Text":"I\u0027ll just write the answer here,"},{"Start":"12:41.490 ","End":"12:44.760","Text":"minus 3 over 2,"},{"Start":"12:44.760 ","End":"12:49.800","Text":"and then x squared square root of x."},{"Start":"12:49.800 ","End":"12:52.570","Text":"That\u0027s it for number 1,"},{"Start":"12:52.570 ","End":"12:58.640","Text":"I just want to rewrite the three that were important that you should memorize."},{"Start":"12:58.640 ","End":"13:06.120","Text":"What we heard is that the derivative of just plain x was 1,"},{"Start":"13:06.120 ","End":"13:09.590","Text":"the other one we had is that the derivative of 1 over"},{"Start":"13:09.590 ","End":"13:16.365","Text":"x was equal to minus 1 over x squared,"},{"Start":"13:16.365 ","End":"13:18.500","Text":"and the other one,"},{"Start":"13:18.500 ","End":"13:24.955","Text":"that\u0027s this one here that we should memorize is the derivative of the square root of x."},{"Start":"13:24.955 ","End":"13:29.900","Text":"That is equal to 1 over twice the square root of x."},{"Start":"13:29.900 ","End":"13:34.110","Text":"These three are worth just remembering."},{"Start":"13:34.110 ","End":"13:36.260","Text":"You don\u0027t have to keep using this formula"},{"Start":"13:36.260 ","End":"13:39.600","Text":"because they all three of them occur quite frequently."},{"Start":"13:39.640 ","End":"13:43.140","Text":"Onto the next one."}],"ID":10435},{"Watched":false,"Name":"Product and Quotient Rules","Duration":"16m 52s","ChapterTopicVideoID":10126,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.765","Text":"This clip is just a continuation of the previous clip on differentiation rules."},{"Start":"00:06.765 ","End":"00:11.940","Text":"In this particular clip we\u0027ll be talking about the sum of functions, the difference,"},{"Start":"00:11.940 ","End":"00:14.580","Text":"the quotient, and the product, and one more,"},{"Start":"00:14.580 ","End":"00:15.855","Text":"which is this one,"},{"Start":"00:15.855 ","End":"00:18.720","Text":"which is a constant times a function."},{"Start":"00:18.720 ","End":"00:21.630","Text":"The rule says that if we have"},{"Start":"00:21.630 ","End":"00:25.225","Text":"a constant times a function and we want to differentiate it,"},{"Start":"00:25.225 ","End":"00:26.900","Text":"the constant, we don\u0027t touch,"},{"Start":"00:26.900 ","End":"00:30.620","Text":"it just stays there and we differentiate the function."},{"Start":"00:30.620 ","End":"00:33.810","Text":"I\u0027ll give a few examples."},{"Start":"00:34.300 ","End":"00:43.980","Text":"If y equals 4x^10,"},{"Start":"00:44.230 ","End":"00:47.030","Text":"4 is my a,"},{"Start":"00:47.030 ","End":"00:49.500","Text":"x^10 is my function,"},{"Start":"00:50.800 ","End":"00:53.480","Text":"the 4 just stays there,"},{"Start":"00:53.480 ","End":"01:01.740","Text":"we don\u0027t touch it, and the derivative of x^10 is 10x^9,"},{"Start":"01:01.740 ","End":"01:05.800","Text":"and so altogether the answer is 40x^9."},{"Start":"01:07.160 ","End":"01:15.745","Text":"Another example, let\u0027s take y equals 6 times the square root of x,"},{"Start":"01:15.745 ","End":"01:21.780","Text":"and in this case our a is 6 and our function of x is the square root of x."},{"Start":"01:21.780 ","End":"01:25.600","Text":"From here we get that y prime equals a,"},{"Start":"01:25.600 ","End":"01:27.295","Text":"which is just the 6."},{"Start":"01:27.295 ","End":"01:31.210","Text":"It stays there, and f prime derivative of the square root of"},{"Start":"01:31.210 ","End":"01:36.100","Text":"x is 1 over twice the square root of x."},{"Start":"01:36.100 ","End":"01:39.265","Text":"We just remember this one of the basic ones,"},{"Start":"01:39.265 ","End":"01:41.050","Text":"and if we simplify this,"},{"Start":"01:41.050 ","End":"01:42.430","Text":"x over 2 is 3,"},{"Start":"01:42.430 ","End":"01:47.670","Text":"so we get 3 over the square root of x."},{"Start":"01:47.670 ","End":"01:49.920","Text":"Another example."},{"Start":"01:49.920 ","End":"01:55.730","Text":"Let\u0027s take y equals just 10x."},{"Start":"01:55.830 ","End":"02:00.150","Text":"In this case, a is 10,"},{"Start":"02:00.150 ","End":"02:07.385","Text":"that\u0027s our constant and the function of x is x. Y prime is the constant a,"},{"Start":"02:07.385 ","End":"02:09.380","Text":"which is 10,"},{"Start":"02:09.380 ","End":"02:16.005","Text":"and then the derivative of x is 1,"},{"Start":"02:16.005 ","End":"02:20.135","Text":"and so the answer is just 10."},{"Start":"02:20.135 ","End":"02:22.440","Text":"Next we come to rule,"},{"Start":"02:22.440 ","End":"02:28.100","Text":"and what it says is that if I have the sum of 2 functions,"},{"Start":"02:28.100 ","End":"02:30.350","Text":"f of x plus g of x,"},{"Start":"02:30.350 ","End":"02:32.240","Text":"and I differentiate that,"},{"Start":"02:32.240 ","End":"02:37.175","Text":"then I differentiate each of the pieces separately and add,"},{"Start":"02:37.175 ","End":"02:39.110","Text":"but it\u0027s not just for sum,"},{"Start":"02:39.110 ","End":"02:40.790","Text":"it\u0027s also the difference."},{"Start":"02:40.790 ","End":"02:42.815","Text":"If I had a minus here,"},{"Start":"02:42.815 ","End":"02:45.440","Text":"f of x minus g of x,"},{"Start":"02:45.440 ","End":"02:47.480","Text":"and I differentiated that,"},{"Start":"02:47.480 ","End":"02:52.340","Text":"then y prime would be a derivative of f minus"},{"Start":"02:52.340 ","End":"02:59.765","Text":"the derivative of g. We\u0027ll call this rule number 9,"},{"Start":"02:59.765 ","End":"03:06.110","Text":"sum and difference, or addition and subtraction functions and the derivative."},{"Start":"03:06.110 ","End":"03:15.580","Text":"As an example, let\u0027s take y equals x^4,"},{"Start":"03:15.580 ","End":"03:22.515","Text":"plus x^10 minus x squared."},{"Start":"03:22.515 ","End":"03:28.325","Text":"Then this will give us the y prime equals."},{"Start":"03:28.325 ","End":"03:31.180","Text":"Now here for x^4,"},{"Start":"03:31.180 ","End":"03:36.465","Text":"derivative of that is 4x cubed."},{"Start":"03:36.465 ","End":"03:39.930","Text":"Because we have a plus here,"},{"Start":"03:39.930 ","End":"03:43.830","Text":"we also put a plus here,"},{"Start":"03:43.830 ","End":"03:47.715","Text":"and then take the derivative of x^10,"},{"Start":"03:47.715 ","End":"03:54.720","Text":"which is 10x^9,"},{"Start":"03:54.720 ","End":"03:58.035","Text":"we lower the exponent by 1,"},{"Start":"03:58.035 ","End":"04:03.795","Text":"and that\u0027s this plus,"},{"Start":"04:03.795 ","End":"04:10.185","Text":"and then we have a minus here so we have a minus here,"},{"Start":"04:10.185 ","End":"04:12.600","Text":"I\u0027ll also color it,"},{"Start":"04:12.600 ","End":"04:15.605","Text":"and for x squared,"},{"Start":"04:15.605 ","End":"04:21.155","Text":"the derivative of that is 2x."},{"Start":"04:21.155 ","End":"04:23.345","Text":"That\u0027s the answer for that one,"},{"Start":"04:23.345 ","End":"04:25.580","Text":"and now another example."},{"Start":"04:25.580 ","End":"04:32.600","Text":"Let\u0027s take y as equaling x squared,"},{"Start":"04:32.600 ","End":"04:41.980","Text":"let\u0027s take minus 1 over x and plus 8."},{"Start":"04:41.980 ","End":"04:46.340","Text":"Then according to the sum and difference rule,"},{"Start":"04:46.340 ","End":"04:48.395","Text":"y prime will equal."},{"Start":"04:48.395 ","End":"04:52.930","Text":"We\u0027ll take the x squared and that will give us 2x."},{"Start":"04:52.930 ","End":"04:55.340","Text":"Now because of the minus here,"},{"Start":"04:55.340 ","End":"04:58.230","Text":"we\u0027ll put a minus here,"},{"Start":"04:58.270 ","End":"05:01.050","Text":"here, and here,"},{"Start":"05:01.050 ","End":"05:08.010","Text":"and the derivative of 1 over x is minus 1 over x squared,"},{"Start":"05:08.010 ","End":"05:12.840","Text":"so here I put minus 1 over x squared,"},{"Start":"05:12.840 ","End":"05:15.000","Text":"let\u0027s put it in brackets,"},{"Start":"05:15.000 ","End":"05:21.510","Text":"and the derivative of 8 is 0,"},{"Start":"05:21.510 ","End":"05:23.640","Text":"but there\u0027s a plus here,"},{"Start":"05:23.640 ","End":"05:26.815","Text":"so we also want a plus here,"},{"Start":"05:26.815 ","End":"05:30.475","Text":"this plus this is plus,"},{"Start":"05:30.475 ","End":"05:37.790","Text":"and I can do a small simplification and write it as 2x minus minus is plus,"},{"Start":"05:37.790 ","End":"05:41.180","Text":"so it\u0027s plus 1 over x squared,"},{"Start":"05:41.180 ","End":"05:44.010","Text":"and the 0, I can omit."},{"Start":"05:44.060 ","End":"05:46.455","Text":"Another example."},{"Start":"05:46.455 ","End":"05:52.180","Text":"Let\u0027s take y as equaling,"},{"Start":"05:52.430 ","End":"05:58.010","Text":"I\u0027ll take 4x^10,"},{"Start":"05:58.670 ","End":"06:01.005","Text":"plus, let\u0027s say,"},{"Start":"06:01.005 ","End":"06:12.505","Text":"8 times square root of x minus 10x and plus 9."},{"Start":"06:12.505 ","End":"06:19.800","Text":"Now sometimes you want to use rule number 9 and rule number 8 combined."},{"Start":"06:20.630 ","End":"06:23.825","Text":"I can first do it in 2 steps,"},{"Start":"06:23.825 ","End":"06:28.055","Text":"and then I can show you how we shorten the process."},{"Start":"06:28.055 ","End":"06:30.410","Text":"Here\u0027s a reminder of rules."},{"Start":"06:30.410 ","End":"06:32.390","Text":"You\u0027ll also be using that when you have"},{"Start":"06:32.390 ","End":"06:35.510","Text":"a constant times a function and then differentiate it,"},{"Start":"06:35.510 ","End":"06:41.304","Text":"the constant just sticks there and you just differentiate the function."},{"Start":"06:41.304 ","End":"06:43.800","Text":"Y prime equals,"},{"Start":"06:43.800 ","End":"06:45.300","Text":"because of rule 8,"},{"Start":"06:45.300 ","End":"06:52.270","Text":"it\u0027s 4 times the derivative of x^10, which is 10x^9."},{"Start":"06:54.650 ","End":"06:59.740","Text":"Then we have a rule 9, the plus here."},{"Start":"07:03.390 ","End":"07:06.760","Text":"Emphasize it with the color."},{"Start":"07:06.760 ","End":"07:14.395","Text":"Then we need 8 times root x to differentiate it and that\u0027s where we use rule 8."},{"Start":"07:14.395 ","End":"07:25.285","Text":"We just write the 8 and it stays then the derivative of root x is 1 over twice root x."},{"Start":"07:25.285 ","End":"07:31.750","Text":"Then we have a minus which is the minus from rule 9,"},{"Start":"07:31.750 ","End":"07:33.295","Text":"the sum and difference."},{"Start":"07:33.295 ","End":"07:35.515","Text":"Then from rule 8,"},{"Start":"07:35.515 ","End":"07:40.040","Text":"It\u0027s a constant 10 times a function x."},{"Start":"07:40.040 ","End":"07:47.355","Text":"Here I put 10 and the derivative of x is 1 and then this plus,"},{"Start":"07:47.355 ","End":"07:49.485","Text":"is this plus,"},{"Start":"07:49.485 ","End":"07:52.930","Text":"so this and this."},{"Start":"07:53.100 ","End":"07:59.305","Text":"The derivative of a constant is just 0."},{"Start":"07:59.305 ","End":"08:06.940","Text":"Then we can just do better simplification 4 times 10 is 40x^9,"},{"Start":"08:06.940 ","End":"08:16.090","Text":"8 over 2 is 4."},{"Start":"08:16.090 ","End":"08:22.225","Text":"So we get 4 over root x."},{"Start":"08:22.225 ","End":"08:27.590","Text":"Then 10 times 1 is 10 and 0 is just 0."},{"Start":"08:27.990 ","End":"08:33.055","Text":"In practice, after you have done a couple of these,"},{"Start":"08:33.055 ","End":"08:36.415","Text":"usually we cut out the middleman."},{"Start":"08:36.415 ","End":"08:45.490","Text":"We would usually go straight from here to here and not bother with this step."},{"Start":"08:45.490 ","End":"08:51.115","Text":"You\u0027d say straightaway 4 and then you\u0027d think 10x^9,"},{"Start":"08:51.115 ","End":"08:53.890","Text":"4 times 10 is 40, and so on."},{"Start":"08:53.890 ","End":"08:56.650","Text":"But yeah, it\u0027s okay to do it in two steps."},{"Start":"08:56.650 ","End":"09:01.460","Text":"Say that in time you tend to do it all in one."},{"Start":"09:01.950 ","End":"09:06.460","Text":"That\u0027s this sum and difference rule."},{"Start":"09:06.460 ","End":"09:10.315","Text":"The next rule is called the product rule."},{"Start":"09:10.315 ","End":"09:13.975","Text":"It\u0027s a very important rule and has its own name."},{"Start":"09:13.975 ","End":"09:19.090","Text":"We just learned about the sum of functions and the difference of functions."},{"Start":"09:19.090 ","End":"09:20.560","Text":"Now we come to the product,"},{"Start":"09:20.560 ","End":"09:22.810","Text":"one function times another and it doesn\u0027t work so"},{"Start":"09:22.810 ","End":"09:25.945","Text":"simply as with the sum and the difference."},{"Start":"09:25.945 ","End":"09:29.920","Text":"What it says is that if we have y,"},{"Start":"09:29.920 ","End":"09:31.540","Text":"which is a product of two functions,"},{"Start":"09:31.540 ","End":"09:33.625","Text":"f of x times g of x,"},{"Start":"09:33.625 ","End":"09:38.545","Text":"but I\u0027ll just write this briefly as f times g just to save space,"},{"Start":"09:38.545 ","End":"09:46.450","Text":"and what it says is that the derivative of y is not just f prime times g prime,"},{"Start":"09:46.450 ","End":"09:47.785","Text":"like you might have guessed."},{"Start":"09:47.785 ","End":"09:49.990","Text":"Something more involved."},{"Start":"09:49.990 ","End":"09:53.935","Text":"What it says is that it\u0027s the sum of two products"},{"Start":"09:53.935 ","End":"09:58.435","Text":"and each one we differentiate one of the functions and leave the other alone."},{"Start":"09:58.435 ","End":"10:01.285","Text":"I\u0027ll write it and it\u0027ll be clearer."},{"Start":"10:01.285 ","End":"10:03.865","Text":"First we take one of them, say,"},{"Start":"10:03.865 ","End":"10:07.960","Text":"differentiate the first times and the second one,"},{"Start":"10:07.960 ","End":"10:11.515","Text":"leave it as it is, plus vice versa,"},{"Start":"10:11.515 ","End":"10:15.880","Text":"meaning f just as it is and"},{"Start":"10:15.880 ","End":"10:22.150","Text":"the derivative of g. Best way to explain this is through an example."},{"Start":"10:22.150 ","End":"10:25.390","Text":"Let\u0027s take the example of,"},{"Start":"10:25.390 ","End":"10:32.050","Text":"let\u0027s take the product of the first one f will be"},{"Start":"10:32.050 ","End":"10:39.445","Text":"2x squared plus 4x plus 1."},{"Start":"10:39.445 ","End":"10:46.900","Text":"The second factor g will be x squared plus 5x."},{"Start":"10:46.900 ","End":"10:52.915","Text":"Perhaps others make a note that this first part is f,"},{"Start":"10:52.915 ","End":"10:55.900","Text":"the second part is g,"},{"Start":"10:55.900 ","End":"11:05.170","Text":"and now we\u0027ll just apply the product rule and say that y prime equals f prime."},{"Start":"11:05.170 ","End":"11:07.435","Text":"So that\u0027s the derivative of this bit,"},{"Start":"11:07.435 ","End":"11:12.445","Text":"and that will just be 4x plus 4."},{"Start":"11:12.445 ","End":"11:17.650","Text":"Didn\u0027t go into more detail than that."},{"Start":"11:17.650 ","End":"11:22.375","Text":"You should be able to manage this on your own."},{"Start":"11:22.375 ","End":"11:29.289","Text":"That\u0027s this part, f prime and then we need g as is,"},{"Start":"11:29.289 ","End":"11:33.160","Text":"so that\u0027s x squared plus 5 x."},{"Start":"11:33.160 ","End":"11:36.580","Text":"Then the other way around we take f as is,"},{"Start":"11:36.580 ","End":"11:42.280","Text":"so 2x squared plus 4x plus 1."},{"Start":"11:42.280 ","End":"11:45.595","Text":"Then g prime, the derivative of this."},{"Start":"11:45.595 ","End":"11:47.560","Text":"I\u0027ll just write it."},{"Start":"11:47.560 ","End":"11:57.740","Text":"It\u0027s x squared gives us 2x and 5x gives us 5."},{"Start":"11:59.150 ","End":"12:02.445","Text":"I could say that this is the answer."},{"Start":"12:02.445 ","End":"12:06.135","Text":"Normally, we would do a bit of simplification,"},{"Start":"12:06.135 ","End":"12:11.830","Text":"but that\u0027s not really the purpose of this exercise."},{"Start":"12:12.020 ","End":"12:15.645","Text":"If you want to multiply"},{"Start":"12:15.645 ","End":"12:22.920","Text":"these this bracket with this bracket and this with this combine like terms,"},{"Start":"12:22.920 ","End":"12:24.900","Text":"and you\u0027ll get something simpler."},{"Start":"12:24.900 ","End":"12:32.305","Text":"In fact, if you really want to go further and check your result,"},{"Start":"12:32.305 ","End":"12:39.400","Text":"you could actually multiply here and figure out what y is as a polynomial and"},{"Start":"12:39.400 ","End":"12:42.880","Text":"then differentiate it and then you could check and"},{"Start":"12:42.880 ","End":"12:46.810","Text":"see if you got the same answer in both the cases."},{"Start":"12:46.810 ","End":"12:49.660","Text":"Anyway, that\u0027s too much work and I\u0027m not going to do"},{"Start":"12:49.660 ","End":"12:53.290","Text":"it and we\u0027ll leave the answer like this."},{"Start":"12:53.290 ","End":"12:55.465","Text":"After the product rule,"},{"Start":"12:55.465 ","End":"12:57.775","Text":"we have the quotient rule,"},{"Start":"12:57.775 ","End":"13:01.240","Text":"and instead of two functions multiplied by each other,"},{"Start":"13:01.240 ","End":"13:04.549","Text":"we have one over the other divided."},{"Start":"13:04.710 ","End":"13:09.595","Text":"If y is equal to f over g,"},{"Start":"13:09.595 ","End":"13:13.735","Text":"and I\u0027m just abbreviating f of x over g of x,"},{"Start":"13:13.735 ","End":"13:19.225","Text":"then the derivative y prime is equal 2."},{"Start":"13:19.225 ","End":"13:21.655","Text":"A little bit more tricky."},{"Start":"13:21.655 ","End":"13:24.100","Text":"You still should remember it."},{"Start":"13:24.100 ","End":"13:26.500","Text":"It\u0027s one of those rules that you have to memorize."},{"Start":"13:26.500 ","End":"13:32.740","Text":"On the denominator, we have the same as before, only squared."},{"Start":"13:32.740 ","End":"13:34.675","Text":"On the numerator,"},{"Start":"13:34.675 ","End":"13:42.265","Text":"we have the derivative of f, the first one,"},{"Start":"13:42.265 ","End":"13:44.305","Text":"the one on the numerator,"},{"Start":"13:44.305 ","End":"13:50.695","Text":"times what was on the denominator just untouched and minus,"},{"Start":"13:50.695 ","End":"13:53.005","Text":"because in the product rule we had a plus in the middle,"},{"Start":"13:53.005 ","End":"13:56.410","Text":"this time it\u0027s minus the numerator,"},{"Start":"13:56.410 ","End":"13:58.345","Text":"just as it was."},{"Start":"13:58.345 ","End":"14:02.995","Text":"But this time we differentiate the denominator."},{"Start":"14:02.995 ","End":"14:08.209","Text":"As an example, let\u0027s take the function y"},{"Start":"14:08.209 ","End":"14:14.675","Text":"equals 1 plus x over the square root of x,"},{"Start":"14:14.675 ","End":"14:17.720","Text":"which is obviously a quotient,"},{"Start":"14:17.720 ","End":"14:20.960","Text":"something over something and the numerator is f and"},{"Start":"14:20.960 ","End":"14:25.910","Text":"the denominator is g. According to our formula,"},{"Start":"14:25.910 ","End":"14:31.445","Text":"y prime equals and I\u0027ll put a dividing line."},{"Start":"14:31.445 ","End":"14:34.635","Text":"For some reason, I like to start with the denominator,"},{"Start":"14:34.635 ","End":"14:41.765","Text":"but you can start where you like and g squared is the square root of x squared."},{"Start":"14:41.765 ","End":"14:43.595","Text":"So that\u0027s just x."},{"Start":"14:43.595 ","End":"14:44.720","Text":"Let\u0027s write that down."},{"Start":"14:44.720 ","End":"14:47.810","Text":"Square root of x squared is x."},{"Start":"14:47.810 ","End":"14:56.500","Text":"Now we have f prime g. This is f, this is g,"},{"Start":"14:56.500 ","End":"15:04.990","Text":"so f prime is 1 and g is the square root of x minus f,"},{"Start":"15:04.990 ","End":"15:10.135","Text":"which is 1 plus x times g prime."},{"Start":"15:10.135 ","End":"15:16.400","Text":"The derivative of square root of x is 1 over twice the square root of x."},{"Start":"15:16.830 ","End":"15:20.650","Text":"This is it except for simplification."},{"Start":"15:20.650 ","End":"15:23.635","Text":"Let\u0027s do a bit of simplification."},{"Start":"15:23.635 ","End":"15:30.550","Text":"I\u0027m going to multiply top and bottom by twice root x and get rid of this fraction."},{"Start":"15:30.550 ","End":"15:34.130","Text":"We\u0027ll get that this is equal 2."},{"Start":"15:35.940 ","End":"15:41.095","Text":"Dividing line. Then here,"},{"Start":"15:41.095 ","End":"15:44.140","Text":"I\u0027ve got the 2 root x disappearing,"},{"Start":"15:44.140 ","End":"15:47.830","Text":"so I just have minus 1 plus x."},{"Start":"15:47.830 ","End":"15:53.270","Text":"Here I have this times 2 root x is going to be just 2,"},{"Start":"15:53.270 ","End":"15:55.715","Text":"root x is 2x,"},{"Start":"15:55.715 ","End":"16:01.965","Text":"and here x times 2 root x."},{"Start":"16:01.965 ","End":"16:12.660","Text":"This is equal to 2x minus x is just x and I still have minus 1,"},{"Start":"16:12.660 ","End":"16:21.130","Text":"and on the denominator 2x root x."},{"Start":"16:21.130 ","End":"16:23.810","Text":"If we wanted to do a check,"},{"Start":"16:23.810 ","End":"16:28.430","Text":"we can actually divide this out and say 1 over"},{"Start":"16:28.430 ","End":"16:34.310","Text":"root x plus x over root x and put it in terms of powers of x. Hopefully,"},{"Start":"16:34.310 ","End":"16:35.510","Text":"if we do it that way,"},{"Start":"16:35.510 ","End":"16:37.685","Text":"we\u0027ll get the same answer."},{"Start":"16:37.685 ","End":"16:41.390","Text":"But I leave that to you."},{"Start":"16:41.500 ","End":"16:45.360","Text":"That\u0027s it for the quotient rule."},{"Start":"16:45.400 ","End":"16:50.060","Text":"In the next clip we\u0027ll continue with the chain rule."},{"Start":"16:50.060 ","End":"16:52.410","Text":"Here we\u0027re done."}],"ID":10438},{"Watched":false,"Name":"Chain Rule","Duration":"23m 58s","ChapterTopicVideoID":10127,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this clip, we\u0027re continuing with differentiation rules."},{"Start":"00:03.900 ","End":"00:06.015","Text":"This time the chain rule."},{"Start":"00:06.015 ","End":"00:12.735","Text":"I\u0027m going to stop with the counting of rules we had up to number 11, the quotient rule."},{"Start":"00:12.735 ","End":"00:18.790","Text":"Anyway, the chain rule is so important and I\u0027m not going to number it."},{"Start":"00:18.800 ","End":"00:25.710","Text":"In general, what it\u0027s good for is for composition of functions."},{"Start":"00:25.710 ","End":"00:33.930","Text":"We would use it when we have y equals f of g of x,"},{"Start":"00:33.930 ","End":"00:37.199","Text":"a function of a function of x."},{"Start":"00:37.199 ","End":"00:42.325","Text":"In this clip, mostly or maybe always,"},{"Start":"00:42.325 ","End":"00:47.825","Text":"our f of x will be of the form x^n,"},{"Start":"00:47.825 ","End":"00:51.350","Text":"like x to the 4th or the 10th,"},{"Start":"00:51.350 ","End":"00:54.340","Text":"or 1 over x squared."},{"Start":"00:54.340 ","End":"00:57.260","Text":"Now I\u0027d like to start with a reminder of"},{"Start":"00:57.260 ","End":"01:03.300","Text":"how we differentiate functions of the form x to the n."},{"Start":"01:03.300 ","End":"01:12.900","Text":"In this case, the derivative y prime is n x to the n minus 1."},{"Start":"01:12.900 ","End":"01:20.045","Text":"For example, if I had y equals x to the 10th,"},{"Start":"01:20.045 ","End":"01:28.865","Text":"then I would say the y prime was 10x to the 10 minus 1, which is 9."},{"Start":"01:28.865 ","End":"01:33.950","Text":"There are 3 special cases I\u0027d like you to memorize,"},{"Start":"01:33.950 ","End":"01:37.460","Text":"even, special case n equals 1,"},{"Start":"01:37.460 ","End":"01:42.840","Text":"special case where n equals 1/2,"},{"Start":"01:42.840 ","End":"01:47.595","Text":"and a special case where n equals minus 1."},{"Start":"01:47.595 ","End":"01:51.165","Text":"In this case, we say,"},{"Start":"01:51.165 ","End":"01:53.990","Text":"we\u0027ll get that if y equals x to the 1,"},{"Start":"01:53.990 ","End":"01:58.860","Text":"which is x, then y prime is simply 1."},{"Start":"01:58.860 ","End":"02:02.265","Text":"It\u0027s going to be 1x to the 1 minus 1."},{"Start":"02:02.265 ","End":"02:03.480","Text":"It comes out 1."},{"Start":"02:03.480 ","End":"02:08.550","Text":"In this case, x to the 1/2 is the square root of x."},{"Start":"02:08.550 ","End":"02:11.569","Text":"If we use this rule,"},{"Start":"02:11.569 ","End":"02:22.450","Text":"it will come down to y prime equals 1 over 2 root x."},{"Start":"02:22.450 ","End":"02:28.020","Text":"With minus 1 means that y is 1 over x."},{"Start":"02:28.020 ","End":"02:37.020","Text":"We will get that y prime is minus 1 over x squared."},{"Start":"02:37.900 ","End":"02:42.665","Text":"This is still introduction and review."},{"Start":"02:42.665 ","End":"02:45.290","Text":"Let\u0027s take an actual case."},{"Start":"02:45.290 ","End":"02:49.070","Text":"Suppose g of x is x squared plus 1."},{"Start":"02:49.070 ","End":"02:53.570","Text":"I want to use this example where f of x is x to the 10th."},{"Start":"02:53.570 ","End":"02:59.880","Text":"In other words, I want to know what happens if y equals instead of x to the 10th,"},{"Start":"02:59.880 ","End":"03:03.765","Text":"x squared plus 1 to the 10th,"},{"Start":"03:03.765 ","End":"03:08.140","Text":"what would y prime equal then?"},{"Start":"03:08.140 ","End":"03:11.510","Text":"I might make a guess and say, well,"},{"Start":"03:11.510 ","End":"03:16.940","Text":"how about 10 times x"},{"Start":"03:16.940 ","End":"03:22.580","Text":"squared plus 1^9,"},{"Start":"03:22.580 ","End":"03:25.895","Text":"as if I replaced x by x squared plus 1."},{"Start":"03:25.895 ","End":"03:27.470","Text":"Would this be true?"},{"Start":"03:27.470 ","End":"03:30.680","Text":"Let me just put a question mark here for the time being,"},{"Start":"03:30.680 ","End":"03:33.650","Text":"and I won\u0027t tell you if this is right or wrong."},{"Start":"03:33.650 ","End":"03:37.710","Text":"I want to take another hypothetical example."},{"Start":"03:37.970 ","End":"03:45.150","Text":"If y equals, say x to the 6,"},{"Start":"03:45.150 ","End":"03:51.240","Text":"then I know from the general rule"},{"Start":"03:51.240 ","End":"03:55.920","Text":"that y prime is 6x to the 5th."},{"Start":"03:55.920 ","End":"03:59.350","Text":"But what if it wasn\u0027t x?"},{"Start":"04:00.560 ","End":"04:05.040","Text":"Instead of x let\u0027s the take g of x to be,"},{"Start":"04:05.040 ","End":"04:07.130","Text":"say 4x plus 5."},{"Start":"04:07.130 ","End":"04:16.110","Text":"In other words, I want to know that if y is 4x plus 5 to the 6th,"},{"Start":"04:16.110 ","End":"04:21.550","Text":"then what is y prime equal?"},{"Start":"04:23.780 ","End":"04:26.895","Text":"Let\u0027s make a guess."},{"Start":"04:26.895 ","End":"04:29.615","Text":"I\u0027ll put a question mark here."},{"Start":"04:29.615 ","End":"04:34.085","Text":"Do you think it would equal 6 times"},{"Start":"04:34.085 ","End":"04:41.400","Text":"4x plus 5 to the 5th?"},{"Start":"04:41.400 ","End":"04:43.530","Text":"As if instead of x,"},{"Start":"04:43.530 ","End":"04:46.810","Text":"I just had 4x plus 5."},{"Start":"04:47.140 ","End":"04:52.615","Text":"I want to do a third hypothetical example?"},{"Start":"04:52.615 ","End":"04:59.920","Text":"We had this 1, and this 1."},{"Start":"04:59.920 ","End":"05:02.210","Text":"Let\u0027s do a third 1."},{"Start":"05:02.820 ","End":"05:06.880","Text":"If I have y equals the square root of x,"},{"Start":"05:06.880 ","End":"05:09.185","Text":"we\u0027ve already covered this case here,"},{"Start":"05:09.185 ","End":"05:15.460","Text":"then we know that y prime is 1 over twice the square root of x."},{"Start":"05:15.460 ","End":"05:25.860","Text":"But what if we had y equals the square root of 2x plus 4."},{"Start":"05:25.860 ","End":"05:29.655","Text":"What would y prime be then?"},{"Start":"05:29.655 ","End":"05:31.410","Text":"You might guess,"},{"Start":"05:31.410 ","End":"05:36.150","Text":"and I\u0027m going to put a question mark here that it\u0027s"},{"Start":"05:36.150 ","End":"05:52.350","Text":"1 over twice the square root of 2x plus 4."},{"Start":"05:52.350 ","End":"05:54.770","Text":"What I can tell you now is"},{"Start":"05:54.770 ","End":"05:58.400","Text":"that the answer to all these 3 questions"},{"Start":"05:58.400 ","End":"05:59.600","Text":"with the question marks,"},{"Start":"05:59.600 ","End":"06:02.615","Text":"the answer to all of them is no."},{"Start":"06:02.615 ","End":"06:09.890","Text":"This is not the right answer for this or for this or for this."},{"Start":"06:09.890 ","End":"06:12.710","Text":"But it\u0027s very close."},{"Start":"06:12.710 ","End":"06:15.215","Text":"It\u0027s what I call a good start."},{"Start":"06:15.215 ","End":"06:18.960","Text":"You\u0027ll see what I mean in a moment."},{"Start":"06:19.520 ","End":"06:22.835","Text":"This is where the chain rule comes in."},{"Start":"06:22.835 ","End":"06:30.800","Text":"What it says is that we can fix this good start by just taking the inner function,"},{"Start":"06:30.800 ","End":"06:32.435","Text":"what we call g of x."},{"Start":"06:32.435 ","End":"06:34.175","Text":"If I just scroll back a moment,"},{"Start":"06:34.175 ","End":"06:37.130","Text":"the inner 1 was g of x."},{"Start":"06:37.130 ","End":"06:42.515","Text":"In this case, our g of x is the x squared plus 1."},{"Start":"06:42.515 ","End":"06:48.815","Text":"If we just multiply by the derivative of the inner function x squared plus 1,"},{"Start":"06:48.815 ","End":"06:50.945","Text":"in this case, that\u0027s 2x,"},{"Start":"06:50.945 ","End":"06:54.330","Text":"then it will give us the right answer."},{"Start":"06:54.520 ","End":"06:58.590","Text":"I can remove the question mark."},{"Start":"06:58.670 ","End":"07:02.090","Text":"In this case to make it right,"},{"Start":"07:02.090 ","End":"07:06.440","Text":"I also take this good start, what we called."},{"Start":"07:06.440 ","End":"07:12.895","Text":"The inner function is 4x plus 5 and multiply by the derivative of that."},{"Start":"07:12.895 ","End":"07:18.140","Text":"That will be the right answer for the derivative of this function."},{"Start":"07:18.140 ","End":"07:22.230","Text":"Over here, the inner function,"},{"Start":"07:22.230 ","End":"07:24.900","Text":"the g of x is 2x plus 4."},{"Start":"07:24.900 ","End":"07:34.300","Text":"All I have to do is multiply by the derivative of 2x plus 4, which is 2."},{"Start":"07:35.090 ","End":"07:38.010","Text":"I guess in this case we could simplify."},{"Start":"07:38.010 ","End":"07:42.550","Text":"We could cancel this 2 with this 2. That doesn\u0027t matter."},{"Start":"07:42.550 ","End":"07:49.655","Text":"Next, I\u0027d like to generalize a little bit the things we\u0027ve been doing so far."},{"Start":"07:49.655 ","End":"07:52.460","Text":"I\u0027m just going to erase what I don\u0027t need."},{"Start":"07:52.460 ","End":"07:56.450","Text":"I don\u0027t need that and I don\u0027t need that."},{"Start":"07:56.450 ","End":"08:01.725","Text":"I want to just generalize what we did in these 3 examples."},{"Start":"08:01.725 ","End":"08:05.140","Text":"What I\u0027d like to do is this inner function."},{"Start":"08:05.140 ","End":"08:07.165","Text":"I also called it g of x earlier,"},{"Start":"08:07.165 ","End":"08:12.820","Text":"just to replace it by a template, a box."},{"Start":"08:12.820 ","End":"08:17.635","Text":"I would write y equals something,"},{"Start":"08:17.635 ","End":"08:24.145","Text":"box^10, where box means some function of x."},{"Start":"08:24.145 ","End":"08:34.610","Text":"Then I would be able to say in general that y prime is 10 times the box^9."},{"Start":"08:35.460 ","End":"08:40.930","Text":"The 2x would be the derivative of what the box was."},{"Start":"08:40.930 ","End":"08:43.840","Text":"In this case, it was x squared plus 1."},{"Start":"08:43.840 ","End":"08:52.030","Text":"Similarly, here, I can generalize this by saying that if y is something,"},{"Start":"08:52.030 ","End":"08:55.480","Text":"some function of x^6,"},{"Start":"08:55.480 ","End":"08:59.050","Text":"then the derivative will be 6 times"},{"Start":"08:59.050 ","End":"09:07.790","Text":"that something to the power of 5 times the derivative of that something."},{"Start":"09:08.760 ","End":"09:19.705","Text":"Here also, if I have y equals the square root of box, some function of x,"},{"Start":"09:19.705 ","End":"09:24.040","Text":"then y prime will equal"},{"Start":"09:24.040 ","End":"09:29.710","Text":"1/2 the square root"},{"Start":"09:29.710 ","End":"09:37.310","Text":"of box times the derivative of box."},{"Start":"09:37.710 ","End":"09:42.460","Text":"Now, this is a generalization for the case where n is 10,"},{"Start":"09:42.460 ","End":"09:44.380","Text":"n is 6, n is 1/2."},{"Start":"09:44.380 ","End":"09:49.820","Text":"I can do 1 more level of generalization."},{"Start":"09:50.130 ","End":"09:55.795","Text":"Say that if y equals box,"},{"Start":"09:55.795 ","End":"10:04.360","Text":"some function of x to the power of a general n then y prime will equal,"},{"Start":"10:04.360 ","End":"10:05.980","Text":"just like we had before,"},{"Start":"10:05.980 ","End":"10:08.215","Text":"n x^n minus 1,"},{"Start":"10:08.215 ","End":"10:12.895","Text":"this time n times box^n minus 1."},{"Start":"10:12.895 ","End":"10:15.430","Text":"But we mustn\u0027t forget the extra bit,"},{"Start":"10:15.430 ","End":"10:19.750","Text":"which is the derivative of box."},{"Start":"10:19.750 ","End":"10:27.650","Text":"This is the general rule that I wanted to get to."},{"Start":"10:28.290 ","End":"10:32.270","Text":"It\u0027s highlighted, not the greatest."},{"Start":"10:32.730 ","End":"10:36.245","Text":"There were special cases,"},{"Start":"10:36.245 ","End":"10:39.030","Text":"which I\u0027m not going to write again."},{"Start":"10:39.030 ","End":"10:41.730","Text":"When n is equal to 1,"},{"Start":"10:41.730 ","End":"10:47.910","Text":"when n is equal to 1/2 and when n is equal to minus 1,"},{"Start":"10:47.910 ","End":"10:52.600","Text":"we could write them out like if n is 1/2,"},{"Start":"10:52.600 ","End":"10:56.920","Text":"then we could get this rule that if y is the square root of box,"},{"Start":"10:56.920 ","End":"10:59.620","Text":"then y prime is 1/2."},{"Start":"10:59.620 ","End":"11:02.515","Text":"Maybe I actually should write them, you know what?"},{"Start":"11:02.515 ","End":"11:07.495","Text":"Wouldn\u0027t hurt. We\u0027ve done the case for n equals 1/2 here."},{"Start":"11:07.495 ","End":"11:10.645","Text":"Let\u0027s do the case where n is 1."},{"Start":"11:10.645 ","End":"11:13.340","Text":"Well, that\u0027s a bit pointless."},{"Start":"11:13.620 ","End":"11:15.820","Text":"But if n is minus 1,"},{"Start":"11:15.820 ","End":"11:17.695","Text":"it would be interesting."},{"Start":"11:17.695 ","End":"11:25.990","Text":"If y is equal to 1/box, that\u0027s box^minus 1."},{"Start":"11:25.990 ","End":"11:32.409","Text":"You see why I got rid of the case 1 because box^1 is just box not interesting."},{"Start":"11:32.409 ","End":"11:37.270","Text":"Then y prime is,"},{"Start":"11:37.270 ","End":"11:42.925","Text":"remember the rule for 1/x was minus 1/x squared."},{"Start":"11:42.925 ","End":"11:46.915","Text":"Only this time it will be minus 1/box squared."},{"Start":"11:46.915 ","End":"11:53.090","Text":"We fix it by multiplying by the derivative of box."},{"Start":"11:54.120 ","End":"11:57.190","Text":"This is the main rule."},{"Start":"11:57.190 ","End":"12:02.155","Text":"This is a special case where"},{"Start":"12:02.155 ","End":"12:07.615","Text":"n is minus 1 and this 1 here is a special case where n is 1/2."},{"Start":"12:07.615 ","End":"12:09.250","Text":"Here n is 1/2,"},{"Start":"12:09.250 ","End":"12:12.775","Text":"here n is minus 1 special cases,"},{"Start":"12:12.775 ","End":"12:15.655","Text":"but you don\u0027t really need them."},{"Start":"12:15.655 ","End":"12:18.140","Text":"This is what we need."},{"Start":"12:18.330 ","End":"12:23.000","Text":"Now, let\u0027s do some more examples."},{"Start":"12:23.940 ","End":"12:28.780","Text":"Here we are again, I just copied this general formula."},{"Start":"12:28.780 ","End":"12:32.255","Text":"I think it looks better without the highlighting."},{"Start":"12:32.255 ","End":"12:41.960","Text":"This was the general rule and there were exceptional cases for n equals 1 1/2 minus 1,"},{"Start":"12:41.960 ","End":"12:44.670","Text":"which we can optionally use or not."},{"Start":"12:44.670 ","End":"12:50.525","Text":"But let\u0027s just stick with this main formula and do some more examples."},{"Start":"12:50.525 ","End":"12:53.560","Text":"Next example, we\u0027ll take y"},{"Start":"12:53.560 ","End":"13:02.590","Text":"equals 1/2x minus 5^3."},{"Start":"13:02.590 ","End":"13:07.375","Text":"Now, we want to put it in this form."},{"Start":"13:07.375 ","End":"13:12.445","Text":"I\u0027ll let the box be 2x minus 5."},{"Start":"13:12.445 ","End":"13:20.050","Text":"But 1 over something to the power of 3 is to the power of minus 3 because"},{"Start":"13:20.050 ","End":"13:23.680","Text":"a positive exponent on the denominators like"},{"Start":"13:23.680 ","End":"13:28.435","Text":"a negative exponent in the numerator like this."},{"Start":"13:28.435 ","End":"13:31.195","Text":"Now, this is our box."},{"Start":"13:31.195 ","End":"13:40.990","Text":"We can use the formula and say that y prime is n which is minus 3 box,"},{"Start":"13:40.990 ","End":"13:46.120","Text":"which is 2x minus 5^n minus 1,"},{"Start":"13:46.120 ","End":"13:48.864","Text":"minus 3 minus 1 is minus 4."},{"Start":"13:48.864 ","End":"13:54.340","Text":"Then we must remember to also put box prime."},{"Start":"13:54.340 ","End":"13:57.745","Text":"Box is the inner function,"},{"Start":"13:57.745 ","End":"14:00.550","Text":"and this is called the inner derivative,"},{"Start":"14:00.550 ","End":"14:03.654","Text":"would be 2. That\u0027s the answer."},{"Start":"14:03.654 ","End":"14:07.615","Text":"Of course, we could simplify it and say minus 6."},{"Start":"14:07.615 ","End":"14:11.425","Text":"Combine the 2 with the minus 3, doesn\u0027t matter."},{"Start":"14:11.425 ","End":"14:15.670","Text":"You could also put this back down on the denominator."},{"Start":"14:15.670 ","End":"14:19.120","Text":"This is fine, but if you want to simplify it,"},{"Start":"14:19.120 ","End":"14:26.140","Text":"you could say this is minus 6 from the minus 2, 10 minus 3."},{"Start":"14:26.140 ","End":"14:34.405","Text":"You can also put the 2x minus 5^plus 4 on the denominator."},{"Start":"14:34.405 ","End":"14:36.550","Text":"That\u0027s another example."},{"Start":"14:36.550 ","End":"14:55.105","Text":"Now, let\u0027s say y equals 1/3x minus 2,"},{"Start":"14:55.105 ","End":"14:57.520","Text":"and I change the color."},{"Start":"14:57.520 ","End":"15:03.385","Text":"This is 1 of our special cases where this is n is minus 1."},{"Start":"15:03.385 ","End":"15:08.965","Text":"We know that the derivative y prime is"},{"Start":"15:08.965 ","End":"15:15.115","Text":"minus 1 over this box,"},{"Start":"15:15.115 ","End":"15:22.495","Text":"inner function to the power of 2 times the inner derivative of box,"},{"Start":"15:22.495 ","End":"15:25.975","Text":"which in this case is just 3."},{"Start":"15:25.975 ","End":"15:28.240","Text":"That\u0027s the answer to that."},{"Start":"15:28.240 ","End":"15:31.675","Text":"Let\u0027s keep on with a few more examples."},{"Start":"15:31.675 ","End":"15:41.590","Text":"How about y equals 1/fourth root"},{"Start":"15:41.590 ","End":"15:51.310","Text":"of 2x squared plus 10x plus 1."},{"Start":"15:51.310 ","End":"15:53.480","Text":"That should do."},{"Start":"15:53.490 ","End":"15:58.915","Text":"First thing I would do would be to write it as an exponent,"},{"Start":"15:58.915 ","End":"16:06.750","Text":"so I would take this 2x squared plus 10x plus 1,"},{"Start":"16:06.750 ","End":"16:09.180","Text":"this is the inner function or the box."},{"Start":"16:09.180 ","End":"16:13.705","Text":"Write this to the power of minus a quarter."},{"Start":"16:13.705 ","End":"16:15.445","Text":"That\u0027s like 2 steps."},{"Start":"16:15.445 ","End":"16:21.025","Text":"The 1 over makes it negative and the fourth root makes it to the power of a quarter."},{"Start":"16:21.025 ","End":"16:23.410","Text":"This is just algebra."},{"Start":"16:23.410 ","End":"16:26.710","Text":"From here, using the formula,"},{"Start":"16:26.710 ","End":"16:31.530","Text":"the chain rule says that y prime is,"},{"Start":"16:31.530 ","End":"16:35.475","Text":"we take n x to the n minus 1,"},{"Start":"16:35.475 ","End":"16:38.310","Text":"so it\u0027s minus a quarter."},{"Start":"16:38.310 ","End":"16:41.090","Text":"But it\u0027s not x, it\u0027s the box,"},{"Start":"16:41.090 ","End":"16:49.975","Text":"so it\u0027s 2x squared plus 10x plus 1 to the power of n minus 1."},{"Start":"16:49.975 ","End":"16:51.910","Text":"I subtract 1 from this,"},{"Start":"16:51.910 ","End":"16:56.620","Text":"I\u0027ll get minus 1 and a quarter or minus 5 quarters,"},{"Start":"16:56.620 ","End":"17:00.580","Text":"and then do not forget the inner derivative,"},{"Start":"17:00.580 ","End":"17:03.730","Text":"which is the derivative of box."},{"Start":"17:03.730 ","End":"17:07.825","Text":"The inner function would be, let\u0027s see,"},{"Start":"17:07.825 ","End":"17:11.470","Text":"2 times 2 is 4, 4x plus,"},{"Start":"17:11.470 ","End":"17:15.400","Text":"derivative of 10 x is 10 and 1 gives me nothing."},{"Start":"17:15.400 ","End":"17:18.940","Text":"This would be the answer to this 1."},{"Start":"17:18.940 ","End":"17:21.730","Text":"I\u0027ll keep going with a couple more."},{"Start":"17:21.730 ","End":"17:31.280","Text":"I know what, we can combine chain rule and product or quotient rule. Let\u0027s try that."},{"Start":"17:31.650 ","End":"17:33.520","Text":"I have 1,"},{"Start":"17:33.520 ","End":"17:44.200","Text":"let\u0027s let y equal x squared plus 1 to the power of 7 times,"},{"Start":"17:44.200 ","End":"17:47.390","Text":"this is where the product rule is going to come in,"},{"Start":"17:47.700 ","End":"17:55.015","Text":"let\u0027s say 4x plus 8,"},{"Start":"17:55.015 ","End":"17:58.570","Text":"so now it\u0027s a bit more involved."},{"Start":"17:58.570 ","End":"18:01.360","Text":"We see that there\u0027s a product,"},{"Start":"18:01.360 ","End":"18:05.260","Text":"and in the first factor in the product,"},{"Start":"18:05.260 ","End":"18:08.740","Text":"we also see that we\u0027re going to need the chain rule."},{"Start":"18:08.740 ","End":"18:12.940","Text":"I\u0027m going to give you a quick reminder of what the product rule says."},{"Start":"18:12.940 ","End":"18:17.470","Text":"If I have a product, let us write it briefly, derivative,"},{"Start":"18:17.470 ","End":"18:22.675","Text":"I differentiate the first and multiply by the second as is,"},{"Start":"18:22.675 ","End":"18:28.250","Text":"and then take the first as is and multiply it by the derivative of the second."},{"Start":"18:28.740 ","End":"18:33.565","Text":"In this case we get that y prime is,"},{"Start":"18:33.565 ","End":"18:35.785","Text":"derivative of the first,"},{"Start":"18:35.785 ","End":"18:38.860","Text":"the derivative of this part."},{"Start":"18:38.860 ","End":"18:42.280","Text":"This part is just the chain rule."},{"Start":"18:42.280 ","End":"18:45.835","Text":"We have box to the power of 7,"},{"Start":"18:45.835 ","End":"18:51.895","Text":"so it\u0027s 7 times box to the power of 6,"},{"Start":"18:51.895 ","End":"18:54.835","Text":"but times box prime,"},{"Start":"18:54.835 ","End":"18:59.320","Text":"which is the derivative of x squared plus 1, which is 2x."},{"Start":"18:59.320 ","End":"19:02.695","Text":"Now all this is just the f prime part."},{"Start":"19:02.695 ","End":"19:04.300","Text":"Now we need the g,"},{"Start":"19:04.300 ","End":"19:09.655","Text":"which is 4x plus 8,"},{"Start":"19:09.655 ","End":"19:11.905","Text":"and now we\u0027re up to the plus."},{"Start":"19:11.905 ","End":"19:14.020","Text":"Now we need the f,"},{"Start":"19:14.020 ","End":"19:20.995","Text":"which is just x squared plus 1 to the power of 7,"},{"Start":"19:20.995 ","End":"19:27.620","Text":"and then we need the g prime derivative of this, which is 4."},{"Start":"19:28.770 ","End":"19:32.800","Text":"Normally I would simplify this, for example,"},{"Start":"19:32.800 ","End":"19:37.030","Text":"x squared plus 1 to the power of 6 could take out the brackets,"},{"Start":"19:37.030 ","End":"19:42.040","Text":"but I\u0027m not going to waste the time on simplification."},{"Start":"19:42.040 ","End":"19:43.570","Text":"I\u0027ll leave you to do that."},{"Start":"19:43.570 ","End":"19:47.080","Text":"Let\u0027s do 1 more example."},{"Start":"19:47.080 ","End":"19:52.220","Text":"This time we\u0027ll combine chain rule with quotient rule."},{"Start":"19:52.680 ","End":"19:57.610","Text":"How about y equals,"},{"Start":"19:57.610 ","End":"20:00.880","Text":"now, we\u0027ll make it as a quotient."},{"Start":"20:00.880 ","End":"20:02.230","Text":"On the numerator,"},{"Start":"20:02.230 ","End":"20:08.440","Text":"I\u0027m going to put in x squared plus 5x,"},{"Start":"20:08.440 ","End":"20:10.900","Text":"and we\u0027ll make that cube,"},{"Start":"20:10.900 ","End":"20:14.290","Text":"so the numerator looks like a case for the chain rule."},{"Start":"20:14.290 ","End":"20:18.445","Text":"For the denominator, I\u0027ll also use an exponent;"},{"Start":"20:18.445 ","End":"20:26.035","Text":"let\u0027s take 4x minus 6 to the power of 8,"},{"Start":"20:26.035 ","End":"20:30.580","Text":"and let\u0027s see if we can differentiate that."},{"Start":"20:30.580 ","End":"20:33.520","Text":"Let me remind you of the quotient rule."},{"Start":"20:33.520 ","End":"20:42.805","Text":"It says that if we have f over g and you want to differentiate that it\u0027s equal to,"},{"Start":"20:42.805 ","End":"20:47.620","Text":"I like to put the denominator first, denominator squared,"},{"Start":"20:47.620 ","End":"20:53.425","Text":"and then it\u0027s the derivative of the numerator times the denominator."},{"Start":"20:53.425 ","End":"20:55.855","Text":"Here it looks a bit like the product rule,"},{"Start":"20:55.855 ","End":"20:57.610","Text":"but there\u0027s a minus."},{"Start":"20:57.610 ","End":"21:05.785","Text":"Minus the numerator times the derivative of the denominator like so,"},{"Start":"21:05.785 ","End":"21:09.490","Text":"so let\u0027s apply it to our case,"},{"Start":"21:09.490 ","End":"21:15.880","Text":"and we get that y prime equals."},{"Start":"21:15.880 ","End":"21:21.790","Text":"Now, I\u0027ll start with the denominator, g squared."},{"Start":"21:21.790 ","End":"21:27.970","Text":"Denominator squared is 4x minus 6 to the 8 squared,"},{"Start":"21:27.970 ","End":"21:30.970","Text":"so I can write straightaway 16."},{"Start":"21:30.970 ","End":"21:36.520","Text":"Something to the power of 8 to the power of 2 is the power of 8 times 2,"},{"Start":"21:36.520 ","End":"21:41.905","Text":"and now here I\u0027ll do the quotient rule,"},{"Start":"21:41.905 ","End":"21:46.465","Text":"f prime, but for this I\u0027ll need the chain rule,"},{"Start":"21:46.465 ","End":"21:49.330","Text":"so it\u0027s something to the power of 3,"},{"Start":"21:49.330 ","End":"21:54.535","Text":"so it is 3 times"},{"Start":"21:54.535 ","End":"22:00.310","Text":"that something box to the power of 2,"},{"Start":"22:00.310 ","End":"22:02.305","Text":"and x to the n minus 1,"},{"Start":"22:02.305 ","End":"22:05.275","Text":"but I need box prime also."},{"Start":"22:05.275 ","End":"22:11.215","Text":"Derivative of x squared plus 5x is 2x plus 5."},{"Start":"22:11.215 ","End":"22:17.904","Text":"That brings me just up to the f prime."},{"Start":"22:17.904 ","End":"22:21.025","Text":"Now I need to throw in the g,"},{"Start":"22:21.025 ","End":"22:30.535","Text":"which is 4x minus 6 to the power of 8."},{"Start":"22:30.535 ","End":"22:35.560","Text":"I\u0027m going to need to make this much longer."},{"Start":"22:35.560 ","End":"22:38.905","Text":"And that\u0027s better,"},{"Start":"22:38.905 ","End":"22:41.365","Text":"and it shifted this a bit."},{"Start":"22:41.365 ","End":"22:43.135","Text":"Now the minus,"},{"Start":"22:43.135 ","End":"22:45.565","Text":"we\u0027re up to here, f,"},{"Start":"22:45.565 ","End":"22:47.380","Text":"which is the numerator,"},{"Start":"22:47.380 ","End":"22:55.315","Text":"is just the x squared plus 5x to the power of 3."},{"Start":"22:55.315 ","End":"22:57.490","Text":"Now I need the g prime,"},{"Start":"22:57.490 ","End":"22:59.950","Text":"which is the derivative of the denominator."},{"Start":"22:59.950 ","End":"23:01.780","Text":"Again, chain rule."},{"Start":"23:01.780 ","End":"23:05.875","Text":"It\u0027s box e to the power of 8,"},{"Start":"23:05.875 ","End":"23:10.285","Text":"so we need 8 times whatever it is,"},{"Start":"23:10.285 ","End":"23:15.610","Text":"the 4x minus 6 to the power of 7,"},{"Start":"23:15.610 ","End":"23:18.790","Text":"and then the inner derivative of the box,"},{"Start":"23:18.790 ","End":"23:20.185","Text":"which is 4x minus 6,"},{"Start":"23:20.185 ","End":"23:23.170","Text":"and it is 4."},{"Start":"23:23.170 ","End":"23:30.490","Text":"First we could simplify like the 4x minus 6 probably,"},{"Start":"23:30.490 ","End":"23:32.920","Text":"but I\u0027m not going to do any simplification."},{"Start":"23:32.920 ","End":"23:34.855","Text":"This is the answer,"},{"Start":"23:34.855 ","End":"23:37.180","Text":"and we\u0027ve run out of time."},{"Start":"23:37.180 ","End":"23:44.290","Text":"I said that I might show you logarithmic and trigonometric cases for chain rule,"},{"Start":"23:44.290 ","End":"23:47.185","Text":"but changed my mind on that."},{"Start":"23:47.185 ","End":"23:49.960","Text":"We ran out of time and we\u0027ll leave it at that."},{"Start":"23:49.960 ","End":"23:58.490","Text":"There will be also more solved exercises following the tutorial. We\u0027re done."}],"ID":10437},{"Watched":false,"Name":"Second Derivatives","Duration":"2m 57s","ChapterTopicVideoID":10134,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.875","Text":"In this clip, I\u0027ll be talking about the second derivative of a function."},{"Start":"00:04.875 ","End":"00:08.265","Text":"Up til now, we know what a derivative is."},{"Start":"00:08.265 ","End":"00:10.190","Text":"Now I\u0027m going to expand on this."},{"Start":"00:10.190 ","End":"00:12.359","Text":"When we have a function y,"},{"Start":"00:12.359 ","End":"00:15.630","Text":"or sometimes we\u0027ll use the f of x notation,"},{"Start":"00:15.630 ","End":"00:17.310","Text":"and I\u0027ll illustrate through an example."},{"Start":"00:17.310 ","End":"00:20.200","Text":"Let\u0027s say this time we have x^4,"},{"Start":"00:20.200 ","End":"00:21.975","Text":"then the derivative,"},{"Start":"00:21.975 ","End":"00:23.939","Text":"and if we\u0027re using the y notation,"},{"Start":"00:23.939 ","End":"00:24.990","Text":"it\u0027s y prime,"},{"Start":"00:24.990 ","End":"00:27.315","Text":"or if we\u0027re using the f notation,"},{"Start":"00:27.315 ","End":"00:29.145","Text":"then it\u0027s f prime of x."},{"Start":"00:29.145 ","End":"00:32.040","Text":"This time it will be 4x cubed."},{"Start":"00:32.040 ","End":"00:36.410","Text":"Now this derivative is the function itself and we could derive it,"},{"Start":"00:36.410 ","End":"00:39.020","Text":"and we could get y double prime,"},{"Start":"00:39.020 ","End":"00:42.500","Text":"that will be 4 times 3x squared,"},{"Start":"00:42.500 ","End":"00:45.095","Text":"in other words, 12x squared."},{"Start":"00:45.095 ","End":"00:48.410","Text":"This is the second derivative which implies that there may be"},{"Start":"00:48.410 ","End":"00:52.625","Text":"a third and fourth derivatives of every level or just can keep going."},{"Start":"00:52.625 ","End":"00:54.460","Text":"Y triple prime,"},{"Start":"00:54.460 ","End":"01:01.235","Text":"third derivative of y would just be 24x and then y quadruple prime,"},{"Start":"01:01.235 ","End":"01:03.770","Text":"we don\u0027t usually write quadruple prime,"},{"Start":"01:03.770 ","End":"01:06.740","Text":"we put a 4 in the brackets,"},{"Start":"01:06.740 ","End":"01:08.270","Text":"which doesn\u0027t mean y^4,"},{"Start":"01:08.270 ","End":"01:10.280","Text":"that means y prime, prime, prime, prime,"},{"Start":"01:10.280 ","End":"01:18.940","Text":"derive 4 times, we get 24 and y fifth derivative is actually equal to 0,"},{"Start":"01:18.940 ","End":"01:22.430","Text":"from 5 onwards it continues to be 0."},{"Start":"01:22.430 ","End":"01:24.810","Text":"You might think that that\u0027s the fate of all functions,"},{"Start":"01:24.810 ","End":"01:26.960","Text":"but now let me give you another example,"},{"Start":"01:26.960 ","End":"01:30.215","Text":"e to the x here as the function itself,"},{"Start":"01:30.215 ","End":"01:37.460","Text":"the first derivative is also e to the x and so the derivative of this will be e to the x,"},{"Start":"01:37.460 ","End":"01:39.200","Text":"and here e to the x,"},{"Start":"01:39.200 ","End":"01:41.105","Text":"e to the x, e to the x."},{"Start":"01:41.105 ","End":"01:43.790","Text":"So they don\u0027t all end up 0."},{"Start":"01:43.790 ","End":"01:46.310","Text":"They also don\u0027t all end up repeating themselves."},{"Start":"01:46.310 ","End":"01:48.215","Text":"I\u0027ll give you another example in a minute."},{"Start":"01:48.215 ","End":"01:51.230","Text":"The 1 I\u0027m most interested in is the second derivative."},{"Start":"01:51.230 ","End":"01:54.960","Text":"Actually, it also has a definition which is double prime,"},{"Start":"01:54.960 ","End":"01:59.300","Text":"second derivative is simply the first derivative of the first derivative,"},{"Start":"01:59.300 ","End":"02:00.880","Text":"it\u0027s y prime, prime."},{"Start":"02:00.880 ","End":"02:03.095","Text":"Or if I\u0027m using the f notation,"},{"Start":"02:03.095 ","End":"02:06.530","Text":"f double prime of x is just"},{"Start":"02:06.530 ","End":"02:09.080","Text":"f prime, prime of x."},{"Start":"02:09.080 ","End":"02:14.435","Text":"I suppose you could write it as f prime of x prime,"},{"Start":"02:14.435 ","End":"02:18.680","Text":"but it just means differentiate and then differentiate again,"},{"Start":"02:18.680 ","End":"02:21.635","Text":"f double prime of x and so on."},{"Start":"02:21.635 ","End":"02:23.750","Text":"At a certain point we stop using primes,"},{"Start":"02:23.750 ","End":"02:25.115","Text":"we use number in brackets."},{"Start":"02:25.115 ","End":"02:28.684","Text":"I\u0027ve actually even seen Roman letters used, for example,"},{"Start":"02:28.684 ","End":"02:32.960","Text":"as y Roman 4 and y Roman 5."},{"Start":"02:32.960 ","End":"02:38.270","Text":"What I\u0027m mostly concentrating on is what we call the second derivative."},{"Start":"02:38.270 ","End":"02:42.290","Text":"This is very useful in calculus in investigating a function."},{"Start":"02:42.290 ","End":"02:46.910","Text":"It\u0027s useful in finding areas of convexity and concavity."},{"Start":"02:46.910 ","End":"02:52.430","Text":"It\u0027s useful in finding out whether a point is a maximum or a minimum."},{"Start":"02:52.430 ","End":"02:54.200","Text":"There\u0027s all uses for it,"},{"Start":"02:54.200 ","End":"02:55.100","Text":"but there you are,"},{"Start":"02:55.100 ","End":"02:58.200","Text":"I\u0027ve introduced you to the second derivative."}],"ID":10439},{"Watched":false,"Name":"The Derivative of Function with Parameters","Duration":"1m 1s","ChapterTopicVideoID":10135,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.270","Text":"In this clip, I\u0027ll talk about differentiating a function with a parameter,"},{"Start":"00:04.270 ","End":"00:06.525","Text":"it could be more than 1 parameter."},{"Start":"00:06.525 ","End":"00:14.760","Text":"Suppose we have y equals ax squared plus bx plus c. Usually we take a,"},{"Start":"00:14.760 ","End":"00:16.560","Text":"b, and c to be constants."},{"Start":"00:16.560 ","End":"00:17.745","Text":"We don\u0027t know what they are."},{"Start":"00:17.745 ","End":"00:19.380","Text":"But when I write something like this,"},{"Start":"00:19.380 ","End":"00:25.170","Text":"it says if I\u0027ve written y equals 4x squared plus 5x plus 3."},{"Start":"00:25.170 ","End":"00:26.850","Text":"In this regard, a, b, and c,"},{"Start":"00:26.850 ","End":"00:29.845","Text":"which are parameters are treated like constants."},{"Start":"00:29.845 ","End":"00:34.775","Text":"So what I get is that y prime is equal to,"},{"Start":"00:34.775 ","End":"00:36.260","Text":"now if I treat this like a constant,"},{"Start":"00:36.260 ","End":"00:39.890","Text":"the constant just stays there and we differentiate x squared,"},{"Start":"00:39.890 ","End":"00:41.440","Text":"we get 2x,"},{"Start":"00:41.440 ","End":"00:43.970","Text":"b being a constant stays there."},{"Start":"00:43.970 ","End":"00:46.675","Text":"The derivative of x is 1,"},{"Start":"00:46.675 ","End":"00:51.395","Text":"c is a constant and the derivative of a constant is 0."},{"Start":"00:51.395 ","End":"00:58.865","Text":"Ultimately, we get basically 2ax plus b and that\u0027s my y prime."},{"Start":"00:58.865 ","End":"01:01.920","Text":"That\u0027s all there is. We\u0027re done."}],"ID":10440},{"Watched":false,"Name":"Exercise 1 - Parts 1-6","Duration":"4m 43s","ChapterTopicVideoID":10129,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this exercise, which is really 12 in 1,"},{"Start":"00:02.820 ","End":"00:06.165","Text":"we just have to find the derivative for each of the function."},{"Start":"00:06.165 ","End":"00:08.820","Text":"First one, f of x is equal to 4,"},{"Start":"00:08.820 ","End":"00:14.280","Text":"and I\u0027ll remind you the derivative of a constant 4 or otherwise, it\u0027s just 0."},{"Start":"00:14.280 ","End":"00:23.775","Text":"Here we write the derivative function f prime of x is equal to 0, constant 0."},{"Start":"00:23.775 ","End":"00:25.080","Text":"We have to remember that."},{"Start":"00:25.080 ","End":"00:33.720","Text":"Now, number 2, g of x is e plus square root of 2 all over 2."},{"Start":"00:33.720 ","End":"00:35.220","Text":"They\u0027re trying to confuse you here with"},{"Start":"00:35.220 ","End":"00:37.250","Text":"all this e and square root and everything,"},{"Start":"00:37.250 ","End":"00:39.395","Text":"but if you look at it, these are just numbers."},{"Start":"00:39.395 ","End":"00:42.140","Text":"e is a number, 2 square root, it\u0027s all numbers."},{"Start":"00:42.140 ","End":"00:51.180","Text":"We\u0027re still in the case of a constant so that g prime of x is 0."},{"Start":"00:51.180 ","End":"00:54.425","Text":"Now in the 3rd exercise,"},{"Start":"00:54.425 ","End":"00:57.545","Text":"we have slightly different."},{"Start":"00:57.545 ","End":"01:06.010","Text":"We have that h of x is equal to x^4."},{"Start":"01:06.010 ","End":"01:11.120","Text":"Now I need to say that the derivative of the function x^n,"},{"Start":"01:11.120 ","End":"01:13.370","Text":"if I differentiate it,"},{"Start":"01:13.370 ","End":"01:18.245","Text":"it comes out as nx to the n minus 1."},{"Start":"01:18.245 ","End":"01:23.630","Text":"In this case, we get that the derivative function h prime is"},{"Start":"01:23.630 ","End":"01:29.090","Text":"put the 4 in front of the x and then lower the 4 by 1, 4x cubed."},{"Start":"01:29.090 ","End":"01:31.040","Text":"Don\u0027t worry, they\u0027ll get hard soon enough."},{"Start":"01:31.040 ","End":"01:33.020","Text":"Meanwhile, enjoy them while they\u0027re easy."},{"Start":"01:33.020 ","End":"01:37.505","Text":"y equals 1 over x squared,"},{"Start":"01:37.505 ","End":"01:41.075","Text":"what we can do here is actually bring it to this form."},{"Start":"01:41.075 ","End":"01:46.400","Text":"Because if you remember 1 over using your laws of exponents, for example,"},{"Start":"01:46.400 ","End":"01:52.685","Text":"1 over a^b in general is a to the power of minus b."},{"Start":"01:52.685 ","End":"01:54.455","Text":"If I use that here,"},{"Start":"01:54.455 ","End":"01:59.200","Text":"then this thing will just be x to the minus 2."},{"Start":"01:59.200 ","End":"02:06.770","Text":"Now, I can use this formula and say that y prime is equal to minus 2,"},{"Start":"02:06.770 ","End":"02:09.620","Text":"which is the exponent in front."},{"Start":"02:09.620 ","End":"02:14.655","Text":"Then reduce it by 1 so it becomes minus 3."},{"Start":"02:14.655 ","End":"02:17.675","Text":"I could leave the answer like this,"},{"Start":"02:17.675 ","End":"02:22.370","Text":"but because the original gave me"},{"Start":"02:22.370 ","End":"02:25.400","Text":"as an exponent in the denominator and"},{"Start":"02:25.400 ","End":"02:29.065","Text":"not negative exponents, so I\u0027d prefer prefer to put it back into that form."},{"Start":"02:29.065 ","End":"02:34.850","Text":"The minus 2 I leave, but x to the minus 3 is 1 over x^3."},{"Start":"02:34.850 ","End":"02:37.950","Text":"This is how I would put the answer."},{"Start":"02:37.950 ","End":"02:40.235","Text":"Now, number 5,"},{"Start":"02:40.235 ","End":"02:44.345","Text":"f of x equals the square root of x."},{"Start":"02:44.345 ","End":"02:47.315","Text":"Notice that just from algebra,"},{"Start":"02:47.315 ","End":"02:55.400","Text":"that the square root of x is x to the power of 1/2."},{"Start":"02:55.400 ","End":"03:01.040","Text":"I can use that and I\u0027ll rewrite the other formula from above"},{"Start":"03:01.040 ","End":"03:07.575","Text":"that the derivative of x^n is nx to the n minus 1,"},{"Start":"03:07.575 ","End":"03:10.810","Text":"because the other one disappeared out of sight."},{"Start":"03:10.810 ","End":"03:15.155","Text":"This is equal to just another way in algebra,"},{"Start":"03:15.155 ","End":"03:19.925","Text":"writing square roots is to say that it\u0027s x to the power of 1/2."},{"Start":"03:19.925 ","End":"03:22.220","Text":"Now if I differentiate,"},{"Start":"03:22.220 ","End":"03:25.670","Text":"I will get from this nx to the n minus 1,"},{"Start":"03:25.670 ","End":"03:34.450","Text":"1/2 x to the 1/2 minus 1 is just minus a 1/2."},{"Start":"03:34.450 ","End":"03:39.650","Text":"Now I want to fix it up because in the original it was in terms of square root signs."},{"Start":"03:39.650 ","End":"03:44.725","Text":"What we do is x to the minus 1/2 is 1 over x to the 1/2,"},{"Start":"03:44.725 ","End":"03:47.025","Text":"which we know is square root of x."},{"Start":"03:47.025 ","End":"03:51.510","Text":"That\u0027s where we have the 1 over the square root of x."},{"Start":"03:51.510 ","End":"03:58.755","Text":"Now all this 1 over this times a 1/2 just gives us 1 over twice the square root of x."},{"Start":"03:58.755 ","End":"04:01.230","Text":"That would be our answer."},{"Start":"04:01.230 ","End":"04:06.269","Text":"Number 6, z as a function of x"},{"Start":"04:06.269 ","End":"04:12.075","Text":"is equal to 1 over the square root of x."},{"Start":"04:12.075 ","End":"04:17.070","Text":"This is equal to 1 over,"},{"Start":"04:17.070 ","End":"04:20.700","Text":"square root of x is, x to the 1/2."},{"Start":"04:20.700 ","End":"04:23.120","Text":"Altogether, if writing it,"},{"Start":"04:23.120 ","End":"04:31.520","Text":"we could also write this as x to the power of minus 1/2 and get rid of that 1 over."},{"Start":"04:31.520 ","End":"04:36.785","Text":"Now, z prime of x,"},{"Start":"04:36.785 ","End":"04:44.610","Text":"the derivative, is going to equal minus a 1/2 x to the power of minus 1 and 1/2."}],"ID":10441},{"Watched":false,"Name":"Exercise 1 - Parts 7-12","Duration":"9m 45s","ChapterTopicVideoID":10130,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.025","Text":"Now 7, I want to,"},{"Start":"00:02.025 ","End":"00:10.580","Text":"again remind you of the famous formula that if you have 4x^ 10th in general,"},{"Start":"00:10.580 ","End":"00:14.085","Text":"ax^n, and you differentiate it,"},{"Start":"00:14.085 ","End":"00:19.950","Text":"what you get is anx^n minus 1,"},{"Start":"00:19.950 ","End":"00:22.980","Text":"which I always say as multiply the exponent"},{"Start":"00:22.980 ","End":"00:26.025","Text":"by the coefficient and reduce the exponent by 1."},{"Start":"00:26.025 ","End":"00:33.500","Text":"The other thing is that if we have some plus or minuses, they just stay."},{"Start":"00:33.500 ","End":"00:38.430","Text":"I\u0027ll also remind you of negative exponents."},{"Start":"00:38.430 ","End":"00:42.590","Text":"1 over x, for example, is x^minus 1."},{"Start":"00:42.590 ","End":"00:49.520","Text":"Given all this, we can say that y prime is equal to 10 times 4 is 40,"},{"Start":"00:49.520 ","End":"00:54.285","Text":"x^10 minus 1 is 9 plus,"},{"Start":"00:54.285 ","End":"00:58.740","Text":"now wait, 1 over x is x^minus 1."},{"Start":"00:58.740 ","End":"01:00.930","Text":"If we take the derivative of that,"},{"Start":"01:00.930 ","End":"01:06.730","Text":"x to the minus 1 prime is going to be minus 1 comes out in front."},{"Start":"01:06.730 ","End":"01:10.620","Text":"The minus 1 also decreases by 1."},{"Start":"01:10.620 ","End":"01:14.435","Text":"What we get is minus 1 over x squared,"},{"Start":"01:14.435 ","End":"01:16.815","Text":"but really should be a minus."},{"Start":"01:16.815 ","End":"01:18.290","Text":"Let me do it there."},{"Start":"01:18.290 ","End":"01:19.775","Text":"That looks a bit better."},{"Start":"01:19.775 ","End":"01:21.605","Text":"Now on to Number 8,"},{"Start":"01:21.605 ","End":"01:26.055","Text":"y equals x plus x squared."},{"Start":"01:26.055 ","End":"01:28.370","Text":"No new things to remember,"},{"Start":"01:28.370 ","End":"01:31.430","Text":"just of course that x is x to the power of 1."},{"Start":"01:31.430 ","End":"01:33.410","Text":"Just bear that in mind."},{"Start":"01:33.410 ","End":"01:37.440","Text":"Y prime is equal,"},{"Start":"01:37.440 ","End":"01:38.625","Text":"well to show you on this side,"},{"Start":"01:38.625 ","End":"01:42.240","Text":"if x is x to the power of 1,"},{"Start":"01:42.240 ","End":"01:43.745","Text":"then if we differentiate,"},{"Start":"01:43.745 ","End":"01:49.895","Text":"x prime is going to be 1 times x to the 1 minus 1."},{"Start":"01:49.895 ","End":"01:53.705","Text":"In other words, it\u0027s just equal to 1."},{"Start":"01:53.705 ","End":"01:55.675","Text":"Here we have 1,"},{"Start":"01:55.675 ","End":"01:57.605","Text":"and here by the usual formula,"},{"Start":"01:57.605 ","End":"02:02.285","Text":"multiply by 2 reduce the power by 1, 1 plus 2x."},{"Start":"02:02.285 ","End":"02:05.735","Text":"That\u0027s what there is to Number 8."},{"Start":"02:05.735 ","End":"02:09.850","Text":"Number 9 is y equals"},{"Start":"02:09.850 ","End":"02:17.765","Text":"4x squared plus 8x cubed minus 5."},{"Start":"02:17.765 ","End":"02:20.945","Text":"These are just the usual formulae."},{"Start":"02:20.945 ","End":"02:25.880","Text":"Y prime is 2 times 4 is 8."},{"Start":"02:25.880 ","End":"02:31.610","Text":"Then reduce the power by 1x to the power of 1 is just x."},{"Start":"02:31.610 ","End":"02:34.565","Text":"Next 3 times 8 is 24,"},{"Start":"02:34.565 ","End":"02:36.255","Text":"lower the power,"},{"Start":"02:36.255 ","End":"02:40.010","Text":"x squared and a constant is 0,"},{"Start":"02:40.010 ","End":"02:41.855","Text":"like we said before."},{"Start":"02:41.855 ","End":"02:44.120","Text":"That\u0027s Number 9."},{"Start":"02:44.120 ","End":"02:47.900","Text":"Next we have Number 10,"},{"Start":"02:47.900 ","End":"02:50.850","Text":"which is y,"},{"Start":"02:50.850 ","End":"02:56.690","Text":"although sometimes people write y of x to show that y really is a function of x."},{"Start":"02:56.690 ","End":"02:58.250","Text":"But there\u0027s nothing new here."},{"Start":"02:58.250 ","End":"03:04.810","Text":"It\u0027s just notation and is 2 over x minus ex."},{"Start":"03:07.510 ","End":"03:12.875","Text":"Now, remember that e is just a number,"},{"Start":"03:12.875 ","End":"03:15.725","Text":"so nothing special about it."},{"Start":"03:15.725 ","End":"03:17.510","Text":"What we get here,"},{"Start":"03:17.510 ","End":"03:19.025","Text":"we have again the,"},{"Start":"03:19.025 ","End":"03:20.630","Text":"we just had it not long ago,"},{"Start":"03:20.630 ","End":"03:24.455","Text":"perhaps I can show you above,"},{"Start":"03:24.455 ","End":"03:33.480","Text":"that the derivative of x to the minus 1 is minus 1x to the minus 2."},{"Start":"03:33.500 ","End":"03:38.179","Text":"We can reuse that or I could just write it again."},{"Start":"03:38.179 ","End":"03:43.460","Text":"Y prime or derivative is equal to,"},{"Start":"03:43.460 ","End":"03:45.875","Text":"let\u0027s just do it again, 2 over x."},{"Start":"03:45.875 ","End":"03:49.925","Text":"That by simple algebra is 2x to the minus 1."},{"Start":"03:49.925 ","End":"03:53.490","Text":"When we differentiate that,"},{"Start":"03:53.810 ","End":"03:56.630","Text":"by taking the derivative,"},{"Start":"03:56.630 ","End":"04:01.955","Text":"we get minus 1 times 2 is minus 2,"},{"Start":"04:01.955 ","End":"04:05.595","Text":"and then x lower it by 1 to the minus 2,"},{"Start":"04:05.595 ","End":"04:10.445","Text":"what this comes out to is minus 2 over x squared."},{"Start":"04:10.445 ","End":"04:13.520","Text":"Because x to the minus 2 is 1 over x squared."},{"Start":"04:13.520 ","End":"04:16.865","Text":"Ex, it could be like 3x or 4x."},{"Start":"04:16.865 ","End":"04:19.650","Text":"It\u0027s just a number."},{"Start":"04:19.810 ","End":"04:23.650","Text":"Then we have the derivative of ax,"},{"Start":"04:23.650 ","End":"04:25.420","Text":"is just x to the 1,"},{"Start":"04:25.420 ","End":"04:27.305","Text":"1 times a is a,"},{"Start":"04:27.305 ","End":"04:30.170","Text":"and it\u0027s x to the 0."},{"Start":"04:30.170 ","End":"04:32.885","Text":"Said this before, because is x to the 1,"},{"Start":"04:32.885 ","End":"04:35.190","Text":"that\u0027s a 1, that\u0027s a prime."},{"Start":"04:35.190 ","End":"04:40.005","Text":"Which is just a,"},{"Start":"04:40.005 ","End":"04:43.850","Text":"because x to the 0 is 1 and our a here is e,"},{"Start":"04:43.850 ","End":"04:49.460","Text":"so it\u0027s just minus e. That\u0027s Number 10."},{"Start":"04:49.460 ","End":"04:54.010","Text":"After 10, we get 11."},{"Start":"04:55.940 ","End":"05:03.135","Text":"11 is where y equals square root of 2,"},{"Start":"05:03.135 ","End":"05:07.650","Text":"x squared plus 2ex."},{"Start":"05:07.650 ","End":"05:11.900","Text":"I see the person who wrote these exercises is trying to be tricky and throw you off."},{"Start":"05:11.900 ","End":"05:16.070","Text":"The square root of 2. The square root doesn\u0027t add any complexity at all."},{"Start":"05:16.070 ","End":"05:23.010","Text":"It\u0027s also just a constant and so is e. I think we can do it in 1 go."},{"Start":"05:23.010 ","End":"05:27.050","Text":"Y prime, but this is a constant times x squared,"},{"Start":"05:27.050 ","End":"05:29.405","Text":"multiply the 2 by the constant."},{"Start":"05:29.405 ","End":"05:32.165","Text":"It\u0027s twice square root of 2,"},{"Start":"05:32.165 ","End":"05:34.025","Text":"lower the power by 1,"},{"Start":"05:34.025 ","End":"05:36.145","Text":"x to the 1."},{"Start":"05:36.145 ","End":"05:41.675","Text":"Here, you should already know that the derivative of x is just 1."},{"Start":"05:41.675 ","End":"05:43.340","Text":"I won\u0027t go over it again."},{"Start":"05:43.340 ","End":"05:46.745","Text":"But in case you just want a reminder of y,"},{"Start":"05:46.745 ","End":"05:49.420","Text":"the derivative of x is 1."},{"Start":"05:49.420 ","End":"05:52.215","Text":"X is just x to the 1."},{"Start":"05:52.215 ","End":"05:54.630","Text":"When you take it prime,"},{"Start":"05:54.630 ","End":"06:01.110","Text":"then we\u0027ve just done it up here anyway. There you go."},{"Start":"06:01.110 ","End":"06:04.425","Text":"It said multiply the 1,"},{"Start":"06:04.425 ","End":"06:05.850","Text":"you lower it by 1,"},{"Start":"06:05.850 ","End":"06:07.530","Text":"and it just comes out to be 1."},{"Start":"06:07.530 ","End":"06:15.810","Text":"In fact, in general, the derivative of ax is a. I\u0027ve done that twice."},{"Start":"06:15.810 ","End":"06:20.535","Text":"That\u0027s the derivative of x is 1 so what we\u0027re left with is the 2e."},{"Start":"06:20.535 ","End":"06:25.805","Text":"That\u0027s the answer. Then we go on to Number 12."},{"Start":"06:25.805 ","End":"06:32.255","Text":"Number 12 is y but as a function of t,"},{"Start":"06:32.255 ","End":"06:35.570","Text":"just like here we had y and they remind you it\u0027s a function of x,"},{"Start":"06:35.570 ","End":"06:39.455","Text":"but that doesn\u0027t add any complexity,"},{"Start":"06:39.455 ","End":"06:47.820","Text":"just another style is equal to 4 over t. Variable is t and not x."},{"Start":"06:47.820 ","End":"06:50.420","Text":"That\u0027s just to break you from the habit of always assuming that"},{"Start":"06:50.420 ","End":"06:54.440","Text":"your independent variable is x could be t,"},{"Start":"06:54.440 ","End":"06:57.590","Text":"often used in physics to denote time,"},{"Start":"06:57.590 ","End":"07:00.140","Text":"4 over t plus cube root of"},{"Start":"07:00.140 ","End":"07:10.710","Text":"t. Let\u0027s see what is y prime of t?"},{"Start":"07:10.710 ","End":"07:18.465","Text":"We do a little bit to the side here. Once again."},{"Start":"07:18.465 ","End":"07:24.390","Text":"Remind you again about the 4 over t. 4 over"},{"Start":"07:24.390 ","End":"07:30.825","Text":"t is 4t^ minus 1."},{"Start":"07:30.825 ","End":"07:33.750","Text":"When you differentiate it,"},{"Start":"07:33.750 ","End":"07:36.695","Text":"4 over t prime,"},{"Start":"07:36.695 ","End":"07:41.575","Text":"is this times this is minus 4t to the minus 2."},{"Start":"07:41.575 ","End":"07:45.390","Text":"That\u0027s minus 4 over t squared."},{"Start":"07:45.390 ","End":"07:48.785","Text":"Here, minus 4 over t squared."},{"Start":"07:48.785 ","End":"07:50.420","Text":"Now, the other 1,"},{"Start":"07:50.420 ","End":"07:58.110","Text":"cube root of t. If you know your algebra,"},{"Start":"07:58.110 ","End":"08:00.225","Text":"it really helps with calculus."},{"Start":"08:00.225 ","End":"08:05.445","Text":"That is t to the power of 1/3."},{"Start":"08:05.445 ","End":"08:08.000","Text":"When we take the derivative of this thing,"},{"Start":"08:08.000 ","End":"08:13.955","Text":"we get 1/3t to the minus 2/3."},{"Start":"08:13.955 ","End":"08:17.155","Text":"Because 1/3 minus 1,"},{"Start":"08:17.155 ","End":"08:19.640","Text":"I\u0027m not going to do it on a calculator or anything."},{"Start":"08:19.640 ","End":"08:22.825","Text":"A 1/3 minus 1 is minus 2/3,"},{"Start":"08:22.825 ","End":"08:27.420","Text":"which is equal to 1 over,"},{"Start":"08:27.420 ","End":"08:28.955","Text":"let\u0027s put the 3 on the bottom."},{"Start":"08:28.955 ","End":"08:37.024","Text":"I think it\u0027s a minus I can also put the t to the power of 2/3 on the bottom also."},{"Start":"08:37.024 ","End":"08:42.965","Text":"However, since the original question was given in terms of a cube root sign,"},{"Start":"08:42.965 ","End":"08:46.470","Text":"I should really put the answer that way also."},{"Start":"08:46.790 ","End":"08:51.230","Text":"This is not going to be a plus after all,"},{"Start":"08:51.230 ","End":"08:52.550","Text":"it\u0027s going to be,"},{"Start":"08:52.550 ","End":"08:53.840","Text":"yes it is, it is a plus,"},{"Start":"08:53.840 ","End":"08:55.775","Text":"it\u0027s not a minus, sorry."},{"Start":"08:55.775 ","End":"09:00.840","Text":"Plus 1 over 3."},{"Start":"09:00.840 ","End":"09:03.340","Text":"But instead of t to the 2/3,"},{"Start":"09:03.340 ","End":"09:09.290","Text":"I can write it as the cube root of t squared."},{"Start":"09:09.290 ","End":"09:19.020","Text":"I\u0027m just using the mathematical formula that the nth root of,"},{"Start":"09:22.010 ","End":"09:24.705","Text":"I don\u0027t know what to say,"},{"Start":"09:24.705 ","End":"09:28.605","Text":"a to the power of m"},{"Start":"09:28.605 ","End":"09:37.445","Text":"is a to the power of m over n and vice versa."},{"Start":"09:37.445 ","End":"09:40.415","Text":"This is the answer for 12."},{"Start":"09:40.415 ","End":"09:45.960","Text":"Actually we\u0027re done with the whole series. That\u0027s it."}],"ID":10442},{"Watched":false,"Name":"Exercise 2 - Parts 1-4","Duration":"11m 58s","ChapterTopicVideoID":10131,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.265","Text":"In the following 12 exercises,"},{"Start":"00:02.265 ","End":"00:04.170","Text":"we just have to find the derivative."},{"Start":"00:04.170 ","End":"00:05.370","Text":"The first 1,"},{"Start":"00:05.370 ","End":"00:12.420","Text":"y equals x squared plus 3 times x minus 1."},{"Start":"00:12.420 ","End":"00:16.440","Text":"The first thing I see and everyone should see is that this is a product,"},{"Start":"00:16.440 ","End":"00:18.435","Text":"so we\u0027re going to use the product rule."},{"Start":"00:18.435 ","End":"00:26.205","Text":"The product rule says that if I have to find the derivative of a product,"},{"Start":"00:26.205 ","End":"00:28.920","Text":"that say, of 2 functions, f and g,"},{"Start":"00:28.920 ","End":"00:34.080","Text":"I\u0027ll just call it f and g. Then their derivative is the derivative of"},{"Start":"00:34.080 ","End":"00:41.165","Text":"the first times the second plus the first times the derivative of the second."},{"Start":"00:41.165 ","End":"00:45.435","Text":"Here\u0027s our f and here\u0027s our g,"},{"Start":"00:45.435 ","End":"00:50.690","Text":"so we get that y prime is equal to f prime,"},{"Start":"00:50.690 ","End":"00:53.540","Text":"that\u0027s x squared plus 3 derivative."},{"Start":"00:53.540 ","End":"00:59.010","Text":"That\u0027s 2x times the other 1 just as is x minus 1,"},{"Start":"00:59.010 ","End":"01:00.375","Text":"and now the reverse."},{"Start":"01:00.375 ","End":"01:02.375","Text":"We take the first 1 as is,"},{"Start":"01:02.375 ","End":"01:04.970","Text":"which is x squared plus 3."},{"Start":"01:04.970 ","End":"01:11.450","Text":"The second 1 is going to be derived and derivative of x minus 1 is just 1."},{"Start":"01:11.450 ","End":"01:13.010","Text":"You could say this is the answer,"},{"Start":"01:13.010 ","End":"01:19.120","Text":"but it\u0027s customary to simplify a bit and not leave it this messy, so let\u0027s continue."},{"Start":"01:19.120 ","End":"01:26.165","Text":"The first 1 if I multiply it out is 2x squared minus 2x,"},{"Start":"01:26.165 ","End":"01:30.410","Text":"and here plus x squared plus 3."},{"Start":"01:30.410 ","End":"01:33.575","Text":"Let\u0027s collect together the x squared terms."},{"Start":"01:33.575 ","End":"01:35.360","Text":"It\u0027s this and this,"},{"Start":"01:35.360 ","End":"01:36.920","Text":"is 3x squared,"},{"Start":"01:36.920 ","End":"01:44.460","Text":"x\u0027s only from here minus 2x and constant numbers from here plus 3."},{"Start":"01:44.460 ","End":"01:48.830","Text":"That\u0027s the answer except that I like"},{"Start":"01:48.830 ","End":"01:53.510","Text":"to point something out that you could do it without the product rule."},{"Start":"01:53.510 ","End":"01:56.555","Text":"You could just open the brackets here"},{"Start":"01:56.555 ","End":"02:01.100","Text":"and convert it into a polynomial and then differentiate."},{"Start":"02:01.100 ","End":"02:05.120","Text":"Obviously, the idea here is to learn the product rule and not to get around it."},{"Start":"02:05.120 ","End":"02:08.960","Text":"That was that. The next 1 is number 2."},{"Start":"02:08.960 ","End":"02:11.690","Text":"Once again, it\u0027s a product and like I said,"},{"Start":"02:11.690 ","End":"02:13.715","Text":"we could just expand it out,"},{"Start":"02:13.715 ","End":"02:15.890","Text":"but that misses the point."},{"Start":"02:15.890 ","End":"02:20.225","Text":"We\u0027re going to do it with the same rule and I\u0027ll just write it again."},{"Start":"02:20.225 ","End":"02:26.735","Text":"Fg prime is f prime g plus fg prime,"},{"Start":"02:26.735 ","End":"02:30.530","Text":"where this is going to be our f and this is going to be our g. I\u0027d also like to"},{"Start":"02:30.530 ","End":"02:35.855","Text":"remind you that the square root of x is x^1/2."},{"Start":"02:35.855 ","End":"02:41.840","Text":"We\u0027ve already previously done the derivative of the square root of x,"},{"Start":"02:41.840 ","End":"02:43.790","Text":"so I won\u0027t bother with all the details."},{"Start":"02:43.790 ","End":"02:46.835","Text":"Again, I\u0027ll just tell you the answer that the square root of x,"},{"Start":"02:46.835 ","End":"02:51.290","Text":"its derivative, is 1 over twice the square root of x."},{"Start":"02:51.290 ","End":"02:55.535","Text":"We have done this before and it\u0027s anyway an easy side exercise."},{"Start":"02:55.535 ","End":"03:01.070","Text":"Taking all this and putting it all this information here,"},{"Start":"03:01.070 ","End":"03:05.390","Text":"we would get that y prime is the derivative of"},{"Start":"03:05.390 ","End":"03:10.630","Text":"the first times the second plus the first times the derivative of the second."},{"Start":"03:10.630 ","End":"03:13.530","Text":"Derivative of the first, 4x plus 10."},{"Start":"03:13.530 ","End":"03:15.765","Text":"That\u0027s easy. That\u0027s 4."},{"Start":"03:15.765 ","End":"03:17.820","Text":"The other 1 as is,"},{"Start":"03:17.820 ","End":"03:22.110","Text":"square root of x minus 1 plus, vice versa."},{"Start":"03:22.110 ","End":"03:23.880","Text":"The first 1 as is,"},{"Start":"03:23.880 ","End":"03:27.965","Text":"4x plus 10, and the derivative of the second."},{"Start":"03:27.965 ","End":"03:33.270","Text":"This is 1 over twice square root of x,"},{"Start":"03:33.270 ","End":"03:36.635","Text":"and that\u0027s basically because the minus 1 gives nothing."},{"Start":"03:36.635 ","End":"03:40.280","Text":"Let\u0027s do a little bit of algebra to clean it up a bit."},{"Start":"03:40.280 ","End":"03:43.745","Text":"See is there anymore simplification we could do."},{"Start":"03:43.745 ","End":"03:47.180","Text":"Well, certainly this and this could be combined,"},{"Start":"03:47.180 ","End":"03:52.520","Text":"so we could get 6 times square root of x"},{"Start":"03:52.520 ","End":"03:59.965","Text":"minus 4 plus 5 square root of x over x."},{"Start":"03:59.965 ","End":"04:02.710","Text":"We\u0027re done with 2,"},{"Start":"04:02.720 ","End":"04:05.880","Text":"and now let\u0027s get onto 3."},{"Start":"04:05.880 ","End":"04:07.820","Text":"This is certainly a product,"},{"Start":"04:07.820 ","End":"04:09.875","Text":"but it\u0027s a product of 3 things."},{"Start":"04:09.875 ","End":"04:14.435","Text":"Whereas our rule really relates only to the product of 2 things."},{"Start":"04:14.435 ","End":"04:18.875","Text":"I suggest we somehow combine 2 of these into 1 and then,"},{"Start":"04:18.875 ","End":"04:23.269","Text":"to me, the natural thing is the first 2 because it gives us something squared."},{"Start":"04:23.269 ","End":"04:27.995","Text":"What I suggest is we write this as y"},{"Start":"04:27.995 ","End":"04:35.450","Text":"equals x minus 1 squared times x minus 2."},{"Start":"04:35.450 ","End":"04:42.240","Text":"Again, we\u0027ll be using the rule that fg prime is f"},{"Start":"04:42.240 ","End":"04:51.620","Text":"prime g plus fg prime but we\u0027ll also be needing the chain rule."},{"Start":"04:51.620 ","End":"04:57.305","Text":"Let me write it in a very specific form."},{"Start":"04:57.305 ","End":"04:59.015","Text":"We have something squared."},{"Start":"04:59.015 ","End":"05:01.610","Text":"If we have something, say,"},{"Start":"05:01.610 ","End":"05:06.840","Text":"box squared and we want to take its derivative,"},{"Start":"05:06.840 ","End":"05:09.465","Text":"the squared function is just 1 of many,"},{"Start":"05:09.465 ","End":"05:11.310","Text":"could have been any function."},{"Start":"05:11.310 ","End":"05:17.720","Text":"The squared is the outer function and the box itself is the inner function."},{"Start":"05:17.720 ","End":"05:20.165","Text":"What we do is we, first of all,"},{"Start":"05:20.165 ","End":"05:23.540","Text":"differentiate the outer function."},{"Start":"05:23.540 ","End":"05:26.360","Text":"Just like I said, if it was x squared,"},{"Start":"05:26.360 ","End":"05:27.770","Text":"you would put 2x."},{"Start":"05:27.770 ","End":"05:31.265","Text":"Box squared you\u0027d put twice box,"},{"Start":"05:31.265 ","End":"05:36.460","Text":"but then you also multiply by the derivative of the inner function, which is the box."},{"Start":"05:36.460 ","End":"05:40.065","Text":"You have to multiply it by box prime."},{"Start":"05:40.065 ","End":"05:43.830","Text":"Here my box is going to be x minus 1."},{"Start":"05:43.830 ","End":"05:54.090","Text":"This whole thing, box squared is the f. My f is box squared or x minus 1 squared,"},{"Start":"05:54.090 ","End":"05:58.260","Text":"and the g is just the x minus 2."},{"Start":"05:58.260 ","End":"06:03.120","Text":"Yeah, x minus 1 squared and g"},{"Start":"06:03.120 ","End":"06:08.100","Text":"becomes here the x minus 2 and then I have my f and g for this."},{"Start":"06:08.100 ","End":"06:14.865","Text":"Here it goes, y prime equals the derivative of the first f prime,"},{"Start":"06:14.865 ","End":"06:16.785","Text":"it\u0027s this thing here."},{"Start":"06:16.785 ","End":"06:18.920","Text":"It\u0027s twice, there\u0027s the box,"},{"Start":"06:18.920 ","End":"06:22.040","Text":"twice the box times the derivative of the box."},{"Start":"06:22.040 ","End":"06:24.395","Text":"The derivative of x minus 1 is 1,"},{"Start":"06:24.395 ","End":"06:27.440","Text":"and then that\u0027s the f prime then times g,"},{"Start":"06:27.440 ","End":"06:29.900","Text":"which is x minus 2,"},{"Start":"06:29.900 ","End":"06:32.660","Text":"plus the other way around,"},{"Start":"06:32.660 ","End":"06:38.039","Text":"which is this untouched x minus 1 squared."},{"Start":"06:38.039 ","End":"06:39.750","Text":"I\u0027m at this point here."},{"Start":"06:39.750 ","End":"06:41.590","Text":"Then finally, g prime,"},{"Start":"06:41.590 ","End":"06:45.315","Text":"that\u0027s x minus 2 prime, which is 1."},{"Start":"06:45.315 ","End":"06:47.520","Text":"Let\u0027s try simplifying this a bit."},{"Start":"06:47.520 ","End":"06:54.960","Text":"From here and here I have twice x squared minus 3x plus 2."},{"Start":"06:54.960 ","End":"06:57.360","Text":"This is, again, throw out the 1,"},{"Start":"06:57.360 ","End":"07:04.105","Text":"it\u0027s just x minus 1 squared is x squared minus 2x plus 1."},{"Start":"07:04.105 ","End":"07:08.685","Text":"This comes out, let\u0027s collect all the x squareds and so on together."},{"Start":"07:08.685 ","End":"07:12.570","Text":"2x squared plus x squared is 3x squared."},{"Start":"07:12.570 ","End":"07:19.020","Text":"Twice minus 3x is minus 6x minus 2x minus 8x."},{"Start":"07:19.020 ","End":"07:21.630","Text":"Here, 2 times 2 is 4,"},{"Start":"07:21.630 ","End":"07:25.410","Text":"and 4 plus 1 is 5."},{"Start":"07:25.410 ","End":"07:29.160","Text":"That\u0027s the answer to number 3."},{"Start":"07:29.160 ","End":"07:37.895","Text":"4 is y equals 4x plus 10 over x squared minus x."},{"Start":"07:37.895 ","End":"07:41.150","Text":"This time, it\u0027s not a product it\u0027s a quotient,"},{"Start":"07:41.150 ","End":"07:42.860","Text":"so we need the quotient rule,"},{"Start":"07:42.860 ","End":"07:46.380","Text":"and the quotient rule, just 1 of those things should memorize."},{"Start":"07:46.380 ","End":"07:50.795","Text":"The quotient rule says that if I take the derivative of a quotient,"},{"Start":"07:50.795 ","End":"07:53.150","Text":"then the derivative of the top times"},{"Start":"07:53.150 ","End":"07:56.765","Text":"the bottom minus the top times the derivative of the bottom,"},{"Start":"07:56.765 ","End":"08:00.630","Text":"and it\u0027s over the bottom squared."},{"Start":"08:00.710 ","End":"08:06.140","Text":"In this case, this is going to be our f. This is going to be the g."},{"Start":"08:06.140 ","End":"08:11.855","Text":"So y prime is going to equal the top derived,"},{"Start":"08:11.855 ","End":"08:17.135","Text":"I\u0027ll just write it as 4x plus 10 derived"},{"Start":"08:17.135 ","End":"08:26.080","Text":"times g x squared minus x minus,"},{"Start":"08:26.080 ","End":"08:28.170","Text":"the other way around,"},{"Start":"08:28.170 ","End":"08:35.715","Text":"which is the top just as is, 4x plus 10."},{"Start":"08:35.715 ","End":"08:40.655","Text":"Then the second 1 is going to be derived x squared minus x"},{"Start":"08:40.655 ","End":"08:47.150","Text":"derived and all this over x squared minus x,"},{"Start":"08:47.150 ","End":"08:49.965","Text":"the denominator all squared."},{"Start":"08:49.965 ","End":"08:54.885","Text":"4x plus 10 prime is just 4,"},{"Start":"08:54.885 ","End":"09:03.265","Text":"times x squared minus x minus 4x plus 10,"},{"Start":"09:03.265 ","End":"09:12.940","Text":"and then I\u0027ll put the 2x minus 1 all over x squared minus x all squared."},{"Start":"09:12.940 ","End":"09:15.390","Text":"This is equal to,"},{"Start":"09:15.390 ","End":"09:17.460","Text":"let\u0027s see now, the top,"},{"Start":"09:17.460 ","End":"09:23.680","Text":"4x squared minus 4x minus,"},{"Start":"09:24.050 ","End":"09:26.910","Text":"let\u0027s put this whole thing,"},{"Start":"09:26.910 ","End":"09:31.510","Text":"expand this, do it quickly."},{"Start":"09:32.210 ","End":"09:37.290","Text":"4x times 2x is 8x squared."},{"Start":"09:37.290 ","End":"09:46.035","Text":"Then x\u0027s I get from here 20x and from here minus 4x, so 16x."},{"Start":"09:46.035 ","End":"09:49.560","Text":"From the last 2, minus 20,"},{"Start":"09:49.560 ","End":"09:53.300","Text":"and then all over, same thing,"},{"Start":"09:53.300 ","End":"09:58.600","Text":"x squared minus x, all squared."},{"Start":"10:05.750 ","End":"10:09.495","Text":"This equals, let\u0027s see,"},{"Start":"10:09.495 ","End":"10:18.735","Text":"4x squared minus 8x squared is minus 4x squared."},{"Start":"10:18.735 ","End":"10:25.295","Text":"Then the x is minus 4x minus 16x minus 20x,"},{"Start":"10:25.295 ","End":"10:30.455","Text":"and then finally minus 20 is plus"},{"Start":"10:30.455 ","End":"10:39.770","Text":"20 all over x squared minus x, all squared."},{"Start":"10:39.770 ","End":"10:42.710","Text":"There\u0027s not really much to simplify here."},{"Start":"10:42.710 ","End":"10:43.955","Text":"I\u0027d leave it like that."},{"Start":"10:43.955 ","End":"10:47.280","Text":"Well, if you\u0027re very fancy, I suppose,"},{"Start":"10:47.280 ","End":"10:54.860","Text":"you could take 4 outside the brackets and I wouldn\u0027t bother."},{"Start":"10:54.860 ","End":"10:58.220","Text":"But just to show you that we could take it 1 more step,"},{"Start":"10:58.220 ","End":"11:03.440","Text":"you could even take minus 4 outside and then be left with x"},{"Start":"11:03.440 ","End":"11:11.190","Text":"squared plus 5x minus 5,"},{"Start":"11:11.190 ","End":"11:16.760","Text":"all over and the question is to leave it as something squared or to expand it."},{"Start":"11:16.760 ","End":"11:19.230","Text":"Well, I would leave it as it is."},{"Start":"11:20.180 ","End":"11:24.870","Text":"x squared minus x squared."},{"Start":"11:24.870 ","End":"11:29.690","Text":"Of course, you could always do something like factorize that and say it\u0027s x,"},{"Start":"11:29.690 ","End":"11:34.140","Text":"x minus 1 squared,"},{"Start":"11:34.140 ","End":"11:35.970","Text":"so it\u0027s x squared,"},{"Start":"11:35.970 ","End":"11:39.795","Text":"x minus 1 squared, and so on."},{"Start":"11:39.795 ","End":"11:41.995","Text":"It\u0027s limit to how much,"},{"Start":"11:41.995 ","End":"11:44.590","Text":"and no pun intended with the word limit,"},{"Start":"11:44.590 ","End":"11:47.435","Text":"there\u0027s how much you can go with these simplifications."},{"Start":"11:47.435 ","End":"11:56.340","Text":"Anyway, this is the answer to number 4. That\u0027s it."}],"ID":10443},{"Watched":false,"Name":"Exercise 2 - Parts 5-8","Duration":"10m 15s","ChapterTopicVideoID":10132,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"Number 5, y equals x squared plus"},{"Start":"00:04.680 ","End":"00:10.950","Text":"4x minus 1 over 2x minus 3."},{"Start":"00:10.950 ","End":"00:15.030","Text":"The first thing I\u0027m going to do is put the quotient formula."},{"Start":"00:15.030 ","End":"00:17.010","Text":"Doing that in here,"},{"Start":"00:17.010 ","End":"00:25.335","Text":"we get that y prime is equal to 2x plus 4."},{"Start":"00:25.335 ","End":"00:35.490","Text":"That\u0027s the f prime and denominator as is 2x minus 3 minus f,"},{"Start":"00:35.490 ","End":"00:40.830","Text":"which is x squared plus 4x minus 1."},{"Start":"00:40.830 ","End":"00:42.270","Text":"Do I have enough room here?"},{"Start":"00:42.270 ","End":"00:43.710","Text":"I think so."},{"Start":"00:43.710 ","End":"00:45.930","Text":"G prime that\u0027s easy,"},{"Start":"00:45.930 ","End":"00:51.045","Text":"that\u0027s just 2 and then all this is over g,"},{"Start":"00:51.045 ","End":"00:55.650","Text":"which is 2x minus 3 squared."},{"Start":"00:55.650 ","End":"00:58.295","Text":"Back to the coloring."},{"Start":"00:58.295 ","End":"01:00.850","Text":"I\u0027m not sure if the colors help or not."},{"Start":"01:00.850 ","End":"01:03.440","Text":"Now we just have to simplify because I mean,"},{"Start":"01:03.440 ","End":"01:04.475","Text":"this is the answer,"},{"Start":"01:04.475 ","End":"01:09.760","Text":"but it\u0027s messy and it\u0027s quite customary to simplify the solution."},{"Start":"01:09.760 ","End":"01:13.290","Text":"2x times 2x is 4x squared."},{"Start":"01:13.290 ","End":"01:14.790","Text":"Now let\u0027s get the x terms."},{"Start":"01:14.790 ","End":"01:17.875","Text":"We got 4 times 2 is 8,"},{"Start":"01:17.875 ","End":"01:21.470","Text":"and we also have minus 3 times 2,"},{"Start":"01:21.470 ","End":"01:24.720","Text":"so we have 8 minus 6 is 2."},{"Start":"01:24.720 ","End":"01:31.620","Text":"So plus 2x and then 4 times minus 3, is minus 12."},{"Start":"01:31.620 ","End":"01:37.940","Text":"All this is the first yellow blue bit here."},{"Start":"01:37.940 ","End":"01:42.575","Text":"Now we\u0027ve got to subtract this bit and I\u0027ll just put it in brackets also,"},{"Start":"01:42.575 ","End":"01:48.130","Text":"2x squared plus 8x minus 2."},{"Start":"01:48.130 ","End":"01:51.410","Text":"Here, no reason to expand it yet. We\u0027ll see."},{"Start":"01:51.410 ","End":"01:52.770","Text":"Let\u0027s keep the option."},{"Start":"01:52.770 ","End":"01:56.565","Text":"I\u0027ll just leave it meanwhile as 2x minus 3 squared."},{"Start":"01:56.565 ","End":"02:02.445","Text":"This equals, now we can combine 4x squared minus 2x squared is 2x squared."},{"Start":"02:02.445 ","End":"02:04.230","Text":"Let\u0027s gather the x terms,"},{"Start":"02:04.230 ","End":"02:10.455","Text":"2x minus 8x is minus 6x and"},{"Start":"02:10.455 ","End":"02:19.425","Text":"then minus 12 minus minus 2 is minus 10,"},{"Start":"02:19.425 ","End":"02:28.360","Text":"I believe, over 2x minus 3 squared."},{"Start":"02:28.360 ","End":"02:35.270","Text":"The only thing that really you might still want to do is take throughout the brackets."},{"Start":"02:35.270 ","End":"02:36.940","Text":"I wouldn\u0027t really bother,"},{"Start":"02:36.940 ","End":"02:42.005","Text":"I could stop here but for those who really like to go as far as they can, 2,"},{"Start":"02:42.005 ","End":"02:49.115","Text":"x squared minus 3x minus 5 over"},{"Start":"02:49.115 ","End":"02:57.575","Text":"2x minus 3 squared and we\u0027re done with number 5."},{"Start":"02:57.575 ","End":"03:05.215","Text":"Number 6 is y equals ex plus 1 over ex minus 1."},{"Start":"03:05.215 ","End":"03:09.900","Text":"Y prime is equal to, well,"},{"Start":"03:09.900 ","End":"03:11.860","Text":"just remember this is the f and this is the g"},{"Start":"03:11.860 ","End":"03:14.420","Text":"because this is a numerator and this is the denominator,"},{"Start":"03:14.420 ","End":"03:16.715","Text":"so we take f prime."},{"Start":"03:16.715 ","End":"03:22.130","Text":"Again, they throw away e. Every time they throw e some people panic."},{"Start":"03:22.130 ","End":"03:24.150","Text":"They think its make exponential something."},{"Start":"03:24.150 ","End":"03:25.880","Text":"E is just a number in this case,"},{"Start":"03:25.880 ","End":"03:27.860","Text":"it\u0027s a constant times x."},{"Start":"03:27.860 ","End":"03:29.870","Text":"The derivative of f is"},{"Start":"03:29.870 ","End":"03:36.020","Text":"just e. Just like as if it was 2x plus 3 the derivative would be 2."},{"Start":"03:36.020 ","End":"03:39.395","Text":"Now times g, which is the denominator,"},{"Start":"03:39.395 ","End":"03:44.520","Text":"which is ex minus 1 minus,"},{"Start":"03:44.520 ","End":"03:52.520","Text":"and then we want the numerator and differentiated just as is,"},{"Start":"03:52.520 ","End":"03:58.880","Text":"ex plus 1 times derivative of the denominator,"},{"Start":"03:58.880 ","End":"04:02.930","Text":"which is just e and then the denominator"},{"Start":"04:02.930 ","End":"04:09.900","Text":"squared ex minus 1 squared."},{"Start":"04:12.530 ","End":"04:14.570","Text":"Next. As I said,"},{"Start":"04:14.570 ","End":"04:15.755","Text":"this is the answer,"},{"Start":"04:15.755 ","End":"04:18.560","Text":"but we would like to simplify it."},{"Start":"04:18.560 ","End":"04:22.620","Text":"I like to start with the bottom."},{"Start":"04:22.620 ","End":"04:24.590","Text":"I always start the easy stuff first."},{"Start":"04:24.590 ","End":"04:26.090","Text":"ex minus 1 squared,"},{"Start":"04:26.090 ","End":"04:27.755","Text":"I\u0027m just going to copy it."},{"Start":"04:27.755 ","End":"04:29.840","Text":"Then I\u0027ll tell you what I\u0027m going to do."},{"Start":"04:29.840 ","End":"04:34.910","Text":"I\u0027m going to take e outside the brackets and in fact then I can take it altogether out of"},{"Start":"04:34.910 ","End":"04:41.120","Text":"the quotient or put it with a bracket with a numerator."},{"Start":"04:41.120 ","End":"04:52.170","Text":"If we go to that that e is like this e and this e gives us this e. What we have left is,"},{"Start":"04:54.070 ","End":"05:01.070","Text":"e times ex minus 1 minus ex"},{"Start":"05:01.070 ","End":"05:06.335","Text":"plus 1 times e is equal to e,"},{"Start":"05:06.335 ","End":"05:08.464","Text":"an I\u0027ll take out of the brackets,"},{"Start":"05:08.464 ","End":"05:19.280","Text":"ex minus 1 minus ex"},{"Start":"05:19.280 ","End":"05:21.930","Text":"plus 1."},{"Start":"05:23.300 ","End":"05:27.570","Text":"Now ex minus 1 less ex plus 1,"},{"Start":"05:27.570 ","End":"05:30.940","Text":"the ex minus the ex cancel,"},{"Start":"05:32.210 ","End":"05:42.915","Text":"and the minus 1 minus 1 just gives us minus 2,"},{"Start":"05:42.915 ","End":"05:45.015","Text":"well, minus 2e,"},{"Start":"05:45.015 ","End":"05:46.590","Text":"e times minus 2,"},{"Start":"05:46.590 ","End":"05:49.060","Text":"we get minus 2e."},{"Start":"05:50.630 ","End":"05:57.560","Text":"Actually would have been better if I just left the minus 2e on the numerator here,"},{"Start":"05:57.560 ","End":"06:02.330","Text":"with the use of the eraser,"},{"Start":"06:02.330 ","End":"06:05.075","Text":"I will just do it like this."},{"Start":"06:05.075 ","End":"06:11.825","Text":"See I can erase that and then just put it as minus 2e over here,"},{"Start":"06:11.825 ","End":"06:18.610","Text":"and that looks like the way I want to leave the answer."},{"Start":"06:19.130 ","End":"06:21.375","Text":"That\u0027s 6."},{"Start":"06:21.375 ","End":"06:23.025","Text":"Then after 6,"},{"Start":"06:23.025 ","End":"06:26.020","Text":"we\u0027re straight away going on to 7."},{"Start":"06:26.020 ","End":"06:28.130","Text":"But in our particular case,"},{"Start":"06:28.130 ","End":"06:36.930","Text":"we have y equals something cubed and that something is 4x plus 10."},{"Start":"06:36.930 ","End":"06:44.285","Text":"What we do to get the derivative is we treat this box as if it was just like an x,"},{"Start":"06:44.285 ","End":"06:45.440","Text":"like this was x cubed,"},{"Start":"06:45.440 ","End":"06:51.405","Text":"then we would say 3x squared and we would say 3 box squared."},{"Start":"06:51.405 ","End":"06:57.395","Text":"Then you take the derivative of the internal derivative,"},{"Start":"06:57.395 ","End":"07:01.415","Text":"which is just box prime."},{"Start":"07:01.415 ","End":"07:05.015","Text":"If I write that down here,"},{"Start":"07:05.015 ","End":"07:08.075","Text":"and remember that box is 4x plus 10,"},{"Start":"07:08.075 ","End":"07:11.090","Text":"what we get is that"},{"Start":"07:11.090 ","End":"07:18.545","Text":"y prime is equal to 3 times box squared,"},{"Start":"07:18.545 ","End":"07:19.925","Text":"so it\u0027s 3,"},{"Start":"07:19.925 ","End":"07:24.600","Text":"4x plus 10 squared."},{"Start":"07:24.600 ","End":"07:27.350","Text":"Here is where we treated it as if it was just something"},{"Start":"07:27.350 ","End":"07:30.785","Text":"cubed and something cubed when you differentiated."},{"Start":"07:30.785 ","End":"07:32.840","Text":"I don\u0027t know if I needed to mention that,"},{"Start":"07:32.840 ","End":"07:38.060","Text":"but the x cubed derivative is 3x squared."},{"Start":"07:38.060 ","End":"07:41.645","Text":"I\u0027m starting to assume already at this stage that you know these things"},{"Start":"07:41.645 ","End":"07:47.490","Text":"without detail explanation. We\u0027re up to here."},{"Start":"07:47.490 ","End":"07:48.710","Text":"We\u0027re up to this point."},{"Start":"07:48.710 ","End":"07:52.055","Text":"Then we have the box prime,"},{"Start":"07:52.055 ","End":"07:56.970","Text":"box is 4x plus 10,"},{"Start":"07:56.970 ","End":"08:00.010","Text":"so it\u0027s prime is just 4."},{"Start":"08:00.740 ","End":"08:02.940","Text":"We just leave the answers."},{"Start":"08:02.940 ","End":"08:05.505","Text":"We combine the 3 and the 4 and get 12."},{"Start":"08:05.505 ","End":"08:13.200","Text":"12 times 4x plus 10 squared or to the power of 2,"},{"Start":"08:13.200 ","End":"08:18.680","Text":"and I would leave this as the answer for Number 7."},{"Start":"08:18.680 ","End":"08:22.230","Text":"I\u0027m continuing onwards."},{"Start":"08:22.230 ","End":"08:26.105","Text":"After 7 we have 8."},{"Start":"08:26.105 ","End":"08:31.180","Text":"In some ways, very similar to the previous exercise,"},{"Start":"08:31.180 ","End":"08:38.815","Text":"where we have y is equal to some box to the fifth,"},{"Start":"08:38.815 ","End":"08:45.510","Text":"where the box is x squared plus 1 and what we get"},{"Start":"08:45.510 ","End":"08:52.300","Text":"is that y prime is the derivative of this,"},{"Start":"08:52.300 ","End":"09:00.210","Text":"which is 5 box to the fourth and then times the internal,"},{"Start":"09:00.210 ","End":"09:02.164","Text":"or derivative, the of the internal function,"},{"Start":"09:02.164 ","End":"09:07.755","Text":"that\u0027s the box prime."},{"Start":"09:07.755 ","End":"09:11.380","Text":"Forgive me, came about a bit crooked, but that\u0027s not."},{"Start":"09:12.320 ","End":"09:22.920","Text":"What we get here is that y prime is equal to 5 box,"},{"Start":"09:22.920 ","End":"09:31.720","Text":"which is x squared plus 1 to the fourth and then times the derivative."},{"Start":"09:32.810 ","End":"09:41.280","Text":"Hang on. That looks a little bit better."},{"Start":"09:41.280 ","End":"09:49.275","Text":"Box prime is the derivative of x squared plus 1 which is 2x."},{"Start":"09:49.275 ","End":"09:52.825","Text":"X squared plus 1 derivative,"},{"Start":"09:52.825 ","End":"09:57.980","Text":"1 goes to nothing here 2x to the 2 minus 1."},{"Start":"09:57.980 ","End":"10:00.125","Text":"But we know these things already."},{"Start":"10:00.125 ","End":"10:02.930","Text":"All I can do is slightly simplify it."},{"Start":"10:02.930 ","End":"10:09.260","Text":"I can combine the 5 with the 2x and get 10x and leave the rest of it as is x"},{"Start":"10:09.260 ","End":"10:16.770","Text":"squared plus 1 to the fourth and that\u0027s the answer to Number 8 here. Done with that."}],"ID":10444},{"Watched":false,"Name":"Exercise 2 - Parts 9-12","Duration":"17m 31s","ChapterTopicVideoID":10133,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"Exercise number 9. There are several ways I could think of doing this."},{"Start":"00:04.680 ","End":"00:07.620","Text":"When we have the square root of something to the power of 3,"},{"Start":"00:07.620 ","End":"00:10.065","Text":"then it will be the same thing."},{"Start":"00:10.065 ","End":"00:16.410","Text":"That\u0027s the x squared plus x plus 1 to the power of 3 to the power of a 1/2."},{"Start":"00:16.410 ","End":"00:19.410","Text":"It\u0027s basically to the power of 3 over 2."},{"Start":"00:19.410 ","End":"00:27.060","Text":"This is a classic example of the chain rule where this thing is in the box."},{"Start":"00:27.060 ","End":"00:35.310","Text":"In our case, if we call this like the square or box to the power of 3 over 2."},{"Start":"00:35.310 ","End":"00:37.405","Text":"If we differentiate that,"},{"Start":"00:37.405 ","End":"00:42.950","Text":"we get 3 over 2 times whatever it is to the power of this thing,"},{"Start":"00:42.950 ","End":"00:44.600","Text":"minus 1 is a half."},{"Start":"00:44.600 ","End":"00:46.790","Text":"But then, people forget,"},{"Start":"00:46.790 ","End":"00:51.860","Text":"but you mustn\u0027t forget to also take the internal derivative,"},{"Start":"00:51.860 ","End":"00:54.200","Text":"the derivative of the internal function,"},{"Start":"00:54.200 ","End":"00:58.070","Text":"which is this thing prime."},{"Start":"00:58.070 ","End":"01:09.675","Text":"This box is just x squared plus x plus 1 here."},{"Start":"01:09.675 ","End":"01:15.625","Text":"What we get is that y prime is equal to"},{"Start":"01:15.625 ","End":"01:20.620","Text":"3 over 2 times boxy x"},{"Start":"01:20.620 ","End":"01:27.590","Text":"squared plus x plus 1 to the power of 1/2."},{"Start":"01:27.590 ","End":"01:30.505","Text":"Then the internal derivative,"},{"Start":"01:30.505 ","End":"01:35.480","Text":"which is simply the derivative of the x squared plus x plus 1."},{"Start":"01:35.480 ","End":"01:39.145","Text":"This, we already can do these things in our head."},{"Start":"01:39.145 ","End":"01:41.455","Text":"X squared gives us 2x,"},{"Start":"01:41.455 ","End":"01:45.520","Text":"x gives us 1 and the 1,"},{"Start":"01:45.520 ","End":"01:48.585","Text":"and it gives us the 0, it\u0027s a constant."},{"Start":"01:48.585 ","End":"01:53.645","Text":"This is the answer. Let\u0027s get on to number 10."},{"Start":"01:53.645 ","End":"01:55.735","Text":"What we have here,"},{"Start":"01:55.735 ","End":"01:58.730","Text":"it looks like first of all, like a product."},{"Start":"01:58.730 ","End":"02:07.605","Text":"Then each of the parts in the product seems to have a function of a function."},{"Start":"02:07.605 ","End":"02:10.330","Text":"It looks like we\u0027re going to use the product rule,"},{"Start":"02:10.330 ","End":"02:15.675","Text":"and the chain rule twice for within each of the factors of the product."},{"Start":"02:15.675 ","End":"02:18.905","Text":"F will be this first part here,"},{"Start":"02:18.905 ","End":"02:21.095","Text":"and g will be the second part."},{"Start":"02:21.095 ","End":"02:30.515","Text":"So we\u0027ll get y prime is equal to 2x plus 1 to the 3."},{"Start":"02:30.515 ","End":"02:37.055","Text":"All this derived prime times the second 1 as is"},{"Start":"02:37.055 ","End":"02:45.790","Text":"4x minus 5 to the 4th plus the first factor as is,"},{"Start":"02:45.790 ","End":"02:49.485","Text":"that\u0027s 2x plus 1 cubed."},{"Start":"02:49.485 ","End":"02:52.260","Text":"The second 1 derived,"},{"Start":"02:52.260 ","End":"03:01.210","Text":"which is 4x minus 5 to the 4th derived or prime."},{"Start":"03:01.460 ","End":"03:04.280","Text":"We have to see, tell me what we\u0027ll do"},{"Start":"03:04.280 ","End":"03:08.630","Text":"the first bit and the last bit with the primes at the side here."},{"Start":"03:08.630 ","End":"03:13.380","Text":"2x plus 1 cubed."},{"Start":"03:13.380 ","End":"03:17.510","Text":"All this derived is equal to?"},{"Start":"03:17.510 ","End":"03:19.340","Text":"This is like a box cubed."},{"Start":"03:19.340 ","End":"03:23.540","Text":"We\u0027ve already d1 enough of these that I think we can just say something cubed."},{"Start":"03:23.540 ","End":"03:27.380","Text":"It\u0027s 3 times that something squared."},{"Start":"03:27.380 ","End":"03:34.470","Text":"But then the derivative of the inside if the 2x plus 1 is just times 2."},{"Start":"03:35.150 ","End":"03:39.345","Text":"This is, if you simplify it,"},{"Start":"03:39.345 ","End":"03:44.355","Text":"6 times 2x plus 1 squared."},{"Start":"03:44.355 ","End":"03:47.250","Text":"Now that\u0027s just this part here."},{"Start":"03:47.250 ","End":"03:50.980","Text":"The second part here,"},{"Start":"03:51.680 ","End":"03:55.090","Text":"I\u0027ll just write the answer of the derived,"},{"Start":"03:55.090 ","End":"04:03.140","Text":"will be 4 times the internal function for x"},{"Start":"04:03.140 ","End":"04:12.560","Text":"minus 5 cubed times the internal derivative, which is 4."},{"Start":"04:12.560 ","End":"04:17.330","Text":"That gives us 16,"},{"Start":"04:17.330 ","End":"04:21.815","Text":"4x minus 5 cubed."},{"Start":"04:21.815 ","End":"04:24.335","Text":"If we put all the bits from here into here,"},{"Start":"04:24.335 ","End":"04:26.045","Text":"we have the first bit."},{"Start":"04:26.045 ","End":"04:30.080","Text":"The 2x plus 1 cubed derived is here."},{"Start":"04:30.080 ","End":"04:39.795","Text":"That is 6 times 2x plus 1 squared times 4x minus 5 to the 4th,"},{"Start":"04:39.795 ","End":"04:45.240","Text":"plus here 2x plus 1 cubed as is."},{"Start":"04:45.240 ","End":"04:50.960","Text":"This bit here came out to 16 times 4x minus 5 cubed."},{"Start":"04:50.960 ","End":"04:54.710","Text":"I think we can do some simplifications here."},{"Start":"04:54.710 ","End":"04:58.250","Text":"Because if you look, there\u0027s 2x plus 1 here and here,"},{"Start":"04:58.250 ","End":"05:01.775","Text":"and 4x minus 5 here and here."},{"Start":"05:01.775 ","End":"05:04.185","Text":"Not quite the same numbers. Let\u0027s see."},{"Start":"05:04.185 ","End":"05:08.170","Text":"The 2x plus 1 I have here squared and here cube,"},{"Start":"05:08.170 ","End":"05:10.595","Text":"so I\u0027ll take out the squared."},{"Start":"05:10.595 ","End":"05:20.865","Text":"What I\u0027ll get is if I can take 2x plus 1 squared outside the brackets."},{"Start":"05:20.865 ","End":"05:22.470","Text":"Then looking at these 2,"},{"Start":"05:22.470 ","End":"05:27.630","Text":"I can also take 4x minus 5 cubed outside the brackets."},{"Start":"05:27.630 ","End":"05:31.120","Text":"In actual fact, I don\u0027t know if it\u0027s worth bothering with I"},{"Start":"05:31.120 ","End":"05:34.260","Text":"could also take it too. Let\u0027s do it."},{"Start":"05:34.260 ","End":"05:40.370","Text":"Also, take the 2 outside the brackets because it comes out of 6 and 16."},{"Start":"05:40.370 ","End":"05:45.650","Text":"What are we left with after we\u0027ve take these outside the brackets?"},{"Start":"05:46.010 ","End":"05:50.824","Text":"We\u0027ve taken 2 out of here, so we\u0027re left with 3."},{"Start":"05:50.824 ","End":"05:54.940","Text":"We\u0027ve taken the whole 2x plus 1 squared out of here,"},{"Start":"05:54.940 ","End":"05:56.865","Text":"so that leaves nothing."},{"Start":"05:56.865 ","End":"06:00.075","Text":"From here I\u0027ve taken 4x minus 5 cubed,"},{"Start":"06:00.075 ","End":"06:01.320","Text":"but I had to the 4th,"},{"Start":"06:01.320 ","End":"06:04.920","Text":"so I still have a missing 4x minus 5."},{"Start":"06:04.920 ","End":"06:11.265","Text":"Similarly, what I have left over here is once I\u0027ve taken the 2 out,"},{"Start":"06:11.265 ","End":"06:14.220","Text":"I\u0027m left with 8 which I\u0027ll put in the beginning."},{"Start":"06:14.220 ","End":"06:19.770","Text":"The 4x minus 5 cubed goes out of here and here."},{"Start":"06:19.770 ","End":"06:24.620","Text":"All we\u0027re left with is here we had 2x plus 1 squared,"},{"Start":"06:24.620 ","End":"06:28.190","Text":"and here cubed, so we\u0027re left with 2x plus 1."},{"Start":"06:28.190 ","End":"06:33.520","Text":"This and this, and this day we just have to see what\u0027s left with the last bracket."},{"Start":"06:33.520 ","End":"06:37.500","Text":"I\u0027ll just copy these, and I\u0027ll put the 2 in the front 2,"},{"Start":"06:37.500 ","End":"06:41.460","Text":"2x plus 1 squared,"},{"Start":"06:41.460 ","End":"06:48.740","Text":"4x minus 5 cubed times the bit in the square brackets."},{"Start":"06:48.740 ","End":"06:50.165","Text":"We can do it in our heads."},{"Start":"06:50.165 ","End":"07:00.360","Text":"3 times 4x is 12x plus 16x 12 plus 16 is 28x."},{"Start":"07:02.180 ","End":"07:08.615","Text":"For the numbers, we have minus 15 plus 8,"},{"Start":"07:08.615 ","End":"07:12.390","Text":"and that gives us minus 7."},{"Start":"07:14.240 ","End":"07:16.490","Text":"Thought it was going to end here,"},{"Start":"07:16.490 ","End":"07:19.145","Text":"but now I see that 7 goes in."},{"Start":"07:19.145 ","End":"07:21.620","Text":"Let\u0027s take the 7 out as well."},{"Start":"07:21.620 ","End":"07:23.000","Text":"Take the 7 out of here,"},{"Start":"07:23.000 ","End":"07:24.620","Text":"put it in front with the 2."},{"Start":"07:24.620 ","End":"07:26.605","Text":"You got 14,"},{"Start":"07:26.605 ","End":"07:31.635","Text":"2x plus 1 squared,"},{"Start":"07:31.635 ","End":"07:35.820","Text":"4x minus 5 cubed."},{"Start":"07:35.820 ","End":"07:43.589","Text":"After I\u0027ve taken the 7 out here,"},{"Start":"07:43.589 ","End":"07:47.500","Text":"I\u0027ve got 2x minus 1."},{"Start":"07:48.960 ","End":"07:54.740","Text":"Is that right? Yeah, that was right 2x minus 1."},{"Start":"07:55.320 ","End":"07:59.410","Text":"That\u0027s my proposed solution to Number 10,"},{"Start":"07:59.410 ","End":"08:02.905","Text":"we\u0027ll go on to Number 11."},{"Start":"08:02.905 ","End":"08:09.775","Text":"Y prime is going to equal the derivative of f,"},{"Start":"08:09.775 ","End":"08:14.185","Text":"which is 2x plus 3."},{"Start":"08:14.185 ","End":"08:16.405","Text":"All this to the 4th."},{"Start":"08:16.405 ","End":"08:18.415","Text":"The derivative of all this,"},{"Start":"08:18.415 ","End":"08:21.205","Text":"that this derivative here."},{"Start":"08:21.205 ","End":"08:24.565","Text":"Next the G, which is the denominator,"},{"Start":"08:24.565 ","End":"08:28.690","Text":"which is just x minus 5,"},{"Start":"08:28.690 ","End":"08:33.580","Text":"to the power of 3 minus f,"},{"Start":"08:33.580 ","End":"08:34.659","Text":"which is the numerator,"},{"Start":"08:34.659 ","End":"08:36.190","Text":"just that it is."},{"Start":"08:36.190 ","End":"08:46.045","Text":"2x plus 3 to the 4th and now the denominator prime,"},{"Start":"08:46.045 ","End":"08:54.470","Text":"which is x minus 5 cubed."},{"Start":"08:59.730 ","End":"09:05.290","Text":"All this derived and now everything"},{"Start":"09:05.290 ","End":"09:11.680","Text":"over the denominator squared and the denominator is something to the power of 3."},{"Start":"09:11.680 ","End":"09:19.120","Text":"If I square it, I just get it to the power of 3 times 2, which is 6."},{"Start":"09:19.120 ","End":"09:26.605","Text":"What I\u0027m saying is that if I have something like"},{"Start":"09:26.605 ","End":"09:34.030","Text":"a cubed and it\u0027s squared by the laws of the exponent its a to the power of 3 times 2,"},{"Start":"09:34.030 ","End":"09:37.190","Text":"which is a to the power of 6."},{"Start":"09:38.910 ","End":"09:43.135","Text":"Here we have to do some work."},{"Start":"09:43.135 ","End":"09:47.395","Text":"Now in the square brackets is where we have the chain rule."},{"Start":"09:47.395 ","End":"09:49.555","Text":"The first 1,"},{"Start":"09:49.555 ","End":"09:52.465","Text":"I\u0027ll take in more detail."},{"Start":"09:52.465 ","End":"10:02.485","Text":"What we have really is something to the power of 4."},{"Start":"10:02.485 ","End":"10:11.965","Text":"All this derived is equal to 4 times whatever it is cubed,"},{"Start":"10:11.965 ","End":"10:14.860","Text":"because that\u0027s like x to the fourth 4x cubed."},{"Start":"10:14.860 ","End":"10:17.769","Text":"But since this is a function,"},{"Start":"10:17.769 ","End":"10:19.510","Text":"this is the inner function."},{"Start":"10:19.510 ","End":"10:25.155","Text":"We have to also take its derivative. Excuse me a second."},{"Start":"10:25.155 ","End":"10:30.960","Text":"Yeah, so square or box is 2x plus 3."},{"Start":"10:30.960 ","End":"10:32.895","Text":"Now since it\u0027s to the 4th,"},{"Start":"10:32.895 ","End":"10:39.085","Text":"the derivative is 4 times 8 cubed times its internal derivative,"},{"Start":"10:39.085 ","End":"10:44.125","Text":"which is equal to 4 times,"},{"Start":"10:44.125 ","End":"10:45.385","Text":"now the square thing,"},{"Start":"10:45.385 ","End":"10:47.725","Text":"this box is 2x plus 3."},{"Start":"10:47.725 ","End":"10:57.010","Text":"It\u0027s 4 times 2x plus 3 to the power of 3 times,"},{"Start":"10:57.010 ","End":"11:02.590","Text":"the internal derivative is the derivative of 2x plus 3,"},{"Start":"11:02.590 ","End":"11:04.525","Text":"which is just 2."},{"Start":"11:04.525 ","End":"11:09.550","Text":"Over here, we have 4 times 2 is 8."},{"Start":"11:09.550 ","End":"11:12.790","Text":"8 times 2x plus"},{"Start":"11:12.790 ","End":"11:30.940","Text":"3 times the x minus 5 cubed."},{"Start":"11:30.940 ","End":"11:32.275","Text":"That\u0027s what I forgot."},{"Start":"11:32.275 ","End":"11:35.920","Text":"Then minus the 2x plus"},{"Start":"11:35.920 ","End":"11:43.165","Text":"3 to the 4th and the derivative of this thing,"},{"Start":"11:43.165 ","End":"11:47.410","Text":"well, similar to this except that here we had a 3."},{"Start":"11:47.410 ","End":"11:53.380","Text":"We have 3 times box to the power of 2."},{"Start":"11:53.380 ","End":"11:56.455","Text":"The internal derivative of this,"},{"Start":"11:56.455 ","End":"12:02.335","Text":"which is x minus 5 its derivative is just 1."},{"Start":"12:02.335 ","End":"12:12.650","Text":"But still everything is going to be over x minus 5 to the 6. Lot of work here."},{"Start":"12:15.780 ","End":"12:19.025","Text":"We\u0027ll continue."},{"Start":"12:19.025 ","End":"12:24.975","Text":"What I get is I want to see what I can take out of both of these terms in the numerator."},{"Start":"12:24.975 ","End":"12:27.320","Text":"I can take out,"},{"Start":"12:27.320 ","End":"12:31.330","Text":"well, from 8 to 3 I don\u0027t have anything in common."},{"Start":"12:31.330 ","End":"12:33.745","Text":"But from this and this,"},{"Start":"12:33.745 ","End":"12:36.980","Text":"I have 2x plus 3 cubed."},{"Start":"12:37.430 ","End":"12:43.894","Text":"I have 2x plus 3 cubed."},{"Start":"12:43.894 ","End":"12:47.230","Text":"From the second term I have x minus 5,"},{"Start":"12:47.230 ","End":"12:49.465","Text":"here there is 3, here there is 2."},{"Start":"12:49.465 ","End":"12:54.665","Text":"The highest I can take out is the smaller 1 is x minus 5 squared."},{"Start":"12:54.665 ","End":"13:00.525","Text":"X minus 5 squared and after I\u0027ve taken all this out,"},{"Start":"13:00.525 ","End":"13:04.420","Text":"here I have left 8."},{"Start":"13:04.420 ","End":"13:12.060","Text":"I have an extra x minus 5."},{"Start":"13:16.030 ","End":"13:24.730","Text":"Here I took out this cubed and this squared."},{"Start":"13:24.730 ","End":"13:30.800","Text":"This x minus 5 squared has come out altogether and all I\u0027m left with a 2x plus 3."},{"Start":"13:30.930 ","End":"13:33.520","Text":"Well, there\u0027s a 3 also."},{"Start":"13:33.520 ","End":"13:42.550","Text":"I\u0027ll put it first and then 2x plus 3, just algebra here."},{"Start":"13:42.550 ","End":"13:50.575","Text":"Over, still I\u0027m carrying on this x minus 5 to the 6th."},{"Start":"13:50.575 ","End":"13:52.585","Text":"Now let\u0027s see what we can do."},{"Start":"13:52.585 ","End":"13:53.680","Text":"Well, this is in order,"},{"Start":"13:53.680 ","End":"13:56.210","Text":"this is an order, this is in order."},{"Start":"13:56.490 ","End":"14:01.915","Text":"Then I know we can cancel. We\u0027ll get to that in a minute."},{"Start":"14:01.915 ","End":"14:04.525","Text":"Did you see the x minus 5 in both places."},{"Start":"14:04.525 ","End":"14:06.595","Text":"We\u0027ll get to that in a moment."},{"Start":"14:06.595 ","End":"14:11.440","Text":"Meanwhile, what I\u0027m trying to do is simplify what\u0027s in the square box."},{"Start":"14:11.440 ","End":"14:15.175","Text":"But you know what? At this point we could already cancel."},{"Start":"14:15.175 ","End":"14:18.130","Text":"Here we can take an x minus 5 squared,"},{"Start":"14:18.130 ","End":"14:24.380","Text":"and out of the x minus 5 to the 6th all we\u0027re left with instead of the 6 is a 4."},{"Start":"14:25.530 ","End":"14:34.539","Text":"We have 2x plus 3 all squared times."},{"Start":"14:34.539 ","End":"14:36.580","Text":"Now I\u0027m going to expand this thing."},{"Start":"14:36.580 ","End":"14:43.615","Text":"8x minus 6x is 2x minus 40,"},{"Start":"14:43.615 ","End":"14:47.410","Text":"minus 9 is minus"},{"Start":"14:47.410 ","End":"14:55.570","Text":"49 over x minus 5 to the 4th."},{"Start":"14:55.570 ","End":"14:57.745","Text":"If I haven\u0027t made a mistake,"},{"Start":"14:57.745 ","End":"15:02.259","Text":"then this is the answer to Number 11."},{"Start":"15:02.259 ","End":"15:06.950","Text":"After 11 comes Number 12."},{"Start":"15:07.740 ","End":"15:12.805","Text":"Now here, mostly it\u0027s going to be the chain rule,"},{"Start":"15:12.805 ","End":"15:16.195","Text":"cause some function of 4x plus 1."},{"Start":"15:16.195 ","End":"15:20.260","Text":"We take the 4x plus 1 and do 1 over the cube root of it."},{"Start":"15:20.260 ","End":"15:26.575","Text":"It\u0027s 4x plus 1 to the power of minus 1/3."},{"Start":"15:26.575 ","End":"15:29.305","Text":"That\u0027s just a rewriting y."},{"Start":"15:29.305 ","End":"15:32.095","Text":"Now I\u0027m going to do the derivative,"},{"Start":"15:32.095 ","End":"15:34.315","Text":"which is y prime,"},{"Start":"15:34.315 ","End":"15:39.205","Text":"which equals minus 4/3 times box,"},{"Start":"15:39.205 ","End":"15:46.855","Text":"which is 4 x plus 1 to the power of minus 4/3."},{"Start":"15:46.855 ","End":"15:50.950","Text":"That could be the answer except that since we were"},{"Start":"15:50.950 ","End":"15:56.705","Text":"using cube roots and not fractional powers,"},{"Start":"15:56.705 ","End":"16:00.170","Text":"I could notice let\u0027s say I call this thing,"},{"Start":"16:00.170 ","End":"16:02.790","Text":"it\u0027s still our a."},{"Start":"16:02.790 ","End":"16:13.285","Text":"That a to the power of minus 4/3 is 1 over a to the 4/3,"},{"Start":"16:13.285 ","End":"16:15.055","Text":"which is 1 over."},{"Start":"16:15.055 ","End":"16:17.950","Text":"Now 4/3 is 1 plus a 1/3."},{"Start":"16:17.950 ","End":"16:23.110","Text":"It\u0027s a times a to the 1/3,"},{"Start":"16:23.110 ","End":"16:32.680","Text":"which is 1 over a and a to the 1/3 is cube root of a. I\u0027ll just write that here where"},{"Start":"16:32.680 ","End":"16:37.990","Text":"4x plus 1 is my a. I have"},{"Start":"16:37.990 ","End":"16:46.135","Text":"the minus 4/3 that I had from before."},{"Start":"16:46.135 ","End":"16:47.720","Text":"They also have this thing,"},{"Start":"16:47.720 ","End":"16:54.230","Text":"so I\u0027ll just elongate the numerator and here we\u0027ll have the a,"},{"Start":"16:54.230 ","End":"17:04.090","Text":"which is 4x plus 1 times the cube root"},{"Start":"17:04.090 ","End":"17:05.965","Text":"of 4x plus 1,"},{"Start":"17:05.965 ","End":"17:12.190","Text":"cube root of 4x plus 1."},{"Start":"17:12.190 ","End":"17:17.220","Text":"In the numerator we have a minus 4."},{"Start":"17:18.040 ","End":"17:24.395","Text":"This looks to me like the answer to Number 12,"},{"Start":"17:24.395 ","End":"17:31.680","Text":"which means that we\u0027re done with the whole set of 12 and that\u0027s it."}],"ID":10445},{"Watched":false,"Name":"Exercise 3","Duration":"2m 6s","ChapterTopicVideoID":29499,"CourseChapterTopicPlaylistID":8710,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In this exercise, we have a quadratic polynomial,"},{"Start":"00:03.390 ","End":"00:06.870","Text":"p of x equals a plus bx plus cx squared."},{"Start":"00:06.870 ","End":"00:10.200","Text":"We have to find all values of the coefficients,"},{"Start":"00:10.200 ","End":"00:16.245","Text":"for which the function p of absolute value of x is differentiable at 0."},{"Start":"00:16.245 ","End":"00:19.020","Text":"Now p of absolute value of x,"},{"Start":"00:19.020 ","End":"00:20.970","Text":"it\u0027s very similar to p of x,"},{"Start":"00:20.970 ","End":"00:22.515","Text":"except in the middle,"},{"Start":"00:22.515 ","End":"00:25.590","Text":"in absolute value of x squared is the same as x squared,"},{"Start":"00:25.590 ","End":"00:27.750","Text":"so we just have this as being different."},{"Start":"00:27.750 ","End":"00:30.225","Text":"The rest of it is differentiable."},{"Start":"00:30.225 ","End":"00:35.460","Text":"There was the function x goes to a plus cx squared is differentiable."},{"Start":"00:35.460 ","End":"00:39.240","Text":"It all boils down to the middle term."},{"Start":"00:39.240 ","End":"00:46.579","Text":"The question is equivalent to when is b absolute value of x differentiable at 0?"},{"Start":"00:46.579 ","End":"00:53.640","Text":"The answer turns out to be if and only if b equals 0, as we\u0027ll show."},{"Start":"00:53.740 ","End":"00:57.170","Text":"The limit as h goes to 0,"},{"Start":"00:57.170 ","End":"01:01.160","Text":"of f of 0 plus h minus f of 0 over h,"},{"Start":"01:01.160 ","End":"01:05.495","Text":"I mean this is the definition of the derivative at 0."},{"Start":"01:05.495 ","End":"01:09.545","Text":"Now, if we plug it in from here, f of x,"},{"Start":"01:09.545 ","End":"01:13.400","Text":"we get the limit of b absolute value of h over"},{"Start":"01:13.400 ","End":"01:19.190","Text":"h. This has a right-hand limit and a left-hand limit,"},{"Start":"01:19.190 ","End":"01:22.189","Text":"the right-hand limit is b,"},{"Start":"01:22.189 ","End":"01:26.540","Text":"because the absolute value of h is equal to h on the right of 0,"},{"Start":"01:26.540 ","End":"01:29.210","Text":"but on the left absolute value of h is minus"},{"Start":"01:29.210 ","End":"01:34.494","Text":"h. We get the limit of minus b and the other side."},{"Start":"01:34.494 ","End":"01:36.735","Text":"Now, for the limit to exist,"},{"Start":"01:36.735 ","End":"01:42.460","Text":"both 1 sided limits have to be equal."},{"Start":"01:42.650 ","End":"01:45.830","Text":"It only exists if b is minus b,"},{"Start":"01:45.830 ","End":"01:48.335","Text":"which is the same as b equals 0."},{"Start":"01:48.335 ","End":"01:52.105","Text":"In this case the limit is 0."},{"Start":"01:52.105 ","End":"01:58.655","Text":"What we can say is that this is differentiable at 0, for b equals 0,"},{"Start":"01:58.655 ","End":"02:00.590","Text":"and for all values,"},{"Start":"02:00.590 ","End":"02:06.660","Text":"meaning arbitrary values of a and c. We\u0027re done."}],"ID":31114}],"Thumbnail":null,"ID":8710},{"Name":"Derivative of Exponents and Logarithmic Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Derivative of Logarithmic Functions","Duration":"2m 1s","ChapterTopicVideoID":10148,"CourseChapterTopicPlaylistID":8711,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.195","Text":"Rule deals with logarithmic functions."},{"Start":"00:03.195 ","End":"00:06.360","Text":"If we use base e,"},{"Start":"00:06.360 ","End":"00:10.515","Text":"in other words, base e is what we call the natural log."},{"Start":"00:10.515 ","End":"00:14.130","Text":"Natural log of x,"},{"Start":"00:14.130 ","End":"00:18.705","Text":"which is actually also the same thing as log to the base e of x,"},{"Start":"00:18.705 ","End":"00:26.445","Text":"then we get that y prime is equal to 1 over x."},{"Start":"00:26.445 ","End":"00:29.400","Text":"But in general, if we didn\u0027t use base e,"},{"Start":"00:29.400 ","End":"00:36.255","Text":"we used a general base log to the base a of x,"},{"Start":"00:36.255 ","End":"00:39.840","Text":"then y prime comes out a bit different."},{"Start":"00:39.840 ","End":"00:48.665","Text":"It comes out as 1 over x times the natural log of a."},{"Start":"00:48.665 ","End":"00:57.125","Text":"For example, if y equals logarithm base 10 of x,"},{"Start":"00:57.125 ","End":"01:01.835","Text":"then the derivative y prime is 1 over"},{"Start":"01:01.835 ","End":"01:07.400","Text":"x times the natural log of 10, and so on."},{"Start":"01:07.400 ","End":"01:12.830","Text":"If we had log to the base 2 of x,"},{"Start":"01:12.830 ","End":"01:21.540","Text":"we would get that y prime is 1 over x times natural log of 2."},{"Start":"01:23.000 ","End":"01:28.200","Text":"If we try instead of a to put e,"},{"Start":"01:28.200 ","End":"01:33.955","Text":"if y equals log to the base e of x,"},{"Start":"01:33.955 ","End":"01:42.740","Text":"we get y prime is 1 over x times natural log of e. But natural log of e is 1,"},{"Start":"01:42.740 ","End":"01:44.530","Text":"so it\u0027s 1 over x."},{"Start":"01:44.530 ","End":"01:49.490","Text":"That\u0027s no surprise because the log to the base e of x is the same as natural log of x,"},{"Start":"01:49.490 ","End":"01:51.545","Text":"so of course we get the same answer."},{"Start":"01:51.545 ","End":"01:55.435","Text":"Again, it shows why we prefer log to the base e."},{"Start":"01:55.435 ","End":"02:02.730","Text":"So many things in calculus become simpler if we use the number e."}],"ID":10453},{"Watched":false,"Name":"The Derivative of Exponential Functions","Duration":"1m 53s","ChapterTopicVideoID":13306,"CourseChapterTopicPlaylistID":8711,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Next is rule for exponential functions."},{"Start":"00:03.570 ","End":"00:08.820","Text":"If y is equal to e^x,"},{"Start":"00:08.820 ","End":"00:10.260","Text":"then it\u0027s derivative,"},{"Start":"00:10.260 ","End":"00:15.495","Text":"y prime, funnily enough turns out to be e^x also,"},{"Start":"00:15.495 ","End":"00:18.240","Text":"the function and the derivative are the same."},{"Start":"00:18.240 ","End":"00:27.720","Text":"However, if we take a more general base like y equals a^x,"},{"Start":"00:27.720 ","End":"00:36.525","Text":"then the derivative is a^x times natural log of a."},{"Start":"00:36.525 ","End":"00:38.910","Text":"The usual restrictions apply,"},{"Start":"00:38.910 ","End":"00:44.090","Text":"that is that a is bigger than 0 and a is not equal to 1."},{"Start":"00:44.090 ","End":"00:49.810","Text":"For example, if we have y equals 2^x,"},{"Start":"00:49.810 ","End":"00:55.715","Text":"2 is another common base for logarithms and exponents and so on,"},{"Start":"00:55.715 ","End":"01:04.025","Text":"then y prime will equal 2^x natural log of 2."},{"Start":"01:04.025 ","End":"01:09.430","Text":"Or if we try y equals 10^x,"},{"Start":"01:09.430 ","End":"01:12.065","Text":"10 is our decimal base,"},{"Start":"01:12.065 ","End":"01:21.140","Text":"then y prime is equal to 10^x natural log of 10."},{"Start":"01:21.140 ","End":"01:24.530","Text":"We\u0027ve tried a equals 2 and 8 equals 10."},{"Start":"01:24.530 ","End":"01:31.700","Text":"What happens if we try a is exactly equal to e. Then using this rule,"},{"Start":"01:31.700 ","End":"01:40.685","Text":"we would get that y prime equals e^x natural log of e. But natural log of e is 1,"},{"Start":"01:40.685 ","End":"01:42.950","Text":"so this just equals e^x,"},{"Start":"01:42.950 ","End":"01:45.320","Text":"which is the same as the first rule."},{"Start":"01:45.320 ","End":"01:53.070","Text":"That\u0027s one of the reasons in calculus we like using the exponential function with base e."}],"ID":13836},{"Watched":false,"Name":"Exercise 1 - Parts 1-4","Duration":"7m 7s","ChapterTopicVideoID":10136,"CourseChapterTopicPlaylistID":8711,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"This exercise is 12 in 1,"},{"Start":"00:02.790 ","End":"00:04.740","Text":"we have to find the first derivative."},{"Start":"00:04.740 ","End":"00:06.390","Text":"In other words, if y is the function,"},{"Start":"00:06.390 ","End":"00:09.329","Text":"then we want y prime."},{"Start":"00:09.329 ","End":"00:17.760","Text":"I\u0027ll do first 4 in this clip and do 4 to a clip. Start with number 1."},{"Start":"00:17.760 ","End":"00:20.670","Text":"Number 1, this is one of those immediate ones."},{"Start":"00:20.670 ","End":"00:22.530","Text":"If y equals e^x,"},{"Start":"00:22.530 ","End":"00:24.840","Text":"it\u0027s 1 of those functions that when you differentiate it,"},{"Start":"00:24.840 ","End":"00:26.775","Text":"it\u0027s just the function itself."},{"Start":"00:26.775 ","End":"00:28.350","Text":"It\u0027s from the table of integrals,"},{"Start":"00:28.350 ","End":"00:31.200","Text":"which is something we know and that\u0027s it."},{"Start":"00:31.200 ","End":"00:33.180","Text":"There\u0027s nothing more to be done."},{"Start":"00:33.180 ","End":"00:35.850","Text":"Let\u0027s get on to number 2 then."},{"Start":"00:35.850 ","End":"00:39.195","Text":"In number 2, we have,"},{"Start":"00:39.195 ","End":"00:47.695","Text":"I\u0027ll copy of the question y equals 4e^x plus 2_x cubed."},{"Start":"00:47.695 ","End":"00:50.510","Text":"Now, we\u0027re going to use some rules here."},{"Start":"00:50.510 ","End":"00:52.415","Text":"I\u0027ll write them at the side,"},{"Start":"00:52.415 ","End":"00:57.250","Text":"after a while you\u0027ll know them without thinking."},{"Start":"00:57.250 ","End":"01:01.370","Text":"These are the rules that if you have a sum and we have a sum here,"},{"Start":"01:01.370 ","End":"01:03.965","Text":"let\u0027s say we have a sum of 2 functions,"},{"Start":"01:03.965 ","End":"01:05.945","Text":"f of x plus g of x,"},{"Start":"01:05.945 ","End":"01:08.120","Text":"then the derivative is"},{"Start":"01:08.120 ","End":"01:11.785","Text":"just the derivative of the first plus the derivative of the second,"},{"Start":"01:11.785 ","End":"01:15.945","Text":"f prime of x plus g prime of x."},{"Start":"01:15.945 ","End":"01:17.750","Text":"Doesn\u0027t work for multiplication,"},{"Start":"01:17.750 ","End":"01:19.520","Text":"but it works for addition and subtraction."},{"Start":"01:19.520 ","End":"01:23.840","Text":"Maybe I\u0027ll even write plus or minus on here, plus or minus."},{"Start":"01:23.840 ","End":"01:29.035","Text":"The other rule is that we have sometimes a constant times a function."},{"Start":"01:29.035 ","End":"01:34.175","Text":"General case, if you have some constant times a function of x,"},{"Start":"01:34.175 ","End":"01:35.780","Text":"then when you differentiate it,"},{"Start":"01:35.780 ","End":"01:37.880","Text":"the constant just sticks."},{"Start":"01:37.880 ","End":"01:44.280","Text":"You just have to differentiate this bit and the constant just tags along."},{"Start":"01:44.290 ","End":"01:48.725","Text":"Now back here we have both of these rules combined."},{"Start":"01:48.725 ","End":"01:51.245","Text":"So what I would say is this,"},{"Start":"01:51.245 ","End":"01:56.930","Text":"the derivative of e^x would be like this is"},{"Start":"01:56.930 ","End":"02:02.375","Text":"my f may be in this rule here."},{"Start":"02:02.375 ","End":"02:07.095","Text":"The derivative of this is e^x because the 4 is a constant,"},{"Start":"02:07.095 ","End":"02:08.580","Text":"it just tags along."},{"Start":"02:08.580 ","End":"02:12.405","Text":"This is just 4e^x, same as here."},{"Start":"02:12.405 ","End":"02:15.360","Text":"Now, there is sum of 2 things, so I put a plus."},{"Start":"02:15.360 ","End":"02:17.510","Text":"Here I have 2_x cubed."},{"Start":"02:17.510 ","End":"02:23.650","Text":"The derivative of x cubed is 3_x squared,"},{"Start":"02:23.650 ","End":"02:25.535","Text":"but the 2 sticks,"},{"Start":"02:25.535 ","End":"02:28.205","Text":"so it\u0027s 2 times 3_x squared."},{"Start":"02:28.205 ","End":"02:30.110","Text":"Now usually we just do this in our heads."},{"Start":"02:30.110 ","End":"02:31.995","Text":"We say 3 times 2 is 6."},{"Start":"02:31.995 ","End":"02:38.310","Text":"So I would write straightaway 4e^x plus 6_x squared."},{"Start":"02:38.310 ","End":"02:45.950","Text":"Now the next, they\u0027re going to disappear from view,"},{"Start":"02:45.950 ","End":"02:48.850","Text":"so I\u0027ll just copy them."},{"Start":"02:48.850 ","End":"02:51.000","Text":"Here\u0027s number 3,"},{"Start":"02:51.000 ","End":"02:55.600","Text":"and I also copied number 4 down below."},{"Start":"02:55.600 ","End":"02:58.745","Text":"This time in number 3,"},{"Start":"02:58.745 ","End":"03:00.170","Text":"we have a product."},{"Start":"03:00.170 ","End":"03:04.280","Text":"Here we have, this is like a multiplication e^x times this."},{"Start":"03:04.280 ","End":"03:06.065","Text":"So I need another rule."},{"Start":"03:06.065 ","End":"03:09.320","Text":"It isn\u0027t just a simple like with plus or minus that you"},{"Start":"03:09.320 ","End":"03:14.250","Text":"can just take this separately and this separately and multiply, no,"},{"Start":"03:14.250 ","End":"03:19.040","Text":"there is product rule that if y equals one function of x say f,"},{"Start":"03:19.040 ","End":"03:21.650","Text":"times another function of x say g,"},{"Start":"03:21.650 ","End":"03:25.835","Text":"then the derivative, not so straightforward."},{"Start":"03:25.835 ","End":"03:27.830","Text":"Each time you differentiate one,"},{"Start":"03:27.830 ","End":"03:30.500","Text":"leave the other alone, and then add,"},{"Start":"03:30.500 ","End":"03:38.180","Text":"so it\u0027s f prime of x. Differentiate the first times the second as is, plus vice versa."},{"Start":"03:38.180 ","End":"03:43.550","Text":"The first one as is times the derivative of the second."},{"Start":"03:43.550 ","End":"03:45.695","Text":"Let\u0027s do that here."},{"Start":"03:45.695 ","End":"03:48.020","Text":"We get that y prime,"},{"Start":"03:48.020 ","End":"03:49.220","Text":"the first bit is f,"},{"Start":"03:49.220 ","End":"03:53.540","Text":"the second bit is g. I get f prime,"},{"Start":"03:53.540 ","End":"03:57.260","Text":"well, it\u0027s e^x so the derivative is just e^x as"},{"Start":"03:57.260 ","End":"04:02.029","Text":"is and the second x squared plus x plus 4."},{"Start":"04:02.029 ","End":"04:05.765","Text":"Then the other way around f as is,"},{"Start":"04:05.765 ","End":"04:07.820","Text":"I know with e^x looks the same,"},{"Start":"04:07.820 ","End":"04:10.055","Text":"the function and the derivative are the same."},{"Start":"04:10.055 ","End":"04:16.130","Text":"But this is the original as is and now I differentiate this and it\u0027s a polynomial."},{"Start":"04:16.130 ","End":"04:21.360","Text":"So x squared gives me 2_x,"},{"Start":"04:21.360 ","End":"04:27.485","Text":"x the derivative is just 1 and the constant is nothing."},{"Start":"04:27.485 ","End":"04:29.900","Text":"So this is what it is,"},{"Start":"04:29.900 ","End":"04:39.555","Text":"but I would like to simplify this and collect e^x and combine these 2 together."},{"Start":"04:39.555 ","End":"04:41.160","Text":"I\u0027ve got this plus this,"},{"Start":"04:41.160 ","End":"04:43.275","Text":"so I have x squared from here,"},{"Start":"04:43.275 ","End":"04:46.330","Text":"x plus 2_x is 3_x,"},{"Start":"04:46.330 ","End":"04:50.225","Text":"and then 4 and 1 is 5,"},{"Start":"04:50.225 ","End":"04:54.815","Text":"so this is the answer, simplified."},{"Start":"04:54.815 ","End":"04:57.845","Text":"That\u0027s 3 down 1 to go."},{"Start":"04:57.845 ","End":"04:59.390","Text":"The last one, well,"},{"Start":"04:59.390 ","End":"05:02.650","Text":"this time it\u0027s a division, a quotient."},{"Start":"05:02.650 ","End":"05:05.300","Text":"I\u0027ll remind you of the quotient rule."},{"Start":"05:05.300 ","End":"05:07.055","Text":"If I have one function,"},{"Start":"05:07.055 ","End":"05:09.230","Text":"f of x over another function,"},{"Start":"05:09.230 ","End":"05:14.225","Text":"call it g of x, then the derivative is equal to,"},{"Start":"05:14.225 ","End":"05:16.685","Text":"it\u0027s a bit more involved."},{"Start":"05:16.685 ","End":"05:18.725","Text":"I could just start with the denominator."},{"Start":"05:18.725 ","End":"05:22.550","Text":"On the denominator, I have the original denominator but squared,"},{"Start":"05:22.550 ","End":"05:26.690","Text":"g of x squared put the brackets possibly."},{"Start":"05:26.690 ","End":"05:31.460","Text":"Then the numerator is similar to the product rule,"},{"Start":"05:31.460 ","End":"05:32.735","Text":"but there\u0027s a minus."},{"Start":"05:32.735 ","End":"05:37.480","Text":"So it\u0027s the derivative of the first times the second"},{"Start":"05:37.480 ","End":"05:44.185","Text":"minus the first times the derivative of the second."},{"Start":"05:44.185 ","End":"05:47.175","Text":"Let\u0027s see how that applies here."},{"Start":"05:47.175 ","End":"05:50.080","Text":"I\u0027ll need to scroll a bit."},{"Start":"05:50.330 ","End":"05:54.500","Text":"We get that y prime is equal to,"},{"Start":"05:54.500 ","End":"05:57.425","Text":"I just like to start with the denominator."},{"Start":"05:57.425 ","End":"06:02.840","Text":"It\u0027s x squared minus x squared and then this rule,"},{"Start":"06:02.840 ","End":"06:06.960","Text":"derivative of the numerator is just e^x."},{"Start":"06:06.960 ","End":"06:08.465","Text":"It\u0027s one of the basics."},{"Start":"06:08.465 ","End":"06:11.510","Text":"Times the denominator as is,"},{"Start":"06:11.510 ","End":"06:16.890","Text":"x squared minus x minus the numerator as"},{"Start":"06:16.890 ","End":"06:24.560","Text":"is and the derivative of the denominator is 2_x minus 1."},{"Start":"06:24.560 ","End":"06:26.570","Text":"I\u0027ll simplify a little bit."},{"Start":"06:26.570 ","End":"06:31.610","Text":"I\u0027ll take e^x as a common factor for the numerator,"},{"Start":"06:31.610 ","End":"06:34.580","Text":"but I\u0027ll leave the denominator as is."},{"Start":"06:34.580 ","End":"06:39.740","Text":"So what I have is the x squared minus x squared."},{"Start":"06:39.740 ","End":"06:41.690","Text":"If I expand it, I don\u0027t think it\u0027ll be simpler,"},{"Start":"06:41.690 ","End":"06:45.950","Text":"it\u0027s better this way and then I\u0027ll have e^x and combine these."},{"Start":"06:45.950 ","End":"06:52.590","Text":"So I have x squared minus x minus 2_x,"},{"Start":"06:52.590 ","End":"06:58.220","Text":"so that\u0027s minus 3_x and then minus, minus is plus."},{"Start":"06:58.220 ","End":"06:59.990","Text":"So if I take e^x out,"},{"Start":"06:59.990 ","End":"07:01.280","Text":"this is what I\u0027m left with,"},{"Start":"07:01.280 ","End":"07:07.410","Text":"and that\u0027s our answer up to number 4 continued in the next clip."}],"ID":10454},{"Watched":false,"Name":"Exercise 1 - Parts 5-8","Duration":"9m 42s","ChapterTopicVideoID":10137,"CourseChapterTopicPlaylistID":8711,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:04.425","Text":"Continuing, we\u0027ve just done 1 through 4,"},{"Start":"00:04.425 ","End":"00:09.375","Text":"now we\u0027ll do 5 through 8 and finding the first derivative."},{"Start":"00:09.375 ","End":"00:12.480","Text":"But I\u0027ve kept the formulas that I wrote previously,"},{"Start":"00:12.480 ","End":"00:15.375","Text":"they will come in handy."},{"Start":"00:15.375 ","End":"00:18.510","Text":"Number 5 is the next,"},{"Start":"00:18.510 ","End":"00:25.435","Text":"y equals e^4x minus 1 plus e^2x."},{"Start":"00:25.435 ","End":"00:28.460","Text":"This time we\u0027re going to need another rule."},{"Start":"00:28.460 ","End":"00:31.760","Text":"The rule that we\u0027re missing here is something which is called"},{"Start":"00:31.760 ","End":"00:36.650","Text":"the chain rule for reasons I won\u0027t go into."},{"Start":"00:36.650 ","End":"00:41.265","Text":"When I have y as a function of a function of x,"},{"Start":"00:41.265 ","End":"00:45.045","Text":"let\u0027s say f of g of x,"},{"Start":"00:45.045 ","End":"00:49.610","Text":"g will be the internal function and f is the external function."},{"Start":"00:49.610 ","End":"00:53.870","Text":"The derivative is the derivative of"},{"Start":"00:53.870 ","End":"00:58.520","Text":"the external function first and the internal we leave as is,"},{"Start":"00:58.520 ","End":"01:01.730","Text":"then we multiply by the derivative of the internal function."},{"Start":"01:01.730 ","End":"01:03.380","Text":"Don\u0027t worry if you don\u0027t understand this."},{"Start":"01:03.380 ","End":"01:04.880","Text":"At this level, it\u0027s abstract."},{"Start":"01:04.880 ","End":"01:07.640","Text":"In the examples, it\u0027ll be much clearer."},{"Start":"01:07.640 ","End":"01:10.750","Text":"This should have been covered in the tutorial anyway."},{"Start":"01:10.750 ","End":"01:15.215","Text":"Now here, let\u0027s look at the first term."},{"Start":"01:15.215 ","End":"01:20.050","Text":"Outwardly, we have a sum and we\u0027re going to use the first rule,"},{"Start":"01:20.050 ","End":"01:21.970","Text":"derivative of this plus derivative of this."},{"Start":"01:21.970 ","End":"01:25.630","Text":"But just looking at the e^4x minus 1,"},{"Start":"01:25.630 ","End":"01:27.730","Text":"I\u0027d like to just circle this."},{"Start":"01:27.730 ","End":"01:29.799","Text":"This is my internal function,"},{"Start":"01:29.799 ","End":"01:37.850","Text":"g. The external function is e^x or e to the power of whatever."},{"Start":"01:37.920 ","End":"01:40.300","Text":"Now when we take the derivative,"},{"Start":"01:40.300 ","End":"01:44.170","Text":"we first take the derivative of the outer function, f prime."},{"Start":"01:44.170 ","End":"01:48.110","Text":"But the derivative of e to the power of is just e to the power of."},{"Start":"01:48.110 ","End":"01:55.085","Text":"This part is the same because the special function e, the exponent."},{"Start":"01:55.085 ","End":"01:58.520","Text":"But now we have to do the g prime part,"},{"Start":"01:58.520 ","End":"02:01.489","Text":"that\u0027s called the inner derivative, the internal derivative."},{"Start":"02:01.489 ","End":"02:05.555","Text":"I\u0027ll take a derivative of 4x minus 1,"},{"Start":"02:05.555 ","End":"02:08.035","Text":"which is just simply 4."},{"Start":"02:08.035 ","End":"02:09.800","Text":"I have to multiply by the 4."},{"Start":"02:09.800 ","End":"02:12.800","Text":"This is the part that sometimes people forget,"},{"Start":"02:12.800 ","End":"02:14.960","Text":"and that\u0027s the inner derivative."},{"Start":"02:14.960 ","End":"02:16.535","Text":"Now in the second 1,"},{"Start":"02:16.535 ","End":"02:18.650","Text":"also a similar situation,"},{"Start":"02:18.650 ","End":"02:22.614","Text":"2x would be the inner function."},{"Start":"02:22.614 ","End":"02:28.190","Text":"This time I get the external derivative is just e to the power of the same thing."},{"Start":"02:28.190 ","End":"02:29.630","Text":"Then the inner derivative,"},{"Start":"02:29.630 ","End":"02:32.810","Text":"derivative 2x is just 2."},{"Start":"02:32.870 ","End":"02:35.010","Text":"We can simplify this a bit."},{"Start":"02:35.010 ","End":"02:36.515","Text":"Maybe put the constants in front,"},{"Start":"02:36.515 ","End":"02:39.020","Text":"but I\u0027m not going to continue with that."},{"Start":"02:39.020 ","End":"02:40.760","Text":"Let\u0027s go on to the next 1,"},{"Start":"02:40.760 ","End":"02:47.075","Text":"which is y equals e to the power of minus x times x plus 1."},{"Start":"02:47.075 ","End":"02:49.775","Text":"I\u0027m going to need a couple of rules here."},{"Start":"02:49.775 ","End":"02:53.330","Text":"I\u0027m going to need the product rule because I have something times something,"},{"Start":"02:53.330 ","End":"02:55.040","Text":"so I\u0027m going to use the product rule."},{"Start":"02:55.040 ","End":"02:57.095","Text":"But when I differentiate this,"},{"Start":"02:57.095 ","End":"03:00.275","Text":"just like in number 5, I\u0027m going to need the chain rule."},{"Start":"03:00.275 ","End":"03:05.135","Text":"Here goes. First of all, the product rule."},{"Start":"03:05.135 ","End":"03:08.300","Text":"I see something times something."},{"Start":"03:08.300 ","End":"03:14.215","Text":"I have to differentiate the first bit and leave the second 1 as is."},{"Start":"03:14.215 ","End":"03:16.430","Text":"The derivative of the first bit,"},{"Start":"03:16.430 ","End":"03:17.660","Text":"just like in number 5,"},{"Start":"03:17.660 ","End":"03:20.450","Text":"we start off with the external derivative e to"},{"Start":"03:20.450 ","End":"03:24.635","Text":"the minus x and because it\u0027s minus x and not x,"},{"Start":"03:24.635 ","End":"03:26.750","Text":"the inner derivative is minus 1."},{"Start":"03:26.750 ","End":"03:28.850","Text":"I have to multiply by minus 1."},{"Start":"03:28.850 ","End":"03:31.520","Text":"That\u0027s the derivative of the first part of the product."},{"Start":"03:31.520 ","End":"03:34.315","Text":"Now back to the product rule,"},{"Start":"03:34.315 ","End":"03:37.830","Text":"I need the second bit as is."},{"Start":"03:37.830 ","End":"03:40.695","Text":"Now, I take the first bit as is,"},{"Start":"03:40.695 ","End":"03:42.100","Text":"e to the minus x,"},{"Start":"03:42.100 ","End":"03:44.285","Text":"and differentiate the second bit,"},{"Start":"03:44.285 ","End":"03:46.625","Text":"which is the derivative of x plus 1,"},{"Start":"03:46.625 ","End":"03:48.725","Text":"which is just 1."},{"Start":"03:48.725 ","End":"03:53.195","Text":"In this case, I will simplify."},{"Start":"03:53.195 ","End":"03:59.855","Text":"What I\u0027ll do is I\u0027ll take e to the minus x outside the brackets,"},{"Start":"03:59.855 ","End":"04:01.835","Text":"and let\u0027s see what we\u0027re left with."},{"Start":"04:01.835 ","End":"04:07.740","Text":"Here I have minus x and"},{"Start":"04:07.740 ","End":"04:13.860","Text":"then I have plus 1."},{"Start":"04:13.860 ","End":"04:20.570","Text":"I\u0027ll write it minus 1 plus 1 but these 2 cancel out,"},{"Start":"04:20.570 ","End":"04:24.215","Text":"so it\u0027s just minus x e to the minus x."},{"Start":"04:24.215 ","End":"04:28.340","Text":"Maybe I\u0027ll write it, minus x e to the minus x,"},{"Start":"04:28.340 ","End":"04:31.550","Text":"and that\u0027s the answer to number 6."},{"Start":"04:31.550 ","End":"04:34.740","Text":"Now let\u0027s get onto the next 1."},{"Start":"04:35.650 ","End":"04:39.340","Text":"I better copy these last 2."},{"Start":"04:39.340 ","End":"04:43.700","Text":"I copied 7 and 8 then Now we can scroll down and maybe I\u0027ll just"},{"Start":"04:43.700 ","End":"04:48.515","Text":"keep the rules up so we can see them."},{"Start":"04:48.515 ","End":"04:53.255","Text":"Number 7, I have a quotient outwardly,"},{"Start":"04:53.255 ","End":"04:58.145","Text":"and I also have a couple of uses for the chain rule here and here."},{"Start":"04:58.145 ","End":"05:01.970","Text":"But the first rule I see is a quotient."},{"Start":"05:01.970 ","End":"05:04.220","Text":"I have y prime equals."},{"Start":"05:04.220 ","End":"05:05.390","Text":"Now for a quotient,"},{"Start":"05:05.390 ","End":"05:10.730","Text":"I need this rule and I like to start with the denominator, it\u0027s more straightforward."},{"Start":"05:10.730 ","End":"05:13.910","Text":"I take the denominator squared."},{"Start":"05:13.910 ","End":"05:19.010","Text":"It\u0027s e^4 plus x squared."},{"Start":"05:19.010 ","End":"05:23.935","Text":"Then I want, the pattern is the derivative of the numerator times denominator."},{"Start":"05:23.935 ","End":"05:27.230","Text":"Let\u0027s start with the derivative of the numerator."},{"Start":"05:27.360 ","End":"05:33.265","Text":"Within this, I need a chain rule and I need a plus minus."},{"Start":"05:33.265 ","End":"05:36.010","Text":"This is second nature of the minus,"},{"Start":"05:36.010 ","End":"05:37.390","Text":"we don\u0027t even mention it."},{"Start":"05:37.390 ","End":"05:39.910","Text":"I\u0027m going to get something minus something."},{"Start":"05:39.910 ","End":"05:41.680","Text":"The first bit with the chain rule,"},{"Start":"05:41.680 ","End":"05:45.670","Text":"I start with e^2x because it\u0027s the exponential function."},{"Start":"05:45.670 ","End":"05:49.490","Text":"Then the inner derivative of 2x is 2."},{"Start":"05:49.490 ","End":"05:52.610","Text":"Then minus 1 for the second bit."},{"Start":"05:52.610 ","End":"05:56.660","Text":"All this is the derivative of the numerator, the f prime,"},{"Start":"05:56.660 ","End":"05:58.190","Text":"and now I need the g,"},{"Start":"05:58.190 ","End":"06:01.595","Text":"which is e^4 plus x."},{"Start":"06:01.595 ","End":"06:04.055","Text":"Then I come to this minus,"},{"Start":"06:04.055 ","End":"06:09.210","Text":"and I want the numerator as is."},{"Start":"06:09.210 ","End":"06:17.020","Text":"The numerator is e^2x minus x."},{"Start":"06:17.020 ","End":"06:20.930","Text":"Now all I need is the derivative of the denominator."},{"Start":"06:20.930 ","End":"06:22.655","Text":"Again, using the chain rule,"},{"Start":"06:22.655 ","End":"06:27.570","Text":"the external function is e^4 plus x."},{"Start":"06:27.570 ","End":"06:30.739","Text":"Now I need to multiply by the inner derivative,"},{"Start":"06:30.739 ","End":"06:34.235","Text":"but the derivative of 4 plus x is just 1."},{"Start":"06:34.235 ","End":"06:35.690","Text":"I write the 1 here,"},{"Start":"06:35.690 ","End":"06:38.310","Text":"although it doesn\u0027t change anything."},{"Start":"06:38.450 ","End":"06:41.925","Text":"I think we should do a bit of simplification."},{"Start":"06:41.925 ","End":"06:46.390","Text":"1 thing, notice that e^4x appears everywhere."},{"Start":"06:46.390 ","End":"06:49.550","Text":"I could like, take it out the brackets in the numerator,"},{"Start":"06:49.550 ","End":"06:54.380","Text":"but then it would cancel with an e^4 plus x in the denominator.I put a line through this,"},{"Start":"06:54.380 ","End":"06:55.975","Text":"a line through this."},{"Start":"06:55.975 ","End":"06:57.725","Text":"Since this is squared,"},{"Start":"06:57.725 ","End":"07:01.800","Text":"I now don\u0027t have it squared just as is."},{"Start":"07:02.050 ","End":"07:04.970","Text":"On the denominator now,"},{"Start":"07:04.970 ","End":"07:09.834","Text":"I just have e^4 plus x."},{"Start":"07:09.834 ","End":"07:13.680","Text":"On the numerator, let\u0027s see what\u0027s left. This is nothing."},{"Start":"07:13.680 ","End":"07:14.850","Text":"I have, here,"},{"Start":"07:14.850 ","End":"07:17.640","Text":"if I put the 2 in front twice e^2x,"},{"Start":"07:17.640 ","End":"07:22.320","Text":"but I also have minus e^2x."},{"Start":"07:22.320 ","End":"07:25.620","Text":"Like combining this term with this term,"},{"Start":"07:25.620 ","End":"07:31.005","Text":"that just leaves me with 1 times e^2x because there\u0027s 2 minus 1 of them."},{"Start":"07:31.005 ","End":"07:34.750","Text":"From here I have a minus 1."},{"Start":"07:35.510 ","End":"07:41.145","Text":"From here I\u0027m left with a minus minus x,"},{"Start":"07:41.145 ","End":"07:44.050","Text":"so it\u0027s plus x."},{"Start":"07:44.420 ","End":"07:47.460","Text":"Now number 8,"},{"Start":"07:47.460 ","End":"07:50.520","Text":"and we\u0027ll need the chain rule mostly."},{"Start":"07:50.520 ","End":"07:55.520","Text":"I\u0027m going to scroll a bit and still keep the chain rule inside."},{"Start":"07:55.520 ","End":"07:58.910","Text":"What I get is y prime, first of all,"},{"Start":"07:58.910 ","End":"08:04.190","Text":"the exponential function, the derivative of it is this thing as is."},{"Start":"08:04.190 ","End":"08:05.270","Text":"That\u0027s the outer function."},{"Start":"08:05.270 ","End":"08:06.995","Text":"That\u0027s the f prime."},{"Start":"08:06.995 ","End":"08:11.675","Text":"Now I need the derivative of the square root of x."},{"Start":"08:11.675 ","End":"08:14.405","Text":"Sometimes if I don\u0027t know it right away,"},{"Start":"08:14.405 ","End":"08:15.830","Text":"I can pause and just say,"},{"Start":"08:15.830 ","End":"08:21.625","Text":"I need the derivative of the square root of x. I might even want to do this at the site."},{"Start":"08:21.625 ","End":"08:26.750","Text":"You might have seen it before and remember that it is 1 of the more basic ones,"},{"Start":"08:26.750 ","End":"08:30.975","Text":"the derivative of it is 1 over twice square root of x."},{"Start":"08:30.975 ","End":"08:32.390","Text":"If it\u0027s 1 over this,"},{"Start":"08:32.390 ","End":"08:36.005","Text":"I can just put e to the square root of x in the numerator."},{"Start":"08:36.005 ","End":"08:38.630","Text":"But let\u0027s assume for a moment that you\u0027ve forgotten how to do this."},{"Start":"08:38.630 ","End":"08:40.895","Text":"I\u0027ll show you quickly at the side."},{"Start":"08:40.895 ","End":"08:47.275","Text":"The square root of x is just x^1/2."},{"Start":"08:47.275 ","End":"08:49.490","Text":"If I use the rules for exponents,"},{"Start":"08:49.490 ","End":"08:53.480","Text":"the derivative of square root of x is going to be"},{"Start":"08:53.480 ","End":"08:59.599","Text":"1/2 times x and then you subtract 1 from the exponent."},{"Start":"08:59.599 ","End":"09:02.605","Text":"If I subtract 1, I\u0027ve got minus 1/2."},{"Start":"09:02.605 ","End":"09:08.640","Text":"Now, this is equal to 1/2 and negative exponents,"},{"Start":"09:08.640 ","End":"09:13.780","Text":"so you can convert to a positive exponent if you throw it into the denominator,"},{"Start":"09:13.780 ","End":"09:15.945","Text":"x to the plus 1/2."},{"Start":"09:15.945 ","End":"09:25.430","Text":"But x^1/2 is the square root of x. I get 1/2 and then 1 over the square root of x,"},{"Start":"09:25.430 ","End":"09:28.170","Text":"which is just what I said,"},{"Start":"09:28.170 ","End":"09:32.510","Text":"1 over twice the square root of x."},{"Start":"09:32.510 ","End":"09:34.100","Text":"It\u0027s best to memorize this,"},{"Start":"09:34.100 ","End":"09:36.830","Text":"it occurs frequently enough."},{"Start":"09:36.830 ","End":"09:42.480","Text":"This is our answer to number 8 and the next clip we continue from number 9."}],"ID":10455},{"Watched":false,"Name":"Exercise 1 - Parts 9-10","Duration":"9m 3s","ChapterTopicVideoID":10138,"CourseChapterTopicPlaylistID":8711,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.835","Text":"We\u0027ve just finished 1 through 8,"},{"Start":"00:02.835 ","End":"00:06.375","Text":"and now we\u0027ll do the remaining ones."},{"Start":"00:06.375 ","End":"00:10.755","Text":"I\u0027ve kept some of the formulas that I wrote."},{"Start":"00:10.755 ","End":"00:15.360","Text":"Let\u0027s continue with number 9."},{"Start":"00:15.360 ","End":"00:23.265","Text":"We have that y equals e to the power of Pi x over x minus 2."},{"Start":"00:23.265 ","End":"00:26.520","Text":"First thing I see is a quotient."},{"Start":"00:26.520 ","End":"00:29.160","Text":"Need the quotient rule."},{"Start":"00:29.160 ","End":"00:32.330","Text":"We\u0027ll use other rules as needed."},{"Start":"00:32.330 ","End":"00:36.500","Text":"What I get is y prime equals."},{"Start":"00:36.500 ","End":"00:38.855","Text":"I like to start with the denominator."},{"Start":"00:38.855 ","End":"00:40.840","Text":"We have the denominator squared,"},{"Start":"00:40.840 ","End":"00:43.640","Text":"so x minus 2 squared."},{"Start":"00:43.640 ","End":"00:46.610","Text":"Then the derivative of the numerator."},{"Start":"00:46.610 ","End":"00:50.330","Text":"For this derivative, I\u0027ll need the chain rule."},{"Start":"00:50.330 ","End":"00:54.370","Text":"This Pi x is like my inner derivative."},{"Start":"00:54.370 ","End":"00:58.575","Text":"The outer derivative, the exponential function is its own derivative,"},{"Start":"00:58.575 ","End":"01:00.485","Text":"so e to the Pi x."},{"Start":"01:00.485 ","End":"01:03.935","Text":"But then multiplied by the inner derivative,"},{"Start":"01:03.935 ","End":"01:06.035","Text":"the derivative of Pi x."},{"Start":"01:06.035 ","End":"01:08.000","Text":"It\u0027s just some constant times x,"},{"Start":"01:08.000 ","End":"01:10.770","Text":"so it\u0027s just that constant."},{"Start":"01:11.260 ","End":"01:14.795","Text":"That\u0027s the derivative of the numerator."},{"Start":"01:14.795 ","End":"01:18.680","Text":"We\u0027re back to the quotient rule."},{"Start":"01:18.680 ","End":"01:21.680","Text":"We\u0027re at this point where we need now the denominator,"},{"Start":"01:21.680 ","End":"01:24.815","Text":"so x minus 2."},{"Start":"01:24.815 ","End":"01:31.510","Text":"Now, I need a minus and I need the numerator as is e to the Pi x."},{"Start":"01:31.510 ","End":"01:35.510","Text":"Then the derivative of the denominator,"},{"Start":"01:35.510 ","End":"01:39.295","Text":"x minus 2 derivative is just 1."},{"Start":"01:39.295 ","End":"01:43.730","Text":"This is the answer except for simplification."},{"Start":"01:43.730 ","End":"01:45.470","Text":"I\u0027ll leave that for you to do."},{"Start":"01:45.470 ","End":"01:49.610","Text":"I\u0027ll just point out that we have e to the Pi x here and e to the Pi x here."},{"Start":"01:49.610 ","End":"01:51.820","Text":"We can take it out of the brackets."},{"Start":"01:51.820 ","End":"01:54.715","Text":"I\u0027ll just leave, I\u0027ll let it be."},{"Start":"01:54.715 ","End":"01:57.365","Text":"Next is number 10,"},{"Start":"01:57.365 ","End":"02:08.725","Text":"which is y equals 1 over the square root of e to the power of 4x plus 1."},{"Start":"02:08.725 ","End":"02:13.055","Text":"There\u0027s more than one way to do number 10."},{"Start":"02:13.055 ","End":"02:15.470","Text":"I see it as a quotient,"},{"Start":"02:15.470 ","End":"02:19.265","Text":"something over something, and I could use the quotient rule."},{"Start":"02:19.265 ","End":"02:21.560","Text":"But there\u0027s another way of doing it."},{"Start":"02:21.560 ","End":"02:22.580","Text":"I\u0027ll just mention it."},{"Start":"02:22.580 ","End":"02:23.660","Text":"I won\u0027t do it that way,"},{"Start":"02:23.660 ","End":"02:26.434","Text":"but I want to point out that"},{"Start":"02:26.434 ","End":"02:32.565","Text":"the square root of something is that thing to the power of 1.5."},{"Start":"02:32.565 ","End":"02:36.200","Text":"When it\u0027s 1 over it makes it to the power of negative 0.5."},{"Start":"02:36.200 ","End":"02:43.905","Text":"Actually, I could write this as e to the 4x plus 1 to the power of minus 0.5."},{"Start":"02:43.905 ","End":"02:48.375","Text":"Then use the chain rule and also the rule for exponents."},{"Start":"02:48.375 ","End":"02:50.045","Text":"Actually, is something that I was using."},{"Start":"02:50.045 ","End":"02:51.835","Text":"I didn\u0027t even write it."},{"Start":"02:51.835 ","End":"02:56.275","Text":"While I\u0027m at it I could have said if y equals x to the power of n,"},{"Start":"02:56.275 ","End":"03:00.560","Text":"then the derivative of y is nx to the n minus 1."},{"Start":"03:00.560 ","End":"03:04.480","Text":"I could use the chain rule plus this rule with n equals minus 0.5."},{"Start":"03:04.480 ","End":"03:05.960","Text":"I won\u0027t do it that way."},{"Start":"03:05.960 ","End":"03:08.780","Text":"That\u0027s an extra exercise for you if you want to."},{"Start":"03:08.780 ","End":"03:11.049","Text":"I\u0027ll do it as a quotient."},{"Start":"03:11.049 ","End":"03:13.710","Text":"Not doing it this way,"},{"Start":"03:13.710 ","End":"03:21.370","Text":"doing it with the quotient rule."},{"Start":"03:21.370 ","End":"03:24.050","Text":"Both methods are good, I just have to choose one."},{"Start":"03:24.050 ","End":"03:25.985","Text":"If it\u0027s a quotient,"},{"Start":"03:25.985 ","End":"03:27.410","Text":"then this is the numerator,"},{"Start":"03:27.410 ","End":"03:29.210","Text":"this is the denominator."},{"Start":"03:29.210 ","End":"03:32.345","Text":"I\u0027ve got y prime equals."},{"Start":"03:32.345 ","End":"03:35.530","Text":"Now, I need the derivative of the numerator."},{"Start":"03:35.530 ","End":"03:38.450","Text":"That\u0027s easy because it\u0027s a constant."},{"Start":"03:38.450 ","End":"03:44.450","Text":"The derivative is 0 times the denominator."},{"Start":"03:44.450 ","End":"03:45.830","Text":"It\u0027s silly to write it,"},{"Start":"03:45.830 ","End":"03:51.665","Text":"but I\u0027ll write it anyway because it\u0027s 0 times e to the 4x plus 1,"},{"Start":"03:51.665 ","End":"03:59.345","Text":"doesn\u0027t matter what it is, it would be 0 minus the numerator as is, which is 1."},{"Start":"03:59.345 ","End":"04:01.970","Text":"Then I need the derivative of the denominator."},{"Start":"04:01.970 ","End":"04:03.650","Text":"Now this is the hard part,"},{"Start":"04:03.650 ","End":"04:08.255","Text":"derivative of the denominator because I have the square root."},{"Start":"04:08.255 ","End":"04:11.000","Text":"We\u0027ve done the derivative of the square root before."},{"Start":"04:11.000 ","End":"04:13.820","Text":"It\u0027s a special case of this rule where n is 0.5."},{"Start":"04:13.820 ","End":"04:15.320","Text":"But let me just quote the result."},{"Start":"04:15.320 ","End":"04:20.465","Text":"We had that if y is equal to the square root of x,"},{"Start":"04:20.465 ","End":"04:24.740","Text":"then y prime was 1 over twice root"},{"Start":"04:24.740 ","End":"04:29.165","Text":"x. I remember we had it and I asked you to please memorize it."},{"Start":"04:29.165 ","End":"04:33.830","Text":"Back here, remember we want the derivative of the denominator,"},{"Start":"04:33.830 ","End":"04:35.945","Text":"the derivative of the square root."},{"Start":"04:35.945 ","End":"04:40.345","Text":"I start off with 1 over twice the square root."},{"Start":"04:40.345 ","End":"04:47.454","Text":"Then I copy what\u0027s in the square root is e to the 4x plus 1."},{"Start":"04:47.454 ","End":"04:54.470","Text":"But I have to also use now the chain rule because it wasn\u0027t square root of x,"},{"Start":"04:54.470 ","End":"04:57.860","Text":"it was square root of something, some function of x,"},{"Start":"04:57.860 ","End":"05:02.525","Text":"the inner function g. I have to multiply now"},{"Start":"05:02.525 ","End":"05:08.125","Text":"by the derivative of e to the 4x plus 1."},{"Start":"05:08.125 ","End":"05:12.200","Text":"Now, this is one of those cases where I don\u0027t want to do it all in 1 step."},{"Start":"05:12.200 ","End":"05:17.250","Text":"I take a break, I just say e to the 4x plus 1 derivative."},{"Start":"05:17.250 ","End":"05:19.520","Text":"Now I can breathe and just say, okay,"},{"Start":"05:19.520 ","End":"05:24.570","Text":"we\u0027ll leave this as a separate exercise at the side and then get back to it."},{"Start":"05:25.060 ","End":"05:27.860","Text":"I almost forgot the denominator."},{"Start":"05:27.860 ","End":"05:31.955","Text":"I usually start with the denominator because I have a tendency to forget it,"},{"Start":"05:31.955 ","End":"05:35.040","Text":"need a denominator squared."},{"Start":"05:35.250 ","End":"05:38.680","Text":"That\u0027s actually turns out neatly because when we"},{"Start":"05:38.680 ","End":"05:41.230","Text":"take a square root of something and then we square it,"},{"Start":"05:41.230 ","End":"05:43.525","Text":"we\u0027ve just got back to the thing itself."},{"Start":"05:43.525 ","End":"05:46.210","Text":"This is e to the 4x plus 1."},{"Start":"05:46.210 ","End":"05:47.955","Text":"That\u0027s this."},{"Start":"05:47.955 ","End":"05:49.960","Text":"Now that\u0027s the quotient rule."},{"Start":"05:49.960 ","End":"05:55.700","Text":"But we still have this bit to do, this side exercise."},{"Start":"05:55.770 ","End":"05:57.850","Text":"Let me return to this."},{"Start":"05:57.850 ","End":"05:59.920","Text":"What I need is the derivative of this thing."},{"Start":"05:59.920 ","End":"06:02.020","Text":"Now, first of all, it\u0027s a sum."},{"Start":"06:02.020 ","End":"06:05.725","Text":"It\u0027s the derivative of e to the 4x plus the derivative of 1."},{"Start":"06:05.725 ","End":"06:10.360","Text":"The 1 gives me nothing so I just have to differentiate e to the 4 x."},{"Start":"06:10.360 ","End":"06:13.135","Text":"We\u0027ve done things like this before. It\u0027s a chain rule."},{"Start":"06:13.135 ","End":"06:15.625","Text":"We started off by saying it\u0027s the exponent,"},{"Start":"06:15.625 ","End":"06:17.585","Text":"so it\u0027s e to the 4x."},{"Start":"06:17.585 ","End":"06:22.930","Text":"Then we remember that 4x is the inner function in the chain rule."},{"Start":"06:22.930 ","End":"06:24.920","Text":"We need the derivative of 4x,"},{"Start":"06:24.920 ","End":"06:27.545","Text":"which is just 4."},{"Start":"06:27.545 ","End":"06:30.515","Text":"Now let\u0027s piece this altogether."},{"Start":"06:30.515 ","End":"06:36.385","Text":"We have y prime equals. The first bit is 0."},{"Start":"06:36.385 ","End":"06:41.900","Text":"Then I have a minus or put the minus in front."},{"Start":"06:41.900 ","End":"06:45.590","Text":"Then I have e to"},{"Start":"06:45.590 ","End":"06:52.910","Text":"the 4x times 4."},{"Start":"06:52.910 ","End":"06:58.260","Text":"Let me put the 4 in front."},{"Start":"06:58.960 ","End":"07:01.385","Text":"That\u0027s this bit here,"},{"Start":"07:01.385 ","End":"07:06.755","Text":"over twice the square root,"},{"Start":"07:06.755 ","End":"07:15.469","Text":"so twice the square root of e^4x plus 1."},{"Start":"07:15.469 ","End":"07:20.000","Text":"But I still have all this over e"},{"Start":"07:20.000 ","End":"07:25.980","Text":"to e^4x plus 1."},{"Start":"07:25.980 ","End":"07:29.290","Text":"I need to scroll since it\u0027s disappearing."},{"Start":"07:29.660 ","End":"07:34.570","Text":"Now simplify it 1 step further,"},{"Start":"07:35.360 ","End":"07:39.970","Text":"2 into 4, goes twice."},{"Start":"07:41.450 ","End":"07:48.160","Text":"What I can do is write this thing as that of a fraction in the top."},{"Start":"07:48.160 ","End":"07:53.110","Text":"I\u0027ll just leave the 2e^4x in"},{"Start":"07:53.110 ","End":"08:00.170","Text":"the numerator and put this denominator into the denominator down here."},{"Start":"08:00.330 ","End":"08:07.680","Text":"What I get is e^4x plus 1,"},{"Start":"08:07.680 ","End":"08:15.670","Text":"I need the brackets times the square root of e^4x plus 1."},{"Start":"08:17.160 ","End":"08:19.300","Text":"I could leave it like this."},{"Start":"08:19.300 ","End":"08:26.240","Text":"I\u0027d like to do one more step and write it as minus 2e^4x."},{"Start":"08:26.340 ","End":"08:31.760","Text":"The denominator I\u0027ll write it and then I\u0027ll explain,"},{"Start":"08:31.760 ","End":"08:38.240","Text":"is e to the power of 4x plus 1 to the power of 3/2."},{"Start":"08:38.240 ","End":"08:39.920","Text":"Now, how did I get that?"},{"Start":"08:39.920 ","End":"08:44.205","Text":"If I look at this thing, e^4x plus 1 here it\u0027s to the power of 1,"},{"Start":"08:44.205 ","End":"08:46.110","Text":"here it\u0027s to the power of 0.5."},{"Start":"08:46.110 ","End":"08:47.580","Text":"Using rules of exponents,"},{"Start":"08:47.580 ","End":"08:52.450","Text":"it\u0027s 1 plus 0.5 and 1 plus 0.5 is 3/2."},{"Start":"08:52.450 ","End":"08:54.305","Text":"This is a bit simpler than this,"},{"Start":"08:54.305 ","End":"08:57.020","Text":"and we\u0027ll leave this as the answer to number 10."},{"Start":"08:57.020 ","End":"09:02.520","Text":"I think we\u0027ll stop here and do number 11 and 12 in the following clip."}],"ID":10456},{"Watched":false,"Name":"Exercise 1 - Parts 11-12","Duration":"7m 29s","ChapterTopicVideoID":10139,"CourseChapterTopicPlaylistID":8711,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"I already copied it out for you, and here it is."},{"Start":"00:03.330 ","End":"00:07.830","Text":"This is number 11. Notice that we have a cube root."},{"Start":"00:07.830 ","End":"00:14.700","Text":"What I\u0027d like to remind you is that the cube root of"},{"Start":"00:14.700 ","End":"00:21.855","Text":"a is the same as a to the power of 1/3."},{"Start":"00:21.855 ","End":"00:28.430","Text":"I\u0027m going to rewrite this as y equals e to"},{"Start":"00:28.430 ","End":"00:35.575","Text":"the x squared plus 1 plus 1 to the power of 1/3."},{"Start":"00:35.575 ","End":"00:40.655","Text":"Now I\u0027m going to differentiate it using exponents."},{"Start":"00:40.655 ","End":"00:44.400","Text":"Y prime is going to equal"},{"Start":"00:44.410 ","End":"00:51.330","Text":"1/3 times e to the x squared plus 1 plus 1."},{"Start":"00:51.330 ","End":"00:53.220","Text":"I lower the power by 1,"},{"Start":"00:53.220 ","End":"00:55.435","Text":"so it\u0027s minus 2/3."},{"Start":"00:55.435 ","End":"00:58.520","Text":"But I also have to multiply by the inner derivative."},{"Start":"00:58.520 ","End":"01:04.250","Text":"For the moment, let me just write it as e to the x squared plus 1 plus 1 prime."},{"Start":"01:04.250 ","End":"01:08.365","Text":"Then it will give me a moment to think about how to differentiate this."},{"Start":"01:08.365 ","End":"01:12.170","Text":"Well differentiating the 1 is no problem that gives 0."},{"Start":"01:12.170 ","End":"01:13.180","Text":"But what about this?"},{"Start":"01:13.180 ","End":"01:16.750","Text":"I want to remind you of a general template"},{"Start":"01:16.750 ","End":"01:20.725","Text":"that if we have the derivative of e to the power of something,"},{"Start":"01:20.725 ","End":"01:23.920","Text":"then it\u0027s equal to e to the power of that same thing,"},{"Start":"01:23.920 ","End":"01:27.010","Text":"but times the derivative of that something."},{"Start":"01:27.010 ","End":"01:31.900","Text":"In our case, this thing will be x squared plus 1."},{"Start":"01:31.900 ","End":"01:35.860","Text":"Other words, I\u0027m going to replace box by x squared plus 1,"},{"Start":"01:35.860 ","End":"01:38.120","Text":"and then I\u0027m going to go back here."},{"Start":"01:38.310 ","End":"01:49.285","Text":"This is equal to 1/3 e to the power of x squared plus 1 plus 1 to the minus 2/3,"},{"Start":"01:49.285 ","End":"01:58.600","Text":"just copying, times e to the power of x squared plus 1."},{"Start":"01:58.600 ","End":"02:00.520","Text":"The 1 gives 0."},{"Start":"02:00.520 ","End":"02:04.540","Text":"But here we have now the inner derivative of x squared plus 1,"},{"Start":"02:04.540 ","End":"02:07.550","Text":"which is just 2x."},{"Start":"02:08.130 ","End":"02:14.350","Text":"I want to remind you another fact from algebra that if I have a negative exponent,"},{"Start":"02:14.350 ","End":"02:16.645","Text":"it means I can put it on a denominator."},{"Start":"02:16.645 ","End":"02:22.985","Text":"In general, if I have a to the power of minus something,"},{"Start":"02:22.985 ","End":"02:25.440","Text":"minus b, let\u0027s say,"},{"Start":"02:25.440 ","End":"02:30.685","Text":"and it\u0027s the same as 1 over a to the power of positive b."},{"Start":"02:30.685 ","End":"02:36.645","Text":"Going back here, I get 1/3."},{"Start":"02:36.645 ","End":"02:39.620","Text":"Instead of to the power of minus 2/3,"},{"Start":"02:39.620 ","End":"02:41.900","Text":"I\u0027ll put it as 1 over."},{"Start":"02:41.900 ","End":"02:44.480","Text":"I tell you what, I\u0027m just going to take the 3"},{"Start":"02:44.480 ","End":"02:46.750","Text":"from the 1/3 and put it in the denominator 2."},{"Start":"02:46.750 ","End":"02:51.385","Text":"Let me erase this and we\u0027ll start again with the denominator,"},{"Start":"02:51.385 ","End":"02:53.785","Text":"where we put the 3 on the bottom,"},{"Start":"02:53.785 ","End":"03:03.170","Text":"and also e to the x squared plus 1 plus 1 to the power of plus 2/3."},{"Start":"03:03.170 ","End":"03:06.310","Text":"The rest of it I can put on the numerator,"},{"Start":"03:06.310 ","End":"03:08.020","Text":"the one I don\u0027t need to bother with,"},{"Start":"03:08.020 ","End":"03:15.955","Text":"and I have e to the power of x squared plus 1 times 2x."},{"Start":"03:15.955 ","End":"03:19.180","Text":"Now, I could leave the answer like this,"},{"Start":"03:19.180 ","End":"03:27.940","Text":"but it\u0027s customary that if the original question was phrased in terms of cube root,"},{"Start":"03:27.940 ","End":"03:31.030","Text":"not to leave the fractional power."},{"Start":"03:31.030 ","End":"03:39.315","Text":"I\u0027m going to do the opposite of what I did up here and say that if I have, in our case,"},{"Start":"03:39.315 ","End":"03:42.515","Text":"a to the power of 2/3,"},{"Start":"03:42.515 ","End":"03:48.785","Text":"that this will equal the cube root of a squared."},{"Start":"03:48.785 ","End":"03:51.790","Text":"Now going back here,"},{"Start":"03:51.790 ","End":"03:55.690","Text":"what I\u0027ll get is let me just put the 2 in front,"},{"Start":"03:55.690 ","End":"03:58.779","Text":"if I put the whole 2x in front it looks better."},{"Start":"03:58.779 ","End":"04:06.825","Text":"2xe to the x squared plus 1 over 3."},{"Start":"04:06.825 ","End":"04:09.240","Text":"Instead of the power of 2/3,"},{"Start":"04:09.240 ","End":"04:19.370","Text":"I\u0027ll take the cube root of e to the x squared plus 1 plus 1 squared."},{"Start":"04:19.370 ","End":"04:23.880","Text":"This is what I will leave as an answer."},{"Start":"04:23.880 ","End":"04:29.790","Text":"I\u0027ll highlight it. There we go."},{"Start":"04:30.030 ","End":"04:34.100","Text":"Now onto Number 12."},{"Start":"04:35.090 ","End":"04:38.430","Text":"I already wrote Number 12 here."},{"Start":"04:38.430 ","End":"04:41.000","Text":"I just copied it to save time,"},{"Start":"04:41.000 ","End":"04:44.060","Text":"and we have to find what is y prime."},{"Start":"04:44.060 ","End":"04:46.220","Text":"Now I see I have a quotient here."},{"Start":"04:46.220 ","End":"04:49.370","Text":"Let me remind you of the quotient rule."},{"Start":"04:49.370 ","End":"04:53.705","Text":"If we have the derivative of a quotient f over g,"},{"Start":"04:53.705 ","End":"05:03.800","Text":"it\u0027s the derivative of the numerator times the denominator minus"},{"Start":"05:03.800 ","End":"05:08.150","Text":"the numerator times the derivative of"},{"Start":"05:08.150 ","End":"05:15.440","Text":"the denominator over the denominator squared."},{"Start":"05:15.440 ","End":"05:17.945","Text":"But I see there\u0027s another formula I need."},{"Start":"05:17.945 ","End":"05:21.560","Text":"I want to remind you just in Number 11,"},{"Start":"05:21.560 ","End":"05:23.000","Text":"in the previous one,"},{"Start":"05:23.000 ","End":"05:30.120","Text":"I\u0027ll just write it again that if we have e to the power of something derived,"},{"Start":"05:30.120 ","End":"05:36.245","Text":"it\u0027s just e to that something times the derivative of that something box,"},{"Start":"05:36.245 ","End":"05:39.870","Text":"which in this case will be minus x squared."},{"Start":"05:41.450 ","End":"05:44.280","Text":"Y prime is equal to,"},{"Start":"05:44.280 ","End":"05:45.770","Text":"now this is F and this is g,"},{"Start":"05:45.770 ","End":"05:47.390","Text":"so it\u0027s f prime."},{"Start":"05:47.390 ","End":"05:50.750","Text":"I already need the derivative of this and I\u0027m going to use this formula."},{"Start":"05:50.750 ","End":"05:55.550","Text":"F prime will be e to the minus x squared."},{"Start":"05:55.550 ","End":"06:01.610","Text":"That\u0027s e to the box times box prime is the derivative of minus x squared,"},{"Start":"06:01.610 ","End":"06:03.935","Text":"which is minus 2x."},{"Start":"06:03.935 ","End":"06:06.065","Text":"All this is the f prime part."},{"Start":"06:06.065 ","End":"06:07.415","Text":"I still need the g,"},{"Start":"06:07.415 ","End":"06:09.325","Text":"which is x,"},{"Start":"06:09.325 ","End":"06:12.520","Text":"minus, that\u0027s this minus f,"},{"Start":"06:12.520 ","End":"06:17.795","Text":"that\u0027s e to the minus x squared times the derivative of g,"},{"Start":"06:17.795 ","End":"06:19.610","Text":"which is just 1,"},{"Start":"06:19.610 ","End":"06:21.995","Text":"because derivative of x is 1."},{"Start":"06:21.995 ","End":"06:25.230","Text":"Then divided by g squared,"},{"Start":"06:25.230 ","End":"06:27.510","Text":"which is x squared."},{"Start":"06:27.510 ","End":"06:29.900","Text":"Let\u0027s tidy up a bit."},{"Start":"06:29.900 ","End":"06:36.080","Text":"What I can do is take e to the minus x squared as a common factor outside the brackets."},{"Start":"06:36.080 ","End":"06:38.350","Text":"E to the minus x squared,"},{"Start":"06:38.350 ","End":"06:40.065","Text":"and what am I left with?"},{"Start":"06:40.065 ","End":"06:47.140","Text":"Minus 2x times x is minus 2x squared."},{"Start":"06:47.720 ","End":"06:51.430","Text":"Then minus 1."},{"Start":"06:51.500 ","End":"06:57.700","Text":"All this over x squared."},{"Start":"06:58.100 ","End":"07:00.370","Text":"This is okay as an answer,"},{"Start":"07:00.370 ","End":"07:02.995","Text":"I think you can slightly tidy it up a bit."},{"Start":"07:02.995 ","End":"07:05.829","Text":"I\u0027d like to take the minus outside the brackets,"},{"Start":"07:05.829 ","End":"07:09.325","Text":"and I prefer to put the polynomial before the exponents."},{"Start":"07:09.325 ","End":"07:18.080","Text":"This will be 2x squared plus 1 times e to the minus x squared,"},{"Start":"07:18.080 ","End":"07:22.185","Text":"all over x squared."},{"Start":"07:22.185 ","End":"07:27.480","Text":"And this is the answer to Number 12."},{"Start":"07:27.480 ","End":"07:30.490","Text":"So we are done."}],"ID":10457},{"Watched":false,"Name":"Exercise 2","Duration":"16m 21s","ChapterTopicVideoID":10147,"CourseChapterTopicPlaylistID":8711,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"This exercise is 4 in 1 and whoops,"},{"Start":"00:03.210 ","End":"00:06.195","Text":"this should say 4 not 12."},{"Start":"00:06.195 ","End":"00:11.355","Text":"Just to remind you, the first derivative is just a regular derivative, so if we have y,"},{"Start":"00:11.355 ","End":"00:14.655","Text":"the first derivative means to find y prime,"},{"Start":"00:14.655 ","End":"00:18.030","Text":"and the second derivative is a derivative of a derivative,"},{"Start":"00:18.030 ","End":"00:20.595","Text":"so that\u0027s y double prime."},{"Start":"00:20.595 ","End":"00:22.815","Text":"Start with number 1."},{"Start":"00:22.815 ","End":"00:27.460","Text":"I\u0027ll copy it. Y equals natural log of x over x."},{"Start":"00:27.460 ","End":"00:29.215","Text":"To find the first derivative,"},{"Start":"00:29.215 ","End":"00:31.200","Text":"we\u0027re going to use the quotient rule,"},{"Start":"00:31.200 ","End":"00:37.820","Text":"obviously, and you should really know the quotient rule by heart."},{"Start":"00:37.820 ","End":"00:39.950","Text":"What we have is, in the denominator,"},{"Start":"00:39.950 ","End":"00:42.635","Text":"we have the original denominator squared,"},{"Start":"00:42.635 ","End":"00:46.160","Text":"and then we have the derivative of the numerator,"},{"Start":"00:46.160 ","End":"00:48.855","Text":"which is 1 over x,"},{"Start":"00:48.855 ","End":"00:51.210","Text":"that\u0027s 1 of the basics, natural log,"},{"Start":"00:51.210 ","End":"00:55.980","Text":"derivative of 1 over x times the denominator,"},{"Start":"00:55.980 ","End":"01:06.665","Text":"and then minus the numerator as is times the derivative of the denominator times 1."},{"Start":"01:06.665 ","End":"01:09.725","Text":"If we simplify this,"},{"Start":"01:09.725 ","End":"01:12.650","Text":"1 over x times x is just 1,"},{"Start":"01:12.650 ","End":"01:19.430","Text":"so this is 1 minus natural log of x over x squared."},{"Start":"01:19.430 ","End":"01:21.080","Text":"That\u0027s the first derivative."},{"Start":"01:21.080 ","End":"01:22.970","Text":"Now, we want the second derivative,"},{"Start":"01:22.970 ","End":"01:25.500","Text":"so y double prime."},{"Start":"01:25.500 ","End":"01:28.340","Text":"Once again, I have a quotient."},{"Start":"01:28.340 ","End":"01:32.000","Text":"I start with denominator squared,"},{"Start":"01:32.000 ","End":"01:34.490","Text":"so that\u0027s x^4th,"},{"Start":"01:34.490 ","End":"01:39.755","Text":"and then I have the derivative of the numerator."},{"Start":"01:39.755 ","End":"01:46.520","Text":"Derivative of the numerator is just minus 1 over x because the 1 gives nothing,"},{"Start":"01:46.520 ","End":"01:49.310","Text":"and the natural log gives 1 over x,"},{"Start":"01:49.310 ","End":"01:54.540","Text":"derivative of the numerator times the denominator,"},{"Start":"01:54.540 ","End":"01:57.165","Text":"and then minus vice versa,"},{"Start":"01:57.165 ","End":"01:59.880","Text":"the numerator as is,"},{"Start":"01:59.880 ","End":"02:06.240","Text":"and the derivative of the denominator, which is 2x."},{"Start":"02:07.690 ","End":"02:10.535","Text":"I\u0027d like to simplify this a bit."},{"Start":"02:10.535 ","End":"02:13.940","Text":"I noticed that I can divide everything by x,"},{"Start":"02:13.940 ","End":"02:16.670","Text":"so if I divide the denominator by x,"},{"Start":"02:16.670 ","End":"02:18.620","Text":"I\u0027ve just got x cubed."},{"Start":"02:18.620 ","End":"02:22.340","Text":"Here I get rid of this x and here I can get rid of"},{"Start":"02:22.340 ","End":"02:27.690","Text":"this x by just crossing out the 2, just x."},{"Start":"02:27.940 ","End":"02:30.890","Text":"This is equal to."},{"Start":"02:30.890 ","End":"02:34.620","Text":"Here I have minus,"},{"Start":"02:34.620 ","End":"02:37.900","Text":"x over x is minus 1,"},{"Start":"02:37.900 ","End":"02:40.175","Text":"and then I have,"},{"Start":"02:40.175 ","End":"02:43.020","Text":"from here, minus 2,"},{"Start":"02:43.020 ","End":"02:46.500","Text":"so altogether, minus 3, and then minus,"},{"Start":"02:46.500 ","End":"02:50.145","Text":"minus is plus 2,"},{"Start":"02:50.145 ","End":"02:56.745","Text":"natural log of x over x cubed."},{"Start":"02:56.745 ","End":"02:59.220","Text":"That\u0027s question 1. Now,"},{"Start":"02:59.220 ","End":"03:00.600","Text":"let\u0027s do number 2."},{"Start":"03:00.600 ","End":"03:02.040","Text":"I\u0027ll do it over here."},{"Start":"03:02.040 ","End":"03:04.320","Text":"We have some room."},{"Start":"03:04.320 ","End":"03:14.270","Text":"This 1 says y equals natural log squared of x minus 1 over natural log of x."},{"Start":"03:14.270 ","End":"03:16.670","Text":"When we put the 2 here,"},{"Start":"03:16.670 ","End":"03:20.640","Text":"it means the natural log of x all squared."},{"Start":"03:21.980 ","End":"03:26.029","Text":"To get y prime, I\u0027m going to use various rules."},{"Start":"03:26.029 ","End":"03:28.250","Text":"For the first term,"},{"Start":"03:28.250 ","End":"03:29.810","Text":"I can use the chain rule."},{"Start":"03:29.810 ","End":"03:31.765","Text":"I have something squared,"},{"Start":"03:31.765 ","End":"03:35.440","Text":"and so it\u0027s twice that something,"},{"Start":"03:35.720 ","End":"03:42.305","Text":"but then I have to use the chain rule and the inner derivative,"},{"Start":"03:42.305 ","End":"03:45.050","Text":"which is the derivative of natural log of x,"},{"Start":"03:45.050 ","End":"03:46.250","Text":"is 1 over x,"},{"Start":"03:46.250 ","End":"03:47.735","Text":"so that\u0027s the first bit,"},{"Start":"03:47.735 ","End":"03:50.450","Text":"minus the second bit."},{"Start":"03:50.450 ","End":"03:53.090","Text":"I could use the quotient rule,"},{"Start":"03:53.090 ","End":"03:55.715","Text":"but I don\u0027t have to use the quotient rule,"},{"Start":"03:55.715 ","End":"03:58.690","Text":"I can use an exponent."},{"Start":"03:58.690 ","End":"04:01.065","Text":"I\u0027ll do this bit at the side,"},{"Start":"04:01.065 ","End":"04:03.620","Text":"the derivative of 1 over natural log of x."},{"Start":"04:03.620 ","End":"04:11.230","Text":"First of all, it is equal to natural log of x^minus 1."},{"Start":"04:11.230 ","End":"04:13.710","Text":"When I differentiate it,"},{"Start":"04:13.710 ","End":"04:15.250","Text":"I\u0027ll do prime,"},{"Start":"04:15.250 ","End":"04:19.995","Text":"I\u0027ve got, by the exponent rule, minus 1,"},{"Start":"04:19.995 ","End":"04:23.555","Text":"natural log of x^minus 2,"},{"Start":"04:23.555 ","End":"04:28.740","Text":"but then I need the chain rule because the inner function is not just the x,"},{"Start":"04:28.740 ","End":"04:30.060","Text":"it\u0027s natural log of x,"},{"Start":"04:30.060 ","End":"04:32.210","Text":"so I need to multiply by 1 over x,"},{"Start":"04:32.210 ","End":"04:34.100","Text":"which is the inner derivative,"},{"Start":"04:34.100 ","End":"04:36.755","Text":"and if we tidy this up,"},{"Start":"04:36.755 ","End":"04:39.535","Text":"this becomes minus 1."},{"Start":"04:39.535 ","End":"04:44.050","Text":"The power of minus 2 and the denominator is the power of plus 2."},{"Start":"04:44.050 ","End":"04:45.535","Text":"I\u0027ll write the 2 here,"},{"Start":"04:45.535 ","End":"04:50.090","Text":"and the x will also go in the denominator."},{"Start":"04:50.320 ","End":"04:55.300","Text":"What I have here is minus,"},{"Start":"04:55.300 ","End":"04:58.480","Text":"minus is plus, that makes it a plus,"},{"Start":"04:58.480 ","End":"05:01.015","Text":"and I have 1 over,"},{"Start":"05:01.015 ","End":"05:02.980","Text":"I think it looks nicer if the x is in"},{"Start":"05:02.980 ","End":"05:07.120","Text":"front and there\u0027s no confusion that it\u0027s x times x or something,"},{"Start":"05:07.120 ","End":"05:11.630","Text":"x times natural log squared of x."},{"Start":"05:11.630 ","End":"05:18.369","Text":"I\u0027d like to generalize this trick of differentiating 1 over something."},{"Start":"05:18.369 ","End":"05:25.925","Text":"Let\u0027s think of natural log of x as just some unknown function of x and call it box."},{"Start":"05:25.925 ","End":"05:29.550","Text":"If I have 1 over box,"},{"Start":"05:30.290 ","End":"05:33.785","Text":"and I want to differentiate this,"},{"Start":"05:33.785 ","End":"05:36.050","Text":"what I can get is,"},{"Start":"05:36.050 ","End":"05:40.460","Text":"just like here, you get minus 1 over box squared,"},{"Start":"05:40.460 ","End":"05:43.810","Text":"so I\u0027ll leave the 1 for a moment,"},{"Start":"05:43.810 ","End":"05:50.170","Text":"box squared, and the 1 over x is box prime."},{"Start":"05:50.170 ","End":"05:55.620","Text":"This gives us a general rule for when we have 1 over something."},{"Start":"05:56.110 ","End":"05:59.855","Text":"It\u0027s going to get in the way. I\u0027m going to put it over here."},{"Start":"05:59.855 ","End":"06:05.130","Text":"Now, let\u0027s continue to y double prime."},{"Start":"06:06.090 ","End":"06:08.740","Text":"For the first term,"},{"Start":"06:08.740 ","End":"06:10.750","Text":"I have a product,"},{"Start":"06:10.750 ","End":"06:12.220","Text":"I could look at it as a quotient."},{"Start":"06:12.220 ","End":"06:13.660","Text":"If I put the x on the bottom,"},{"Start":"06:13.660 ","End":"06:15.325","Text":"I\u0027ll leave it as a product."},{"Start":"06:15.325 ","End":"06:20.890","Text":"The product rule says I\u0027ll take the 2 with the natural log of x,"},{"Start":"06:20.890 ","End":"06:23.260","Text":"the derivative of the first bit,"},{"Start":"06:23.260 ","End":"06:26.784","Text":"which is just 2 over x,"},{"Start":"06:26.784 ","End":"06:32.970","Text":"times the second bit as is plus this 1 as is,"},{"Start":"06:32.970 ","End":"06:35.745","Text":"twice natural log of x,"},{"Start":"06:35.745 ","End":"06:39.045","Text":"times the derivative of 1 over x,"},{"Start":"06:39.045 ","End":"06:41.790","Text":"and the derivative of 1 over x,"},{"Start":"06:41.790 ","End":"06:47.335","Text":"either using this rule or using the fact that it\u0027s x^minus 1,"},{"Start":"06:47.335 ","End":"06:50.330","Text":"it\u0027s minus 1 over x squared."},{"Start":"06:50.330 ","End":"06:51.740","Text":"That\u0027s the derivative of that."},{"Start":"06:51.740 ","End":"06:53.510","Text":"I\u0027ll put it in brackets."},{"Start":"06:53.510 ","End":"06:55.750","Text":"That\u0027s up to the plus,"},{"Start":"06:55.750 ","End":"07:01.490","Text":"and I\u0027m going to use this rule that we just wrote with that the box being,"},{"Start":"07:01.490 ","End":"07:03.890","Text":"this time, this thing here,"},{"Start":"07:03.890 ","End":"07:05.690","Text":"so I have 1 over box,"},{"Start":"07:05.690 ","End":"07:08.375","Text":"so I get minus,"},{"Start":"07:08.375 ","End":"07:10.760","Text":"and I\u0027ll start with the denominator,"},{"Start":"07:10.760 ","End":"07:13.970","Text":"box squared, this thing squared,"},{"Start":"07:13.970 ","End":"07:15.950","Text":"I can just square each bit separately,"},{"Start":"07:15.950 ","End":"07:19.460","Text":"x squared, natural log^4th,"},{"Start":"07:19.460 ","End":"07:22.055","Text":"x squared squared is to the 4th."},{"Start":"07:22.055 ","End":"07:27.930","Text":"Then on the top, I\u0027ll just write it as this thing prime,"},{"Start":"07:27.930 ","End":"07:34.640","Text":"and it gives me time to pause and just do this separately."},{"Start":"07:34.640 ","End":"07:37.430","Text":"Meanwhile, let me just simplify this."},{"Start":"07:37.430 ","End":"07:41.780","Text":"What we have here is, let\u0027s see,"},{"Start":"07:41.780 ","End":"07:49.075","Text":"2 over x times 1 over x is 2 over x squared,"},{"Start":"07:49.075 ","End":"07:51.990","Text":"and then here I have 2 times minus 1,"},{"Start":"07:51.990 ","End":"07:53.745","Text":"so I have minus."},{"Start":"07:53.745 ","End":"07:59.700","Text":"In the numerator, 2 natural log x and on the denominator,"},{"Start":"07:59.700 ","End":"08:06.075","Text":"x squared, and then minus this bit here."},{"Start":"08:06.075 ","End":"08:10.620","Text":"I\u0027ll copy x squared natural log^4th x,"},{"Start":"08:10.620 ","End":"08:17.930","Text":"and this gives me a chance to do this derivative and put it here on the numerator."},{"Start":"08:17.930 ","End":"08:20.180","Text":"I\u0027m looking at this,"},{"Start":"08:20.180 ","End":"08:22.400","Text":"and I see a product that I got,"},{"Start":"08:22.400 ","End":"08:24.650","Text":"x times something,"},{"Start":"08:24.650 ","End":"08:26.240","Text":"so I\u0027ll use the product rule."},{"Start":"08:26.240 ","End":"08:27.760","Text":"This is the first bit."},{"Start":"08:27.760 ","End":"08:32.125","Text":"This is the second bit. The first bit prime is 1,"},{"Start":"08:32.125 ","End":"08:34.260","Text":"and the second bit as is,"},{"Start":"08:34.260 ","End":"08:36.615","Text":"natural log squared x,"},{"Start":"08:36.615 ","End":"08:40.830","Text":"plus vice versa, x as is,"},{"Start":"08:40.830 ","End":"08:44.255","Text":"and then the derivative of natural log squared x."},{"Start":"08:44.255 ","End":"08:45.920","Text":"We\u0027ve done this before."},{"Start":"08:45.920 ","End":"08:48.305","Text":"If you look at the passage from here to here,"},{"Start":"08:48.305 ","End":"08:50.660","Text":"we\u0027ve got 2 natural log of x,"},{"Start":"08:50.660 ","End":"08:52.075","Text":"1 over x,"},{"Start":"08:52.075 ","End":"08:58.670","Text":"so this times 2 natural log x 1 over x."},{"Start":"08:58.670 ","End":"09:01.490","Text":"We\u0027ve done it before, so I might as well copy."},{"Start":"09:02.520 ","End":"09:04.690","Text":"We could simplify this."},{"Start":"09:04.690 ","End":"09:06.055","Text":"I\u0027m just going to do a little bit."},{"Start":"09:06.055 ","End":"09:12.805","Text":"Say this x cancels with this x and obviously I don\u0027t need this 1 here."},{"Start":"09:12.805 ","End":"09:14.920","Text":"I could break it up,"},{"Start":"09:14.920 ","End":"09:16.210","Text":"try to simplify."},{"Start":"09:16.210 ","End":"09:21.190","Text":"I think this is good enough and I\u0027ll leave this as a result for Number 2."},{"Start":"09:21.190 ","End":"09:24.700","Text":"Before I scroll, I\u0027d better copy"},{"Start":"09:24.700 ","End":"09:29.830","Text":"the questions for 3 and 4. Let\u0027s see."},{"Start":"09:29.830 ","End":"09:32.500","Text":"Number 3 is e^ 2x,"},{"Start":"09:32.500 ","End":"09:36.265","Text":"natural log of x squared plus 4."},{"Start":"09:36.265 ","End":"09:41.170","Text":"Here, natural log of x over e^ x."},{"Start":"09:41.170 ","End":"09:45.350","Text":"Now I\u0027m free to continue, I mean to scroll."},{"Start":"09:45.390 ","End":"09:50.740","Text":"Let\u0027s see. This is what y equals."},{"Start":"09:50.740 ","End":"09:54.100","Text":"I better write the y equals."},{"Start":"09:54.100 ","End":"09:56.950","Text":"Next is this Number 3."},{"Start":"09:56.950 ","End":"09:59.725","Text":"The first thing I see is a product."},{"Start":"09:59.725 ","End":"10:02.785","Text":"I\u0027m going to start using the product rule."},{"Start":"10:02.785 ","End":"10:07.735","Text":"Y prime is equal to derivative of the first."},{"Start":"10:07.735 ","End":"10:10.330","Text":"We\u0027ve seen many times the exponent."},{"Start":"10:10.330 ","End":"10:11.515","Text":"so it\u0027s a chain rule."},{"Start":"10:11.515 ","End":"10:13.510","Text":"It\u0027s e^ 2x,"},{"Start":"10:13.510 ","End":"10:16.420","Text":"but times the inner derivative, which is 2."},{"Start":"10:16.420 ","End":"10:23.500","Text":"It has derivative of the first and the second as is natural log of x squared plus 4,"},{"Start":"10:23.500 ","End":"10:27.535","Text":"plus the first bit as is e^ 2x."},{"Start":"10:27.535 ","End":"10:30.265","Text":"Now I need the derivative of this."},{"Start":"10:30.265 ","End":"10:32.695","Text":"The derivative of the natural log,"},{"Start":"10:32.695 ","End":"10:34.540","Text":"we\u0027ve seen this a few times."},{"Start":"10:34.540 ","End":"10:38.140","Text":"We basically say it\u0027s 1 over whatever this is."},{"Start":"10:38.140 ","End":"10:39.610","Text":"Again, we\u0027re using the chain rule."},{"Start":"10:39.610 ","End":"10:42.445","Text":"It\u0027s 1 over x squared plus 4."},{"Start":"10:42.445 ","End":"10:45.235","Text":"But we need the inner derivative."},{"Start":"10:45.235 ","End":"10:49.850","Text":"The derivative of x squared plus 4 is 2x."},{"Start":"10:50.100 ","End":"10:53.080","Text":"Now this one\u0027s a bit tricky."},{"Start":"10:53.080 ","End":"10:56.080","Text":"We have a product and the product here."},{"Start":"10:56.080 ","End":"10:59.454","Text":"1 of the factors in the product is a quotient."},{"Start":"10:59.454 ","End":"11:03.490","Text":"I\u0027m going to use the product rule twice as well as the quotient rule."},{"Start":"11:03.490 ","End":"11:09.730","Text":"Y double-prime, I\u0027m thinking of the 2 as being put upfront."},{"Start":"11:09.730 ","End":"11:12.055","Text":"I\u0027ve got 2^ 2x."},{"Start":"11:12.055 ","End":"11:20.680","Text":"The derivative of 2e^ 2x is 2e^ 2x and then times an inner derivative which is 2."},{"Start":"11:20.680 ","End":"11:28.420","Text":"Then natural log of x squared plus 4 as is,"},{"Start":"11:28.420 ","End":"11:36.910","Text":"plus the other way round 2e^ 2x as is."},{"Start":"11:36.910 ","End":"11:41.275","Text":"Then the derivative of natural log of x squared plus 4."},{"Start":"11:41.275 ","End":"11:44.530","Text":"We already did that earlier."},{"Start":"11:44.530 ","End":"11:47.200","Text":"This is what it came out to be."},{"Start":"11:47.200 ","End":"11:49.090","Text":"I\u0027m just copying that here."},{"Start":"11:49.090 ","End":"11:54.115","Text":"So 2x over x squared plus 4."},{"Start":"11:54.115 ","End":"11:57.790","Text":"All this just came out of the bit up to the plus,"},{"Start":"11:57.790 ","End":"12:00.505","Text":"so we\u0027re going to have to continue on the next line."},{"Start":"12:00.505 ","End":"12:05.140","Text":"Plus and then here is a product."},{"Start":"12:05.140 ","End":"12:12.745","Text":"The derivative of the first is e^ 2x times 2,"},{"Start":"12:12.745 ","End":"12:23.080","Text":"and then the second 1 as is 2x over x squared plus 4 plus e^ 2x as is."},{"Start":"12:23.080 ","End":"12:26.510","Text":"Now I need the derivative of this."},{"Start":"12:27.510 ","End":"12:30.685","Text":"Let me just circle this so I don\u0027t lose track."},{"Start":"12:30.685 ","End":"12:35.275","Text":"This is what I want the derivative of and I\u0027m going to use the quotient rule on this."},{"Start":"12:35.275 ","End":"12:41.710","Text":"For the quotient, we start off like this as I do like do the denominator bit first,"},{"Start":"12:41.710 ","End":"12:45.925","Text":"it\u0027s squared, then the derivative of the numerator,"},{"Start":"12:45.925 ","End":"12:49.225","Text":"which is 2 times the denominator,"},{"Start":"12:49.225 ","End":"12:53.320","Text":"x squared plus 4 to the minus here,"},{"Start":"12:53.320 ","End":"12:57.830","Text":"the numerator as is 2x."},{"Start":"12:59.550 ","End":"13:04.675","Text":"Then the derivative of the denominator,"},{"Start":"13:04.675 ","End":"13:07.780","Text":"which is also 2x."},{"Start":"13:07.780 ","End":"13:10.270","Text":"Now that\u0027s y double-prime."},{"Start":"13:10.270 ","End":"13:12.460","Text":"Just need some simplification."},{"Start":"13:12.460 ","End":"13:17.410","Text":"You know what? This is really not an exercise in algebra."},{"Start":"13:17.410 ","End":"13:20.125","Text":"I\u0027ll just indicate that optionally,"},{"Start":"13:20.125 ","End":"13:22.615","Text":"you should do some simplification."},{"Start":"13:22.615 ","End":"13:25.345","Text":"But I won\u0027t bother with that."},{"Start":"13:25.345 ","End":"13:29.620","Text":"I want to go straight on to Number 4."},{"Start":"13:29.620 ","End":"13:33.190","Text":"Here, the first thing that\u0027s obvious,"},{"Start":"13:33.190 ","End":"13:35.050","Text":"we have a quotient."},{"Start":"13:35.050 ","End":"13:39.160","Text":"Y prime, using the quotient rule,"},{"Start":"13:39.160 ","End":"13:41.635","Text":"I do the denominator squared."},{"Start":"13:41.635 ","End":"13:43.540","Text":"I can write each of the x squared,"},{"Start":"13:43.540 ","End":"13:47.035","Text":"but I can write it as e to the 2x using the rules of exponents."},{"Start":"13:47.035 ","End":"13:52.435","Text":"Now I have the derivative of the numerator,"},{"Start":"13:52.435 ","End":"13:57.820","Text":"which is 1 over x times the denominator,"},{"Start":"13:57.820 ","End":"14:03.055","Text":"e^ x minus the other way round numerator as is,"},{"Start":"14:03.055 ","End":"14:06.760","Text":"and the derivative of the denominator,"},{"Start":"14:06.760 ","End":"14:14.620","Text":"which is also e^ x. I could divide top and bottom by e^ x."},{"Start":"14:14.620 ","End":"14:16.480","Text":"If I divide the top by e^ x,"},{"Start":"14:16.480 ","End":"14:18.355","Text":"the common factor just comes out."},{"Start":"14:18.355 ","End":"14:22.225","Text":"I\u0027ve got 1 over x minus natural log of x."},{"Start":"14:22.225 ","End":"14:24.640","Text":"If I\u0027d left this as e^ x squared,"},{"Start":"14:24.640 ","End":"14:29.035","Text":"it\u0027s e^ x times e^ x. I\u0027m just left with a single e^ x"},{"Start":"14:29.035 ","End":"14:33.910","Text":"after I\u0027ve divided by e^ x. Symbolically,"},{"Start":"14:33.910 ","End":"14:36.670","Text":"I could just say I erase this and this and here I divided,"},{"Start":"14:36.670 ","End":"14:39.745","Text":"so I just got knocked out the 2."},{"Start":"14:39.745 ","End":"14:42.895","Text":"Now I have a quotient again."},{"Start":"14:42.895 ","End":"14:49.585","Text":"Y double prime is equal to, once again,"},{"Start":"14:49.585 ","End":"14:51.250","Text":"the quotient, I mean, yeah,"},{"Start":"14:51.250 ","End":"14:56.380","Text":"the quotient rule gives me the denominator squared e^ 2x."},{"Start":"14:56.380 ","End":"15:00.430","Text":"Now, derivative of the numerator,"},{"Start":"15:00.430 ","End":"15:03.370","Text":"I have minus 1 over x squared."},{"Start":"15:03.370 ","End":"15:05.575","Text":"We\u0027ve done this 1 before and here,"},{"Start":"15:05.575 ","End":"15:07.150","Text":"minus 1 over x,"},{"Start":"15:07.150 ","End":"15:09.235","Text":"that\u0027s the derivative of the numerator,"},{"Start":"15:09.235 ","End":"15:11.110","Text":"times the denominator,"},{"Start":"15:11.110 ","End":"15:13.060","Text":"part of the quotient rule,"},{"Start":"15:13.060 ","End":"15:17.720","Text":"minus the other way round numerator."},{"Start":"15:19.410 ","End":"15:21.610","Text":"I\u0027ll just put it in brackets."},{"Start":"15:21.610 ","End":"15:27.980","Text":"Numerator left as is minus 1 over x minus natural log of x."},{"Start":"15:28.350 ","End":"15:31.375","Text":"The derivative of the denominator,"},{"Start":"15:31.375 ","End":"15:33.250","Text":"just e^ x, 1 of those,"},{"Start":"15:33.250 ","End":"15:35.990","Text":"that\u0027s its own derivative."},{"Start":"15:36.570 ","End":"15:42.850","Text":"Once again, I can divide top and bottom by e^ x so I just knock out the 2 here."},{"Start":"15:42.850 ","End":"15:45.295","Text":"Let\u0027s see what I\u0027m left with."},{"Start":"15:45.295 ","End":"15:54.925","Text":"Minus 1 over x squared minus 1 over x minus 1 over x is minus 2 over x."},{"Start":"15:54.925 ","End":"16:04.060","Text":"Then minus minus is plus natural log x over e^ x."},{"Start":"16:04.060 ","End":"16:07.720","Text":"Suppose we could write this differently."},{"Start":"16:07.720 ","End":"16:08.950","Text":"I don\u0027t know if it\u0027ll be simpler."},{"Start":"16:08.950 ","End":"16:11.170","Text":"Multiply top and bottom by x squared,"},{"Start":"16:11.170 ","End":"16:12.610","Text":"maybe right."},{"Start":"16:12.610 ","End":"16:17.125","Text":"This is e^ minus x. I think this is good enough."},{"Start":"16:17.125 ","End":"16:22.370","Text":"I declare that we\u0027ve done all 4."}],"ID":10458}],"Thumbnail":null,"ID":8711},{"Name":"Trigonometric Derivatives","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Derivative of Trigonometric Functions","Duration":"2m 16s","ChapterTopicVideoID":10142,"CourseChapterTopicPlaylistID":8712,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"I\u0027ll finish off here with some trigonometric rules,"},{"Start":"00:03.000 ","End":"00:04.960","Text":"there will be 4 of these."},{"Start":"00:04.960 ","End":"00:10.814","Text":"We\u0027ll talk about y equals sine x,"},{"Start":"00:10.814 ","End":"00:14.805","Text":"y equals cosine x,"},{"Start":"00:14.805 ","End":"00:20.670","Text":"y equals tangent x"},{"Start":"00:20.670 ","End":"00:25.755","Text":"and y equals cotangent of x,"},{"Start":"00:25.755 ","End":"00:32.250","Text":"4 most basic and we\u0027ll settle for these 4."},{"Start":"00:32.250 ","End":"00:34.260","Text":"Okay, I\u0027ll just give them to you,"},{"Start":"00:34.260 ","End":"00:37.980","Text":"there\u0027s nothing more to be done,"},{"Start":"00:37.980 ","End":"00:39.990","Text":"I just tell you sine x,"},{"Start":"00:39.990 ","End":"00:47.400","Text":"its derivative is cosine x. Y equals cosine x,"},{"Start":"00:47.400 ","End":"00:50.770","Text":"its derivative is minus sine x."},{"Start":"00:50.770 ","End":"00:53.170","Text":"If it\u0027s tangent x,"},{"Start":"00:53.170 ","End":"01:01.740","Text":"we derive it and get 1 over cosine squared x"},{"Start":"01:01.740 ","End":"01:03.880","Text":"and if y is cotangent x,"},{"Start":"01:03.880 ","End":"01:11.990","Text":"then y prime is 1 over sine squared x,"},{"Start":"01:11.990 ","End":"01:15.450","Text":"not quite, there\u0027s a minus in front of it."},{"Start":"01:16.410 ","End":"01:19.540","Text":"There are hundreds more rules,"},{"Start":"01:19.540 ","End":"01:24.835","Text":"but these are the main ones for the most basic functions."},{"Start":"01:24.835 ","End":"01:28.585","Text":"What we\u0027ll do in the next clip is talk"},{"Start":"01:28.585 ","End":"01:32.169","Text":"about how to combine some of these basic functions."},{"Start":"01:32.169 ","End":"01:37.330","Text":"For example, we might want the sum or product,"},{"Start":"01:37.330 ","End":"01:40.690","Text":"the difference, the quotient of some functions."},{"Start":"01:40.690 ","End":"01:44.875","Text":"We would want the function to the power of a function,"},{"Start":"01:44.875 ","End":"01:47.770","Text":"composition of functions, a function of"},{"Start":"01:47.770 ","End":"01:50.285","Text":"a function and so forth."},{"Start":"01:50.285 ","End":"01:52.360","Text":"There are the basic rules to some of"},{"Start":"01:52.360 ","End":"01:55.060","Text":"the basic functions and then there are rules which are more general,"},{"Start":"01:55.060 ","End":"01:58.210","Text":"which say how to combine 2 or more."},{"Start":"01:58.210 ","End":"02:01.870","Text":"There will be a rule for example,"},{"Start":"02:01.870 ","End":"02:04.990","Text":"for a product of 2 functions where if we know"},{"Start":"02:04.990 ","End":"02:06.460","Text":"the derivative of each of them,"},{"Start":"02:06.460 ","End":"02:10.065","Text":"how we can get the derivative of the product and so on."},{"Start":"02:10.065 ","End":"02:15.420","Text":"But meanwhile, for this clip, we\u0027re done here."}],"ID":10446},{"Watched":false,"Name":"Exercise 1- Parts 1-6","Duration":"13m 34s","ChapterTopicVideoID":10140,"CourseChapterTopicPlaylistID":8712,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In these exercises, there\u0027s 12 of them."},{"Start":"00:03.750 ","End":"00:07.155","Text":"We have to find the first derivative of the function."},{"Start":"00:07.155 ","End":"00:09.540","Text":"It just means differentiate them."},{"Start":"00:09.540 ","End":"00:12.000","Text":"Let\u0027s go. Number 1,"},{"Start":"00:12.000 ","End":"00:14.160","Text":"f of x equals sine x."},{"Start":"00:14.160 ","End":"00:18.780","Text":"Well, this is one of the basics and it comes from a formula that you should"},{"Start":"00:18.780 ","End":"00:25.800","Text":"remember that the derivative of sine x is cosine x."},{"Start":"00:25.800 ","End":"00:27.390","Text":"It\u0027s a fundamental,"},{"Start":"00:27.390 ","End":"00:34.270","Text":"and so I think to do f prime of x is cosine x."},{"Start":"00:34.550 ","End":"00:37.110","Text":"Next, number 2,"},{"Start":"00:37.110 ","End":"00:39.330","Text":"g of x is sine 4x."},{"Start":"00:39.330 ","End":"00:42.010","Text":"Let\u0027s scroll up a bit."},{"Start":"00:49.390 ","End":"00:52.590","Text":"That\u0027s number 2,"},{"Start":"00:54.520 ","End":"00:59.580","Text":"g of x is sine 4x."},{"Start":"01:00.710 ","End":"01:03.345","Text":"Different letter g instead of f,"},{"Start":"01:03.345 ","End":"01:06.830","Text":"no point getting used to just f all the time."},{"Start":"01:07.130 ","End":"01:10.515","Text":"Here we have the chain rule."},{"Start":"01:10.515 ","End":"01:13.705","Text":"If I was to write it specifically for sine,"},{"Start":"01:13.705 ","End":"01:18.300","Text":"then I would say that if I have sine of"},{"Start":"01:18.300 ","End":"01:25.005","Text":"something box then its derivative it\u0027s not exactly cosine box."},{"Start":"01:25.005 ","End":"01:29.150","Text":"We also have to multiply by the internal derivative,"},{"Start":"01:29.150 ","End":"01:31.895","Text":"which is box prime."},{"Start":"01:31.895 ","End":"01:36.700","Text":"In our case, we have that the box is 4x."},{"Start":"01:36.700 ","End":"01:40.120","Text":"What\u0027s inside here? That\u0027s 4x."},{"Start":"01:41.150 ","End":"01:50.205","Text":"We get that g prime of x is first of all the cosine of the 4x,"},{"Start":"01:50.205 ","End":"01:54.375","Text":"and I\u0027ll put brackets just to separate things."},{"Start":"01:54.375 ","End":"01:59.160","Text":"Then the derivative of the 4x; the internal,"},{"Start":"01:59.160 ","End":"02:01.785","Text":"is 4 times 4,"},{"Start":"02:01.785 ","End":"02:08.280","Text":"and perhaps better written simply as 4 cosine 4x."},{"Start":"02:08.280 ","End":"02:11.370","Text":"Slightly neater that way."},{"Start":"02:11.370 ","End":"02:14.865","Text":"Now, number 3."},{"Start":"02:14.865 ","End":"02:20.350","Text":"Number 3 is cosine of 0.5x."},{"Start":"02:26.810 ","End":"02:31.940","Text":"I should give it a letter really."},{"Start":"02:31.940 ","End":"02:35.675","Text":"I didn\u0027t give a name of the function,"},{"Start":"02:35.675 ","End":"02:41.940","Text":"so let\u0027s just say y equals just to be better that way."},{"Start":"02:41.940 ","End":"02:44.805","Text":"Y prime is equal to."},{"Start":"02:44.805 ","End":"02:46.820","Text":"Now, again, we have a thing with a box."},{"Start":"02:46.820 ","End":"02:52.230","Text":"Since the derivative of cosine is minus sine,"},{"Start":"02:52.250 ","End":"02:58.455","Text":"cosine x derivative is minus sine x."},{"Start":"02:58.455 ","End":"03:00.735","Text":"If we use the chain rule,"},{"Start":"03:00.735 ","End":"03:07.260","Text":"then the cosine of box derivative is"},{"Start":"03:07.260 ","End":"03:14.465","Text":"minus sine of box times the internal derivative of that."},{"Start":"03:14.465 ","End":"03:17.615","Text":"In this case; the cosine,"},{"Start":"03:17.615 ","End":"03:22.260","Text":"we have minus sine of the same thing,"},{"Start":"03:23.360 ","End":"03:30.320","Text":"but we have to multiply by the internal derivative and the derivative of"},{"Start":"03:30.320 ","End":"03:39.550","Text":"0.5x it\u0027s a like a constant times x is just that constant, times 0.5."},{"Start":"03:39.740 ","End":"03:42.060","Text":"Again, this is better written."},{"Start":"03:42.060 ","End":"03:44.990","Text":"Probably, it just looks a bit neater if I write minus"},{"Start":"03:44.990 ","End":"03:53.130","Text":"0.5 sine of 0.5x."},{"Start":"03:53.390 ","End":"03:56.550","Text":"That\u0027s number 3. Now,"},{"Start":"03:56.550 ","End":"04:01.000","Text":"let\u0027s get on to number 4."},{"Start":"04:02.000 ","End":"04:08.080","Text":"Number 4, y equals sine squared x."},{"Start":"04:10.760 ","End":"04:19.120","Text":"Now, remember that sine squared x is sine of x all squared."},{"Start":"04:20.780 ","End":"04:24.690","Text":"Because the notation can be confusing,"},{"Start":"04:24.750 ","End":"04:27.580","Text":"the squared although it\u0027s written here,"},{"Start":"04:27.580 ","End":"04:31.334","Text":"really means the sine of x all squared."},{"Start":"04:31.334 ","End":"04:34.855","Text":"Again, we could use the chain rule,"},{"Start":"04:34.855 ","End":"04:39.140","Text":"but this time the form is of the square function."},{"Start":"04:39.140 ","End":"04:48.545","Text":"If the derivative of x squared is 2x,"},{"Start":"04:48.545 ","End":"04:51.500","Text":"then the derivative of something else squared;"},{"Start":"04:51.500 ","End":"04:56.265","Text":"box squared derivative will be twice that box,"},{"Start":"04:56.265 ","End":"05:01.970","Text":"but I also have to multiply by the derivative of the internal function."},{"Start":"05:01.970 ","End":"05:06.090","Text":"In this case, the box would be sine x."},{"Start":"05:06.920 ","End":"05:12.975","Text":"What I get here is y prime is 2 sine x;"},{"Start":"05:12.975 ","End":"05:18.825","Text":"from the box squared is twice box 2 sine x."},{"Start":"05:18.825 ","End":"05:25.335","Text":"2 sine x,"},{"Start":"05:25.335 ","End":"05:29.290","Text":"but then times the box prime."},{"Start":"05:30.140 ","End":"05:34.425","Text":"That\u0027s the internal derivative which is sine x."},{"Start":"05:34.425 ","End":"05:36.360","Text":"Let\u0027s put brackets here,"},{"Start":"05:36.360 ","End":"05:39.105","Text":"2 times cosine x."},{"Start":"05:39.105 ","End":"05:40.920","Text":"That was from a previous formula,"},{"Start":"05:40.920 ","End":"05:43.515","Text":"the derivative of sine x is cosine x."},{"Start":"05:43.515 ","End":"05:46.005","Text":"That\u0027s the answer, 2 sine x cosine"},{"Start":"05:46.005 ","End":"05:50.265","Text":"x. I could do it without the brackets, it doesn\u0027t matter."},{"Start":"05:50.265 ","End":"05:54.460","Text":"Next is number 5."},{"Start":"05:56.600 ","End":"06:07.720","Text":"Number 5 is that f of x is equal to cosine^ 5x."},{"Start":"06:18.800 ","End":"06:24.505","Text":"The 4 here actually means the whole thing to the fourth; cosine of 5x^4."},{"Start":"06:24.505 ","End":"06:26.920","Text":"This is a really good example of"},{"Start":"06:26.920 ","End":"06:30.425","Text":"the chain rule because actually it\u0027s the chain within the chain."},{"Start":"06:30.425 ","End":"06:32.325","Text":"There are actually 3 functions."},{"Start":"06:32.325 ","End":"06:35.690","Text":"We start with the x, then Phi of x is a function."},{"Start":"06:35.690 ","End":"06:37.780","Text":"We take the cosine of Phi of x,"},{"Start":"06:37.780 ","End":"06:40.295","Text":"and then we raise all that to the fourth."},{"Start":"06:40.295 ","End":"06:46.030","Text":"Let me see if we can do it now without all these intermediary steps."},{"Start":"06:46.030 ","End":"06:48.805","Text":"Just go straight at it."},{"Start":"06:48.805 ","End":"06:51.455","Text":"F prime of x,"},{"Start":"06:51.455 ","End":"06:53.130","Text":"we go from the outside."},{"Start":"06:53.130 ","End":"06:57.080","Text":"in The outside function is to the power of 4."},{"Start":"06:57.080 ","End":"06:59.120","Text":"Just like it was x^4,"},{"Start":"06:59.120 ","End":"07:00.860","Text":"it would be 4x cubed."},{"Start":"07:00.860 ","End":"07:07.590","Text":"We start off with 4 times something cubed."},{"Start":"07:07.590 ","End":"07:16.560","Text":"The something is the derivative of cosine 5x,"},{"Start":"07:16.560 ","End":"07:24.675","Text":"so we take the cosine 5x derivative."},{"Start":"07:24.675 ","End":"07:27.105","Text":"I better put that first derivative,"},{"Start":"07:27.105 ","End":"07:30.395","Text":"and this whole thing is going to be cubed."},{"Start":"07:30.395 ","End":"07:34.040","Text":"Instead of just 4x cubed,"},{"Start":"07:34.040 ","End":"07:38.690","Text":"it\u0027s 4 of the inside derivative cubed."},{"Start":"07:38.690 ","End":"07:41.600","Text":"Now we have to open it up layer by layer."},{"Start":"07:41.600 ","End":"07:44.660","Text":"The next layer to open up is to find what is"},{"Start":"07:44.660 ","End":"07:49.050","Text":"the derivative of cosine of 5x and go more inwardly."},{"Start":"07:50.740 ","End":"07:56.475","Text":"I\u0027ll just write the formulas that I\u0027m using without the box,"},{"Start":"07:56.475 ","End":"08:01.670","Text":"then if we have x^4 its derivative is 4x cubed."},{"Start":"08:01.670 ","End":"08:05.119","Text":"I\u0027ll just put an arrow for a moment for derivative."},{"Start":"08:05.119 ","End":"08:09.710","Text":"The next thing we need to know is that we\u0027re"},{"Start":"08:09.710 ","End":"08:13.730","Text":"going to have the derivative of cosine Phi of x,"},{"Start":"08:13.730 ","End":"08:16.235","Text":"so we\u0027re first going to have a derivative of cosine."},{"Start":"08:16.235 ","End":"08:22.685","Text":"A derivative of cosine as we said before is minus sine,"},{"Start":"08:22.685 ","End":"08:26.330","Text":"but it\u0027s not x, it\u0027s 5x."},{"Start":"08:26.330 ","End":"08:35.790","Text":"So it\u0027s 4 times."},{"Start":"08:39.920 ","End":"08:48.150","Text":"I know all this is going to be cubed, minus sine 5x."},{"Start":"08:51.530 ","End":"08:56.595","Text":"Then all these times the derivative of 5x;"},{"Start":"08:56.595 ","End":"09:02.110","Text":"the internal derivative, and still cubed."},{"Start":"09:02.300 ","End":"09:07.105","Text":"Now, the derivative of 5x is just 5."},{"Start":"09:07.105 ","End":"09:09.245","Text":"If this is just 5,"},{"Start":"09:09.245 ","End":"09:12.985","Text":"I can start now working my way out again."},{"Start":"09:12.985 ","End":"09:17.780","Text":"It\u0027s 4 times"},{"Start":"09:19.200 ","End":"09:29.120","Text":"minus sine 5x times 5."},{"Start":"09:29.120 ","End":"09:33.845","Text":"All this cubed and now there\u0027s no more derivatives so we just expand."},{"Start":"09:33.845 ","End":"09:35.420","Text":"This is equal too."},{"Start":"09:35.420 ","End":"09:38.990","Text":"Now a minus to the power of 3 is still a minus,"},{"Start":"09:38.990 ","End":"09:45.885","Text":"so I can put that minus right at the front and the 4 can go not with the 5."},{"Start":"09:45.885 ","End":"09:51.585","Text":"Here we have 5 cubed times sine 5x cubed. You know what?"},{"Start":"09:51.585 ","End":"09:53.570","Text":"I won\u0027t expand it just yet,"},{"Start":"09:53.570 ","End":"09:56.000","Text":"I\u0027ll write minus, then the 4,"},{"Start":"09:56.000 ","End":"09:58.115","Text":"then the 5 cubed,"},{"Start":"09:58.115 ","End":"10:02.000","Text":"and then I have the Sine 5x cubed."},{"Start":"10:07.280 ","End":"10:10.830","Text":"If I just write it out,"},{"Start":"10:10.830 ","End":"10:16.875","Text":"5 cubed is 125."},{"Start":"10:16.875 ","End":"10:19.920","Text":"Times 4, makes it 500."},{"Start":"10:19.920 ","End":"10:23.020","Text":"It\u0027s minus 500."},{"Start":"10:23.810 ","End":"10:28.670","Text":"Sine 5x cubed is usually written as you\u0027ve seen above,"},{"Start":"10:28.670 ","End":"10:31.770","Text":"as sine cubed 5x."},{"Start":"10:32.180 ","End":"10:37.165","Text":"That\u0027s the derivative in number 5 and that was a bit of work,"},{"Start":"10:37.165 ","End":"10:38.960","Text":"so I\u0027ll just go over it again."},{"Start":"10:38.960 ","End":"10:41.315","Text":"We have cosine something to the fourth,"},{"Start":"10:41.315 ","End":"10:44.165","Text":"so it\u0027s 4 something cubed."},{"Start":"10:44.165 ","End":"10:46.080","Text":"Then we have a cosine,"},{"Start":"10:46.080 ","End":"10:51.530","Text":"so we get minus sine but again we have an inner derivative and that"},{"Start":"10:51.530 ","End":"10:55.070","Text":"5x has to also be differentiated and"},{"Start":"10:55.070 ","End":"10:59.970","Text":"it\u0027s 5 and then we just do the math and expand it out."},{"Start":"11:01.360 ","End":"11:06.300","Text":"The next one will be number 6."},{"Start":"11:10.880 ","End":"11:14.055","Text":"Where is number 6? I\u0027m looking at it."},{"Start":"11:14.055 ","End":"11:18.720","Text":"It\u0027s this time an unusual name for a function, it\u0027s z."},{"Start":"11:18.720 ","End":"11:21.570","Text":"For English speakers and for Americans,"},{"Start":"11:21.570 ","End":"11:24.330","Text":"it\u0027s z. I\u0027ll use the American, call it z."},{"Start":"11:24.330 ","End":"11:33.120","Text":"Z of x is equal to the square root of sine 2x."},{"Start":"11:36.440 ","End":"11:40.690","Text":"Again, I\u0027m not going to write down all these little boxes,"},{"Start":"11:40.690 ","End":"11:42.820","Text":"they might even be more confusing than they are."},{"Start":"11:42.820 ","End":"11:43.840","Text":"As far as I\u0027m concerned,"},{"Start":"11:43.840 ","End":"11:47.065","Text":"it\u0027s easier to go straight at it with the chain rule."},{"Start":"11:47.065 ","End":"11:50.810","Text":"The first thing to see is it\u0027s a square root."},{"Start":"11:51.020 ","End":"11:56.970","Text":"If I just had a plain square root of x;"},{"Start":"11:56.970 ","End":"12:00.740","Text":"and not a square root because it\u0027s something complicated and I took its derivative,"},{"Start":"12:00.740 ","End":"12:04.795","Text":"it\u0027s well-known that it\u0027s 1 over twice square root of x."},{"Start":"12:04.795 ","End":"12:06.400","Text":"If you don\u0027t remember,"},{"Start":"12:06.400 ","End":"12:10.675","Text":"you can always compute it at the side as x^1/2."},{"Start":"12:10.675 ","End":"12:13.585","Text":"I\u0027m not going to do this, I\u0027m Just going to quote it."},{"Start":"12:13.585 ","End":"12:17.075","Text":"What we do in our case is we have the square root."},{"Start":"12:17.075 ","End":"12:24.510","Text":"We have 1 over twice the square root of sine 2x."},{"Start":"12:24.510 ","End":"12:28.770","Text":"It\u0027s just emulating this but with sine 2x instead of x,"},{"Start":"12:28.770 ","End":"12:36.555","Text":"but now we have to multiply by the inner function\u0027s derivative and we have sine 2x."},{"Start":"12:36.555 ","End":"12:39.255","Text":"Now sine 2x if you differentiate it,"},{"Start":"12:39.255 ","End":"12:46.290","Text":"the sine, if it was just plain sine x that would go to cosine x."},{"Start":"12:46.290 ","End":"12:47.880","Text":"That\u0027s a start for us."},{"Start":"12:47.880 ","End":"12:51.570","Text":"That times cosine 2x,"},{"Start":"12:51.570 ","End":"12:52.830","Text":"but it isn\u0027t x."},{"Start":"12:52.830 ","End":"12:55.880","Text":"It\u0027s 2x, so we still have an internal function which is the"},{"Start":"12:55.880 ","End":"12:59.815","Text":"2x and then we multiply all that by 2."},{"Start":"12:59.815 ","End":"13:04.295","Text":"If we just put all this together as a nice fraction,"},{"Start":"13:04.295 ","End":"13:07.475","Text":"we get that this is equal to,"},{"Start":"13:07.475 ","End":"13:10.650","Text":"put one fraction sine."},{"Start":"13:11.800 ","End":"13:16.320","Text":"The 2 in the numerator,"},{"Start":"13:16.320 ","End":"13:18.195","Text":"2 in the denominator cancel,"},{"Start":"13:18.195 ","End":"13:22.935","Text":"so we have cosine 2x over"},{"Start":"13:22.935 ","End":"13:29.235","Text":"the square root of sine 2x."},{"Start":"13:29.235 ","End":"13:32.950","Text":"That\u0027s the answer for number 6."}],"ID":10447},{"Watched":false,"Name":"Exercise 1 - Parts 7-12","Duration":"21m 8s","ChapterTopicVideoID":10141,"CourseChapterTopicPlaylistID":8712,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"In this set of exercises and we just finished number 6,"},{"Start":"00:04.680 ","End":"00:07.725","Text":"so let\u0027s continue with the number 7,"},{"Start":"00:07.725 ","End":"00:11.100","Text":"which is y equals"},{"Start":"00:11.100 ","End":"00:18.435","Text":"sine x cosine 3x,"},{"Start":"00:18.435 ","End":"00:26.740","Text":"and I put a little dot there because it\u0027s a multiplication."},{"Start":"00:26.780 ","End":"00:30.615","Text":"This is the product rule."},{"Start":"00:30.615 ","End":"00:32.010","Text":"Let me scroll up first bit,"},{"Start":"00:32.010 ","End":"00:34.365","Text":"give us some space."},{"Start":"00:34.365 ","End":"00:36.855","Text":"We\u0027re going to need some formulae,"},{"Start":"00:36.855 ","End":"00:41.020","Text":"so just draw little margin here."},{"Start":"00:42.200 ","End":"00:47.630","Text":"We\u0027re going to need to remember the product formula where"},{"Start":"00:47.630 ","End":"00:57.305","Text":"f times g prime is f prime g plus fg prime."},{"Start":"00:57.305 ","End":"01:03.590","Text":"I\u0027ll also remind you that the derivative of sine is cosine,"},{"Start":"01:03.590 ","End":"01:12.135","Text":"and that the derivative of cosine is minus sine."},{"Start":"01:12.135 ","End":"01:15.400","Text":"These are the only things you really need."},{"Start":"01:15.400 ","End":"01:23.245","Text":"A chain rule will do automatically without specifically writing it."},{"Start":"01:23.245 ","End":"01:26.180","Text":"We need y prime."},{"Start":"01:26.180 ","End":"01:29.150","Text":"Now, the first thing we see is the product."},{"Start":"01:29.150 ","End":"01:32.930","Text":"I\u0027ll even put brackets to emphasize that this is going to be f and this is going"},{"Start":"01:32.930 ","End":"01:38.340","Text":"to be g. I\u0027ll even indicate that here. The can be clearer."},{"Start":"01:38.340 ","End":"01:47.280","Text":"F prime is the derivative of sine x and cosine x times g,"},{"Start":"01:47.280 ","End":"01:53.520","Text":"which is cosine 3x plus f,"},{"Start":"01:53.520 ","End":"01:56.080","Text":"which is sine x,"},{"Start":"01:56.620 ","End":"01:59.179","Text":"and then g prime."},{"Start":"01:59.179 ","End":"02:01.910","Text":"Now you would think that the derivative of cosine is"},{"Start":"02:01.910 ","End":"02:10.640","Text":"minus sine and in a way it\u0027s close to minus sine 3x,"},{"Start":"02:10.640 ","End":"02:12.455","Text":"except that because it\u0027s 3x,"},{"Start":"02:12.455 ","End":"02:15.460","Text":"instead of x, we need an internal derivative."},{"Start":"02:15.460 ","End":"02:21.970","Text":"The internal derivative of 3x is just 3 times 3."},{"Start":"02:23.270 ","End":"02:27.800","Text":"Now let\u0027s just collect things together with a bit of algebra."},{"Start":"02:27.800 ","End":"02:32.010","Text":"Here we have cosine x cosine 3x."},{"Start":"02:32.390 ","End":"02:38.165","Text":"Next, but they even like brackets because otherwise you don\u0027t know where 1 thing begins."},{"Start":"02:38.165 ","End":"02:40.190","Text":"I\u0027ll write it in brackets."},{"Start":"02:40.190 ","End":"02:42.530","Text":"Although mostly it\u0027s optional."},{"Start":"02:42.530 ","End":"02:46.625","Text":"Cosine 3x plus,"},{"Start":"02:46.625 ","End":"02:52.755","Text":"now here we have sine x minus sine 3x."},{"Start":"02:52.755 ","End":"02:58.185","Text":"It\u0027s minus sine squared without even a plus at all."},{"Start":"02:58.185 ","End":"03:01.690","Text":"Let me just change that to a minus,"},{"Start":"03:01.700 ","End":"03:07.420","Text":"there that\u0027s the minus from the sine x, sine 3x."},{"Start":"03:14.510 ","End":"03:23.220","Text":"Then we\u0027re going to have another minus with this 3 minus 3 sine x."},{"Start":"03:27.050 ","End":"03:31.460","Text":"That\u0027s basically it might have even left it the"},{"Start":"03:31.460 ","End":"03:35.630","Text":"way it was in the line above, that was 7."},{"Start":"03:35.630 ","End":"03:42.360","Text":"Next we have number 8,"},{"Start":"03:42.360 ","End":"03:44.130","Text":"just scroll up a bit."},{"Start":"03:44.130 ","End":"03:54.615","Text":"Number 8 is y equals,"},{"Start":"03:54.615 ","End":"04:07.330","Text":"so fraction, and it\u0027s sine x minus 1 over"},{"Start":"04:09.530 ","End":"04:14.800","Text":"cosine 2x plus 2."},{"Start":"04:16.340 ","End":"04:27.285","Text":"The derivative of sine and cosine I have already written here,"},{"Start":"04:27.285 ","End":"04:33.170","Text":"what I\u0027m missing is the quotient rule that f over"},{"Start":"04:33.170 ","End":"04:41.465","Text":"g prime is equal to f"},{"Start":"04:41.465 ","End":"04:51.590","Text":"prime times g minus fg prime over g squared."},{"Start":"04:51.590 ","End":"04:56.545","Text":"Let\u0027s apply that here and get what y prime is equal to,"},{"Start":"04:56.545 ","End":"05:00.785","Text":"so we have the derivative of the numerator."},{"Start":"05:00.785 ","End":"05:05.360","Text":"Sine gives us cosine and the 1 gives nothing,"},{"Start":"05:05.360 ","End":"05:08.825","Text":"so that\u0027s just cosine x."},{"Start":"05:08.825 ","End":"05:11.570","Text":"That\u0027s the f prime times g,"},{"Start":"05:11.570 ","End":"05:13.330","Text":"which is a denominator,"},{"Start":"05:13.330 ","End":"05:20.990","Text":"cosine 2x plus 2 minus f,"},{"Start":"05:20.990 ","End":"05:30.620","Text":"which is sine x minus 1 and g prime,"},{"Start":"05:30.620 ","End":"05:38.460","Text":"g prime is the derivative of the denominator."},{"Start":"05:38.460 ","End":"05:42.400","Text":"I might need a little bit more room here, hang on."},{"Start":"05:42.400 ","End":"05:44.580","Text":"Now a bit more space."},{"Start":"05:44.580 ","End":"05:47.250","Text":"We were up to the f and now we need"},{"Start":"05:47.250 ","End":"05:53.535","Text":"the g prime and g prime is the derivative of the denominator."},{"Start":"05:53.535 ","End":"05:56.550","Text":"Since it\u0027s a plus 2 each 1 separately,"},{"Start":"05:56.550 ","End":"05:58.890","Text":"the 2 gives nothing so that we can ignore."},{"Start":"05:58.890 ","End":"06:01.660","Text":"We just need the derivative of cosine 2x."},{"Start":"06:01.660 ","End":"06:03.610","Text":"That was plain cosine."},{"Start":"06:03.610 ","End":"06:12.630","Text":"We\u0027d have minus sine of 2x."},{"Start":"06:12.630 ","End":"06:18.230","Text":"But it\u0027s not x, it\u0027s 2x so we have to take the internal function derivative,"},{"Start":"06:18.230 ","End":"06:20.880","Text":"which is a derivative of 2x,"},{"Start":"06:20.880 ","End":"06:24.790","Text":"which is times 2."},{"Start":"06:26.780 ","End":"06:31.280","Text":"Of course lastly, the g squared,"},{"Start":"06:31.280 ","End":"06:37.595","Text":"which is cosine 2x plus 2 all squared."},{"Start":"06:37.595 ","End":"06:41.210","Text":"Now let\u0027s see if we can simplify any of this stuff."},{"Start":"06:41.210 ","End":"06:44.405","Text":"I\u0027ll just expand it out and then we\u0027ll see where we are."},{"Start":"06:44.405 ","End":"06:51.920","Text":"Cosine x cosine 2x plus 2"},{"Start":"06:51.920 ","End":"06:56.060","Text":"cosine x minus"},{"Start":"06:56.060 ","End":"07:02.700","Text":"sine x."},{"Start":"07:02.700 ","End":"07:05.950","Text":"The minus with the 2 is a minus 2."},{"Start":"07:06.220 ","End":"07:09.540","Text":"Let me just write something at the side."},{"Start":"07:10.360 ","End":"07:18.815","Text":"What we have beside the sine x minus 1 is minus and a minus is a plus."},{"Start":"07:18.815 ","End":"07:21.860","Text":"What we really have for this bit,"},{"Start":"07:21.860 ","End":"07:27.845","Text":"and I\u0027ll just mark which I mean this bit here."},{"Start":"07:27.845 ","End":"07:30.230","Text":"If I write it at the side here,"},{"Start":"07:30.230 ","End":"07:31.850","Text":"this minus, minus and 2,"},{"Start":"07:31.850 ","End":"07:33.560","Text":"I can pull together as plus 2."},{"Start":"07:33.560 ","End":"07:37.265","Text":"I\u0027m just writing 2 plus for emphasis, the plus 2."},{"Start":"07:37.265 ","End":"07:40.830","Text":"Then I have sine 2x,"},{"Start":"07:41.900 ","End":"07:48.975","Text":"and then I have sine x minus 1."},{"Start":"07:48.975 ","End":"07:52.995","Text":"If I put all that together,"},{"Start":"07:52.995 ","End":"07:59.290","Text":"then I\u0027ll go back to my regular color."},{"Start":"07:59.290 ","End":"08:05.905","Text":"I\u0027m going to have plus sine 2 sine x time 2x,"},{"Start":"08:05.905 ","End":"08:13.850","Text":"I\u0027m running out of space."},{"Start":"08:13.880 ","End":"08:18.970","Text":"Then finally minus 2 sine 2x,"},{"Start":"08:22.550 ","End":"08:27.614","Text":"I did it, over the same thing again,"},{"Start":"08:27.614 ","End":"08:29.130","Text":"but it was squared,"},{"Start":"08:29.130 ","End":"08:33.840","Text":"so I take cosine 2x plus 2."},{"Start":"08:33.840 ","End":"08:37.915","Text":"Possibly could be some simplifications done here."},{"Start":"08:37.915 ","End":"08:41.085","Text":"Any event this is the derivative and I\u0027ll settle for that."},{"Start":"08:41.085 ","End":"08:43.140","Text":"Let\u0027s call number 8 done,"},{"Start":"08:43.140 ","End":"08:44.730","Text":"and then go on"},{"Start":"08:44.730 ","End":"08:54.855","Text":"to number 9, go up a bit."},{"Start":"08:54.855 ","End":"09:00.100","Text":"Number 9 is y equals"},{"Start":"09:08.780 ","End":"09:15.280","Text":"x cubed times sine 4x."},{"Start":"09:19.230 ","End":"09:23.755","Text":"We just need the product rule again."},{"Start":"09:23.755 ","End":"09:30.655","Text":"Have it handy, is quite quickly f prime g plus f g prime."},{"Start":"09:30.655 ","End":"09:38.510","Text":"We also might have to remember that the derivative of sine x is cosine x."},{"Start":"09:40.290 ","End":"09:43.030","Text":"Back to here,"},{"Start":"09:43.030 ","End":"09:47.110","Text":"what we get is that y prime equals now the first thing"},{"Start":"09:47.110 ","End":"09:50.770","Text":"is that it\u0027s a product at this times this,"},{"Start":"09:50.770 ","End":"09:56.110","Text":"where we call this 1 f and this 1 whole thing g. Let\u0027s start with"},{"Start":"09:56.110 ","End":"10:05.530","Text":"that f prime is x cubed prime which is 3x squared and then times g,"},{"Start":"10:05.530 ","End":"10:12.805","Text":"which is sine 4x plus f which is"},{"Start":"10:12.805 ","End":"10:17.260","Text":"x cubed times the derivative of"},{"Start":"10:17.260 ","End":"10:23.500","Text":"g. The derivative of g is the 1 that\u0027s slightly tricky because you have a tendency,"},{"Start":"10:23.500 ","End":"10:28.660","Text":"when you see sine to put the derivative as cosine 4x."},{"Start":"10:28.660 ","End":"10:32.740","Text":"This is not quite right because this 4x,"},{"Start":"10:32.740 ","End":"10:34.465","Text":"I\u0027ll emphasize it with brackets,"},{"Start":"10:34.465 ","End":"10:38.050","Text":"is the inner function because of the chain rule,"},{"Start":"10:38.050 ","End":"10:41.245","Text":"we have to multiply by the derivative of the inner function."},{"Start":"10:41.245 ","End":"10:47.080","Text":"Derivative of this of course is 4 and that\u0027s basically the answer,"},{"Start":"10:47.080 ","End":"10:50.350","Text":"except that I would put the 4 in front,"},{"Start":"10:50.350 ","End":"10:53.755","Text":"let\u0027s just rewrite it quickly,"},{"Start":"10:53.755 ","End":"10:58.690","Text":"3x squared sine 4x plus"},{"Start":"10:58.690 ","End":"11:04.360","Text":"4x cubed cosine of 4x,"},{"Start":"11:04.360 ","End":"11:08.215","Text":"we need the brackets there just to tidy it up a bit."},{"Start":"11:08.215 ","End":"11:10.480","Text":"After number 9,"},{"Start":"11:10.480 ","End":"11:13.360","Text":"we get number 10,"},{"Start":"11:13.360 ","End":"11:19.825","Text":"and number 10 is"},{"Start":"11:19.825 ","End":"11:23.395","Text":"this times not just y to y of x is"},{"Start":"11:23.395 ","End":"11:28.015","Text":"to remind you once in a while that y is a function of x really."},{"Start":"11:28.015 ","End":"11:32.530","Text":"y of x is equal to natural log of"},{"Start":"11:32.530 ","End":"11:42.280","Text":"the cosine x of cosine and we can\u0027t just write cosine x without the brackets,"},{"Start":"11:42.280 ","End":"11:46.465","Text":"but here they\u0027ve chosen to write with the brackets."},{"Start":"11:46.465 ","End":"11:50.695","Text":"Now here\u0027s clearly a case of the chain rule."},{"Start":"11:50.695 ","End":"11:54.530","Text":"Because we have a function of a function,"},{"Start":"12:00.060 ","End":"12:06.120","Text":"we need to remember the derivatives of natural log and of cosine."},{"Start":"12:06.120 ","End":"12:12.795","Text":"If it\u0027s cosine x derivative it\u0027s minus sine x,"},{"Start":"12:12.795 ","End":"12:15.275","Text":"and if it\u0027s natural log of x,"},{"Start":"12:15.275 ","End":"12:17.395","Text":"derivative is 1 over x."},{"Start":"12:17.395 ","End":"12:24.230","Text":"These are some of the basics that we just have to memorize or look at the formula sheet."},{"Start":"12:24.240 ","End":"12:27.955","Text":"y prime of x."},{"Start":"12:27.955 ","End":"12:31.780","Text":"Now out in the outside we see the natural logarithm,"},{"Start":"12:31.780 ","End":"12:34.045","Text":"we want to put 1 over."},{"Start":"12:34.045 ","End":"12:38.980","Text":"I\u0027ll just leave the 1 out for a second it\u0027s 1 over what\u0027s inside,"},{"Start":"12:38.980 ","End":"12:43.720","Text":"which is cosine of x."},{"Start":"12:43.720 ","End":"12:47.170","Text":"1 over times the internal derivative,"},{"Start":"12:47.170 ","End":"12:50.485","Text":"which is times minus sine x."},{"Start":"12:50.485 ","End":"12:53.410","Text":"But don\u0027t I just skip a step instead of"},{"Start":"12:53.410 ","End":"12:57.295","Text":"saying 1 over this times this will just put that into the numerator."},{"Start":"12:57.295 ","End":"13:02.410","Text":"It\u0027s minus sine x and there\u0027s no need for it."},{"Start":"13:02.410 ","End":"13:04.630","Text":"Well done the brackets there."},{"Start":"13:04.630 ","End":"13:06.160","Text":"We\u0027ll put the brackets there."},{"Start":"13:06.160 ","End":"13:08.725","Text":"I could leave it like that."},{"Start":"13:08.725 ","End":"13:12.940","Text":"But we could also see that the sine over the cosine is"},{"Start":"13:12.940 ","End":"13:19.750","Text":"the tangent after we can write minus tangent of x."},{"Start":"13:19.750 ","End":"13:25.195","Text":"Although I have seen books where instead of tangent they write it as TG."},{"Start":"13:25.195 ","End":"13:29.180","Text":"I think in this course we write it like that."},{"Start":"13:31.050 ","End":"13:35.155","Text":"That\u0027s it for number 10."},{"Start":"13:35.155 ","End":"13:36.850","Text":"Only 2 more to go,"},{"Start":"13:36.850 ","End":"13:40.150","Text":"11 and 12, so we\u0027ll go to 11 next."},{"Start":"13:40.150 ","End":"13:50.600","Text":"In 11, we have y equals"},{"Start":"13:52.380 ","End":"13:56.590","Text":"e to the power of sine"},{"Start":"13:56.590 ","End":"14:05.860","Text":"2x times"},{"Start":"14:05.860 ","End":"14:08.390","Text":"natural log of x,"},{"Start":"14:16.920 ","End":"14:20.590","Text":"let\u0027s continue the line here."},{"Start":"14:20.590 ","End":"14:22.465","Text":"Let see what we\u0027ll need."},{"Start":"14:22.465 ","End":"14:24.025","Text":"Well, obviously it\u0027s a product,"},{"Start":"14:24.025 ","End":"14:27.910","Text":"distinct times distinct, so we need the product rule,"},{"Start":"14:27.910 ","End":"14:34.375","Text":"f g prime is f prime g plus f g prime."},{"Start":"14:34.375 ","End":"14:36.760","Text":"Let\u0027s see what else will need,"},{"Start":"14:36.760 ","End":"14:45.490","Text":"will need e to the power of derivative is the old basics."},{"Start":"14:45.490 ","End":"14:48.650","Text":"We Will need natural log,"},{"Start":"14:49.560 ","End":"14:51.670","Text":"things we should remember,"},{"Start":"14:51.670 ","End":"14:53.755","Text":"but just in case you don\u0027t,"},{"Start":"14:53.755 ","End":"14:57.445","Text":"I\u0027m writing them again for you, 1 over x,"},{"Start":"14:57.445 ","End":"15:01.225","Text":"and we also have a sine function somewhere in there."},{"Start":"15:01.225 ","End":"15:04.810","Text":"The derivative of sine x is"},{"Start":"15:04.810 ","End":"15:10.490","Text":"cosine x. I think these should do as well together with the chain rule."},{"Start":"15:12.630 ","End":"15:19.630","Text":"Calling the first bit f and the second bit we\u0027ll"},{"Start":"15:19.630 ","End":"15:27.080","Text":"call that g and we\u0027ll get that y prime equals f prime,"},{"Start":"15:29.580 ","End":"15:32.335","Text":"we have to use the chain rule here."},{"Start":"15:32.335 ","End":"15:37.340","Text":"The derivative of e to the something is e to the something,"},{"Start":"15:38.220 ","End":"15:42.650","Text":"but times the derivative of that something."},{"Start":"15:44.250 ","End":"15:47.740","Text":"We have to write the derivative of sine 2x."},{"Start":"15:47.740 ","End":"15:51.775","Text":"[LAUGHTER] Now, for the derivative of sine 2x,"},{"Start":"15:51.775 ","End":"15:53.800","Text":"we know it\u0027s a sine of something."},{"Start":"15:53.800 ","End":"16:00.115","Text":"If it\u0027s a sine of something like the sine of 2x and the derivative would be cosine 2x."},{"Start":"16:00.115 ","End":"16:03.970","Text":"But that\u0027s not where it ends because the 2x itself is"},{"Start":"16:03.970 ","End":"16:08.450","Text":"an inner function and we have to take the derivative of it."},{"Start":"16:08.700 ","End":"16:10.885","Text":"e to the sine 2x,"},{"Start":"16:10.885 ","End":"16:13.915","Text":"then this is the inner function, derive it."},{"Start":"16:13.915 ","End":"16:19.345","Text":"Sine is cosine, but there\u0027s still another inner function 2x, that\u0027s 2."},{"Start":"16:19.345 ","End":"16:25.135","Text":"So far, we\u0027ve just done the f prime bit times the g,"},{"Start":"16:25.135 ","End":"16:32.170","Text":"natural log of x plus the other way around."},{"Start":"16:32.170 ","End":"16:34.585","Text":"This 1 stays as it is,"},{"Start":"16:34.585 ","End":"16:41.120","Text":"which is e to the power of sine 2x,"},{"Start":"16:41.160 ","End":"16:44.440","Text":"and g gets derived,"},{"Start":"16:44.440 ","End":"16:48.200","Text":"differentiated, and that\u0027s 1 over x."},{"Start":"16:49.260 ","End":"16:51.940","Text":"This is the answer,"},{"Start":"16:51.940 ","End":"16:54.820","Text":"but customary to simplify."},{"Start":"16:54.820 ","End":"16:59.275","Text":"I mean, right away I see I have e to the sine 2x and both of them."},{"Start":"16:59.275 ","End":"17:01.705","Text":"I\u0027d like to take that outside the brackets."},{"Start":"17:01.705 ","End":"17:10.390","Text":"Let\u0027s take e to the power of sine 2x outside the brackets and see what we\u0027re left with."},{"Start":"17:10.390 ","End":"17:15.620","Text":"Here we have this and this so it\u0027s 2,"},{"Start":"17:15.930 ","End":"17:18.715","Text":"constants usually written upfront."},{"Start":"17:18.715 ","End":"17:21.205","Text":"2 cosine 2x,"},{"Start":"17:21.205 ","End":"17:23.410","Text":"natural log of x,"},{"Start":"17:23.410 ","End":"17:24.820","Text":"that\u0027s the first bit."},{"Start":"17:24.820 ","End":"17:31.480","Text":"The second bit after I\u0027ve taken this factor out is plus 1 over x."},{"Start":"17:31.480 ","End":"17:35.575","Text":"That will be the answer for number 11,"},{"Start":"17:35.575 ","End":"17:38.785","Text":"leaving us just with 12."},{"Start":"17:38.785 ","End":"17:47.920","Text":"There is something wrong with the way 12 is written it seems to be a t"},{"Start":"17:47.920 ","End":"17:57.540","Text":"instead of an x or an x instead of the t. Tell you what we\u0027ll go with t,"},{"Start":"17:57.540 ","End":"18:05.640","Text":"assuming they wanted to use different letters and practice with them."},{"Start":"18:05.640 ","End":"18:10.449","Text":"Let\u0027s say that number 12 is just a typo."},{"Start":"18:10.449 ","End":"18:13.510","Text":"A typist must have missed something."},{"Start":"18:13.510 ","End":"18:17.650","Text":"We\u0027ll take it as number 12,"},{"Start":"18:17.650 ","End":"18:21.745","Text":"will take it as y of t,"},{"Start":"18:21.745 ","End":"18:28.600","Text":"and we\u0027ll put t in the other places is sine of cosine,"},{"Start":"18:29.060 ","End":"18:37.990","Text":"let\u0027s put t. I\u0027ll just emphasize that I\u0027m using letter t rather than x."},{"Start":"18:41.000 ","End":"18:46.665","Text":"The formulas will need basically,"},{"Start":"18:46.665 ","End":"18:51.880","Text":"I mean, that\u0027s the chain rule and the derivative of sine."},{"Start":"18:51.880 ","End":"18:54.565","Text":"If it was sine of plain old x,"},{"Start":"18:54.565 ","End":"18:58.855","Text":"then it would be cosine of x,"},{"Start":"18:58.855 ","End":"19:03.310","Text":"and the derivative of cosine x or"},{"Start":"19:03.310 ","End":"19:08.845","Text":"any letter cosine of whatever"},{"Start":"19:08.845 ","End":"19:15.310","Text":"was q it would be minus sine q or whatever,"},{"Start":"19:15.310 ","End":"19:19.610","Text":"but it\u0027s x. Silly me,"},{"Start":"19:21.800 ","End":"19:26.430","Text":"take the eraser, erase that."},{"Start":"19:26.430 ","End":"19:34.370","Text":"Back to Pen minus sine of x. There we are."},{"Start":"19:34.710 ","End":"19:43.490","Text":"This chain rule, we get that y prime of t is equal to,"},{"Start":"19:43.490 ","End":"19:46.160","Text":"first of all, we derive the sine,"},{"Start":"19:46.160 ","End":"19:50.885","Text":"that\u0027s the outer function and the sine derivative is cosine."},{"Start":"19:50.885 ","End":"19:54.620","Text":"First of all, we have cosine of whatever was inside before,"},{"Start":"19:54.620 ","End":"19:59.990","Text":"which was cosine of t. But then we have to"},{"Start":"19:59.990 ","End":"20:02.890","Text":"multiply by the derivative of the internal function and"},{"Start":"20:02.890 ","End":"20:05.965","Text":"the internal function was cosine t,"},{"Start":"20:05.965 ","End":"20:11.875","Text":"so its derivative is minus sine t. It\u0027s minus times"},{"Start":"20:11.875 ","End":"20:20.690","Text":"minus sine of t and now I\u0027ll just put brackets here too."},{"Start":"20:25.620 ","End":"20:31.570","Text":"That\u0027s basically all 1 can do."},{"Start":"20:31.570 ","End":"20:33.550","Text":"To simplify I mean,"},{"Start":"20:33.550 ","End":"20:35.350","Text":"you could put the minus upfront,"},{"Start":"20:35.350 ","End":"20:39.200","Text":"and maybe it\u0027s getting rid of redundant brackets."},{"Start":"20:39.200 ","End":"20:44.920","Text":"But really what it stays as not even like to write the sine t upfront."},{"Start":"20:44.920 ","End":"20:54.670","Text":"This is minus sine t and then cosine of"},{"Start":"20:54.670 ","End":"21:03.070","Text":"cosine t. That\u0027s it"},{"Start":"21:03.070 ","End":"21:06.740","Text":"for number 12 and that\u0027s the end of the series."}],"ID":10448}],"Thumbnail":null,"ID":8712},{"Name":"Derivative of Power Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Derivative of a Function to the Power of a Function","Duration":"21m 52s","ChapterTopicVideoID":10146,"CourseChapterTopicPlaylistID":8713,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.850","Text":"In this clip, I\u0027m going to show you how to differentiate"},{"Start":"00:03.850 ","End":"00:08.755","Text":"a function of the form f of x to the power of g of x,"},{"Start":"00:08.755 ","End":"00:12.505","Text":"some function of x to the power of another function of x."},{"Start":"00:12.505 ","End":"00:15.115","Text":"We haven\u0027t actually done this till now."},{"Start":"00:15.115 ","End":"00:18.730","Text":"We\u0027ve had stuff like a constant,"},{"Start":"00:18.730 ","End":"00:22.915","Text":"say 4 to the power of a function of x."},{"Start":"00:22.915 ","End":"00:26.650","Text":"For example, 4 to the power of 2x minus 1."},{"Start":"00:26.650 ","End":"00:29.095","Text":"This we know how to do."},{"Start":"00:29.095 ","End":"00:33.550","Text":"We\u0027ve also seen the other way around where we have something on"},{"Start":"00:33.550 ","End":"00:40.944","Text":"the base like cosine x and to the power of a constant,"},{"Start":"00:40.944 ","End":"00:42.820","Text":"like to the power of 4."},{"Start":"00:42.820 ","End":"00:44.620","Text":"This we\u0027ve done, this we\u0027ve done."},{"Start":"00:44.620 ","End":"00:50.340","Text":"What we haven\u0027t done is something which involves both."},{"Start":"00:50.340 ","End":"00:53.775","Text":"Whether the function like 4x plus 1 here,"},{"Start":"00:53.775 ","End":"00:57.245","Text":"and it\u0027s also to the power of something involving x."},{"Start":"00:57.245 ","End":"00:58.865","Text":"This we don\u0027t know how to do."},{"Start":"00:58.865 ","End":"01:02.630","Text":"This is like f of x^g of x and I\u0027ll give another example."},{"Start":"01:02.630 ","End":"01:07.325","Text":"What we haven\u0027t done, just put a little line here."},{"Start":"01:07.325 ","End":"01:09.005","Text":"This we have done,"},{"Start":"01:09.005 ","End":"01:10.445","Text":"this we have done,"},{"Start":"01:10.445 ","End":"01:12.830","Text":"and this we haven\u0027t done."},{"Start":"01:12.830 ","End":"01:16.910","Text":"Sine x to the power of natural log of x,"},{"Start":"01:16.910 ","End":"01:23.295","Text":"for example, we haven\u0027t learned how to do."},{"Start":"01:23.295 ","End":"01:26.055","Text":"Let\u0027s take another example."},{"Start":"01:26.055 ","End":"01:33.730","Text":"Some function of x squared plus 1 to the power of e^x."},{"Start":"01:34.730 ","End":"01:37.790","Text":"This is what I\u0027m going to show you how to do today."},{"Start":"01:37.790 ","End":"01:39.500","Text":"There are several methods,"},{"Start":"01:39.500 ","End":"01:43.220","Text":"and I\u0027ll choose 1 and we\u0027ll do some examples,"},{"Start":"01:43.220 ","End":"01:46.800","Text":"because the formula is a bit cumbersome."},{"Start":"01:46.800 ","End":"01:48.780","Text":"I\u0027ve already given it away."},{"Start":"01:48.780 ","End":"01:51.510","Text":"It\u0027s going to be using a formula."},{"Start":"01:52.580 ","End":"01:57.410","Text":"I\u0027ve erased the examples and now I\u0027m going to show you"},{"Start":"01:57.410 ","End":"02:00.200","Text":"the formula that I started to mention."},{"Start":"02:00.200 ","End":"02:04.615","Text":"There is a formula for solving this sort of thing."},{"Start":"02:04.615 ","End":"02:06.500","Text":"It goes like this."},{"Start":"02:06.500 ","End":"02:10.205","Text":"Let\u0027s start off with first of all, the original form,"},{"Start":"02:10.205 ","End":"02:18.445","Text":"where y is f of x to the power of g of x. I\u0027ll tell you what y\u0027 is."},{"Start":"02:18.445 ","End":"02:23.060","Text":"I\u0027ll do it slowly because I\u0027m hoping you\u0027ll memorize this."},{"Start":"02:23.060 ","End":"02:27.170","Text":"Well, it starts off with the same thing with f of"},{"Start":"02:27.170 ","End":"02:31.625","Text":"x to the power of g of x. That\u0027s the beginning."},{"Start":"02:31.625 ","End":"02:36.800","Text":"Then we open up square brackets or any other shape bracket you like,"},{"Start":"02:36.800 ","End":"02:39.295","Text":"and inside here,"},{"Start":"02:39.295 ","End":"02:44.405","Text":"we write what was in the exponent here, g of x."},{"Start":"02:44.405 ","End":"02:51.510","Text":"Then natural logarithm of what was in the base."},{"Start":"02:53.200 ","End":"02:59.865","Text":"All this, we have to differentiate."},{"Start":"02:59.865 ","End":"03:02.220","Text":"This is not the answer."},{"Start":"03:02.220 ","End":"03:04.780","Text":"This is a template of what to do,"},{"Start":"03:04.780 ","End":"03:06.170","Text":"because after we\u0027ve written this,"},{"Start":"03:06.170 ","End":"03:10.710","Text":"we still have to do the differentiation of what\u0027s in the square brackets."},{"Start":"03:11.110 ","End":"03:15.605","Text":"Let\u0027s use a marker and the function f,"},{"Start":"03:15.605 ","End":"03:17.030","Text":"which is at the bottom."},{"Start":"03:17.030 ","End":"03:18.995","Text":"I\u0027ll do in this green."},{"Start":"03:18.995 ","End":"03:29.195","Text":"It appears here and it also appears inside here with the natural logarithm."},{"Start":"03:29.195 ","End":"03:34.740","Text":"On the other hand, the exponent function g"},{"Start":"03:34.750 ","End":"03:40.790","Text":"that came from here appears again here and also,"},{"Start":"03:40.790 ","End":"03:44.850","Text":"at the very beginning inside the square brackets."},{"Start":"03:45.440 ","End":"03:47.700","Text":"That\u0027s just the formula."},{"Start":"03:47.700 ","End":"03:50.349","Text":"What we still need as an example."},{"Start":"03:50.349 ","End":"03:52.370","Text":"But before the example,"},{"Start":"03:52.370 ","End":"03:53.645","Text":"just once more,"},{"Start":"03:53.645 ","End":"03:57.995","Text":"we have a function to the power of another function and its derivative is,"},{"Start":"03:57.995 ","End":"04:01.055","Text":"we copy the original as is."},{"Start":"04:01.055 ","End":"04:02.960","Text":"Then inside the square brackets,"},{"Start":"04:02.960 ","End":"04:04.790","Text":"we have to differentiate something"},{"Start":"04:04.790 ","End":"04:08.120","Text":"and that something is some kind of scramble of f and g,"},{"Start":"04:08.120 ","End":"04:11.530","Text":"and what it is is the top 1 comes first,"},{"Start":"04:11.530 ","End":"04:16.220","Text":"and secondly, the natural logarithm of the bottom 1."},{"Start":"04:16.220 ","End":"04:20.825","Text":"That\u0027s how you can try and remember it. Onto the example."},{"Start":"04:20.825 ","End":"04:26.115","Text":"The example is y equals 4x plus 1,"},{"Start":"04:26.115 ","End":"04:27.540","Text":"which is a function of x,"},{"Start":"04:27.540 ","End":"04:29.215","Text":"that\u0027s my f of x,"},{"Start":"04:29.215 ","End":"04:31.010","Text":"to the power of x squared,"},{"Start":"04:31.010 ","End":"04:33.650","Text":"which is my g of x."},{"Start":"04:34.250 ","End":"04:38.735","Text":"The derivative y\u0027 will be first of all,"},{"Start":"04:38.735 ","End":"04:41.810","Text":"this thing here copied."},{"Start":"04:42.880 ","End":"04:47.035","Text":"Let\u0027s copy and I\u0027ll also highlight it."},{"Start":"04:47.035 ","End":"04:53.670","Text":"The f of x in green and the x squared in turquoise."},{"Start":"04:53.670 ","End":"05:00.260","Text":"Then we put the square brackets with what\u0027s written in here,"},{"Start":"05:00.260 ","End":"05:03.305","Text":"is to, first of all, take the turquoise,"},{"Start":"05:03.305 ","End":"05:12.920","Text":"which is the x squared times natural log of what was in the green,"},{"Start":"05:12.920 ","End":"05:17.810","Text":"which is f of x, which is 4x plus 1."},{"Start":"05:18.760 ","End":"05:22.220","Text":"Also, I\u0027ll highlight it."},{"Start":"05:22.220 ","End":"05:26.690","Text":"We need to take the derivative of this product."},{"Start":"05:26.690 ","End":"05:28.940","Text":"I\u0027ll go over it again."},{"Start":"05:28.940 ","End":"05:32.845","Text":"We start off with the function to the power of function,"},{"Start":"05:32.845 ","End":"05:36.320","Text":"green to the power of turquoise or whatever,"},{"Start":"05:36.320 ","End":"05:38.135","Text":"the bottoms to the power of top,"},{"Start":"05:38.135 ","End":"05:42.500","Text":"and the derivative is the same bottom to the power of top just copied."},{"Start":"05:42.500 ","End":"05:47.315","Text":"Then there\u0027s a square bracket with a derivative sine and inside it,"},{"Start":"05:47.315 ","End":"05:50.450","Text":"it contains the product of what was in"},{"Start":"05:50.450 ","End":"05:54.680","Text":"the top times the logarithm of what was in the bottom."},{"Start":"05:54.680 ","End":"06:00.710","Text":"You could probably make up some kind of poem or mnemonic for yourself,"},{"Start":"06:00.710 ","End":"06:09.485","Text":"like bottom to the top times the derivative of top times logarithm of bottom,"},{"Start":"06:09.485 ","End":"06:11.880","Text":"or something like that."},{"Start":"06:12.080 ","End":"06:16.635","Text":"Now, we still haven\u0027t done the differentiation there\u0027s a prime here."},{"Start":"06:16.635 ","End":"06:20.480","Text":"There\u0027s actually a couple of formulas I want to bring in,"},{"Start":"06:20.480 ","End":"06:22.850","Text":"in case you\u0027ve forgotten them."},{"Start":"06:22.850 ","End":"06:27.110","Text":"1 of the formulas is the product rule,"},{"Start":"06:27.110 ","End":"06:32.180","Text":"which says that if I have a function of x times a function of x,"},{"Start":"06:32.180 ","End":"06:34.495","Text":"and I want to differentiate that,"},{"Start":"06:34.495 ","End":"06:36.440","Text":"just call them u and v for short,"},{"Start":"06:36.440 ","End":"06:38.300","Text":"I haven\u0027t even put in the x\u0027s,"},{"Start":"06:38.300 ","End":"06:44.390","Text":"but the derivative of a product is the derivative of the first times the second,"},{"Start":"06:44.390 ","End":"06:48.454","Text":"plus the first times the derivative of the second."},{"Start":"06:48.454 ","End":"06:51.380","Text":"That\u0027s the product rule and you should know this very well by now."},{"Start":"06:51.380 ","End":"06:55.100","Text":"The other 1 is the template form of the logarithm rule"},{"Start":"06:55.100 ","End":"06:59.345","Text":"that when we differentiate the logarithm of something\u0027s box,"},{"Start":"06:59.345 ","End":"07:02.420","Text":"then the derivative is 1 over the box,"},{"Start":"07:02.420 ","End":"07:04.100","Text":"but times box prime."},{"Start":"07:04.100 ","End":"07:08.720","Text":"I usually prefer to write it with the box prime up here."},{"Start":"07:08.720 ","End":"07:14.460","Text":"I\u0027ll show you. There, it\u0027s a little bit simpler like this."},{"Start":"07:14.480 ","End":"07:16.595","Text":"Now, back to this."},{"Start":"07:16.595 ","End":"07:22.220","Text":"We were just going to expand y\u0027 to do the differentiation here."},{"Start":"07:22.220 ","End":"07:27.420","Text":"I\u0027ll repeat the y\u0027 equals,"},{"Start":"07:27.420 ","End":"07:29.780","Text":"and I think I\u0027ll stop with the highlighting."},{"Start":"07:29.780 ","End":"07:31.535","Text":"It\u0027s just too much."},{"Start":"07:31.535 ","End":"07:37.160","Text":"Sorry. There. 4x plus 1^x squared,"},{"Start":"07:37.160 ","End":"07:39.950","Text":"that\u0027s just copying the beginning."},{"Start":"07:39.950 ","End":"07:46.310","Text":"Then I\u0027m differentiating and I\u0027m using the product rule as well as this rule."},{"Start":"07:46.310 ","End":"07:47.900","Text":"Here I see it as a product."},{"Start":"07:47.900 ","End":"07:49.180","Text":"This is u, this is v,"},{"Start":"07:49.180 ","End":"07:51.830","Text":"so I need u\u0027 first,"},{"Start":"07:51.830 ","End":"07:53.555","Text":"which is x squared prime,"},{"Start":"07:53.555 ","End":"07:55.405","Text":"which is 2x,"},{"Start":"07:55.405 ","End":"07:57.915","Text":"and the other 1 as is,"},{"Start":"07:57.915 ","End":"08:03.010","Text":"natural log of 4x plus 1."},{"Start":"08:03.070 ","End":"08:07.070","Text":"That\u0027s the first part of the product rule."},{"Start":"08:07.070 ","End":"08:16.160","Text":"Plus u, which is the first part of this product."},{"Start":"08:16.160 ","End":"08:18.310","Text":"This is the x squared."},{"Start":"08:18.310 ","End":"08:24.210","Text":"Well, this is the u and all this natural log for x plus 1,"},{"Start":"08:24.210 ","End":"08:26.250","Text":"that\u0027s v. So it\u0027s x squared is u."},{"Start":"08:26.250 ","End":"08:31.220","Text":"Now, v\u0027 is the derivative of the natural log of this."},{"Start":"08:31.220 ","End":"08:32.720","Text":"I look over here,"},{"Start":"08:32.720 ","End":"08:36.830","Text":"and what I do with this is I make it into a fraction with a derivative on top"},{"Start":"08:36.830 ","End":"08:38.775","Text":"and it itself on the bottom."},{"Start":"08:38.775 ","End":"08:41.255","Text":"This is times,"},{"Start":"08:41.255 ","End":"08:44.670","Text":"here\u0027s the fraction part on the bottom,"},{"Start":"08:44.670 ","End":"08:47.280","Text":"4x plus 1 just as is,"},{"Start":"08:47.280 ","End":"08:49.145","Text":"and on the top, the derivative of that,"},{"Start":"08:49.145 ","End":"08:51.115","Text":"which is just 4."},{"Start":"08:51.115 ","End":"08:52.940","Text":"Now there\u0027s no more prime."},{"Start":"08:52.940 ","End":"08:55.560","Text":"We\u0027ve already done the differentiation and this,"},{"Start":"08:55.560 ","End":"08:57.180","Text":"in fact, is the answer."},{"Start":"08:57.180 ","End":"08:59.870","Text":"Although, personally I would simplify it."},{"Start":"08:59.870 ","End":"09:01.580","Text":"That\u0027s if you have the time."},{"Start":"09:01.580 ","End":"09:05.055","Text":"Technically, this is the solution."},{"Start":"09:05.055 ","End":"09:06.940","Text":"Usually, 1 simplifies."},{"Start":"09:06.940 ","End":"09:09.020","Text":"I won\u0027t do that here."},{"Start":"09:10.240 ","End":"09:18.795","Text":"Second example will be y equals x to the power of 2x."},{"Start":"09:18.795 ","End":"09:23.570","Text":"This will give us the y\u0027 equals."},{"Start":"09:23.570 ","End":"09:29.600","Text":"Once again, we copy the original function as is,"},{"Start":"09:29.600 ","End":"09:31.625","Text":"which is x^2x,"},{"Start":"09:31.625 ","End":"09:34.460","Text":"then we open up a square bracket."},{"Start":"09:34.460 ","End":"09:40.260","Text":"What we do is we put the function at the top,"},{"Start":"09:40.260 ","End":"09:43.130","Text":"which is the 2x in the beginning,"},{"Start":"09:43.130 ","End":"09:47.390","Text":"times the natural logarithm of whatever was at the bottom,"},{"Start":"09:47.390 ","End":"09:51.120","Text":"which is just the natural log of f of x,"},{"Start":"09:51.120 ","End":"09:53.295","Text":"which is the green, well, it\u0027s not colored yet,"},{"Start":"09:53.295 ","End":"09:57.280","Text":"of x, but I will color it now."},{"Start":"09:57.280 ","End":"10:00.335","Text":"There we go. A big oops,"},{"Start":"10:00.335 ","End":"10:02.885","Text":"that I forgot to put the prime here."},{"Start":"10:02.885 ","End":"10:06.920","Text":"It means to differentiate this using the product rule again."},{"Start":"10:06.920 ","End":"10:14.550","Text":"What we get is that y\u0027 is equal to x to the power of 2x."},{"Start":"10:14.550 ","End":"10:17.390","Text":"Then using the product rule,"},{"Start":"10:17.390 ","End":"10:19.640","Text":"we get the derivative of the first,"},{"Start":"10:19.640 ","End":"10:25.070","Text":"which is 2 times the second as is natural log of x,"},{"Start":"10:25.070 ","End":"10:31.490","Text":"plus the first as is times the derivative of natural log of x,"},{"Start":"10:31.490 ","End":"10:34.040","Text":"which is just 1/x,"},{"Start":"10:34.040 ","End":"10:35.975","Text":"and that\u0027s the answer,"},{"Start":"10:35.975 ","End":"10:38.660","Text":"but I would strongly recommend simplification."},{"Start":"10:38.660 ","End":"10:40.399","Text":"Here it begs to be simplified,"},{"Start":"10:40.399 ","End":"10:42.530","Text":"but we\u0027re not into simplification today,"},{"Start":"10:42.530 ","End":"10:45.695","Text":"just the derivative and that\u0027s it."},{"Start":"10:45.695 ","End":"10:50.855","Text":"Now, I remembered there is another way of remembering this formula."},{"Start":"10:50.855 ","End":"10:52.160","Text":"Instead of using colors,"},{"Start":"10:52.160 ","End":"10:53.300","Text":"I could use shapes."},{"Start":"10:53.300 ","End":"10:55.500","Text":"I\u0027ll show you what I mean."},{"Start":"10:56.000 ","End":"10:59.180","Text":"I\u0027ll get us a little bit more space here."},{"Start":"10:59.180 ","End":"11:04.250","Text":"What I could do is think of the green colored as, say,"},{"Start":"11:04.250 ","End":"11:10.290","Text":"a square, and this like my f of x will be written as a square,"},{"Start":"11:10.290 ","End":"11:11.880","Text":"and the g of x, the blue,"},{"Start":"11:11.880 ","End":"11:13.665","Text":"will be written as a triangle."},{"Start":"11:13.665 ","End":"11:17.600","Text":"I can remember this formula as saying"},{"Start":"11:17.600 ","End":"11:23.750","Text":"that the derivative of something to the power of something,"},{"Start":"11:23.750 ","End":"11:27.770","Text":"in this case, square to the power of triangle is,"},{"Start":"11:27.770 ","End":"11:30.845","Text":"and I\u0027m looking at this formula while I\u0027m writing this,"},{"Start":"11:30.845 ","End":"11:35.360","Text":"to say that it\u0027s equal to square to the power of triangle"},{"Start":"11:35.360 ","End":"11:37.790","Text":"or box to the power of pyramid,"},{"Start":"11:37.790 ","End":"11:42.410","Text":"or whatever, times square brackets,"},{"Start":"11:42.410 ","End":"11:47.670","Text":"triangle, natural log of square."},{"Start":"11:49.980 ","End":"11:54.640","Text":"All this, square brackets, prime."},{"Start":"11:54.640 ","End":"11:57.925","Text":"So I now color it,"},{"Start":"11:57.925 ","End":"12:02.710","Text":"and there\u0027s no real difference between writing the formula this way or this way,"},{"Start":"12:02.710 ","End":"12:06.790","Text":"just to some people might be easier when they solve this exercise,"},{"Start":"12:06.790 ","End":"12:09.880","Text":"for example, to see something to the power of something,"},{"Start":"12:09.880 ","End":"12:11.830","Text":"so it\u0027s square to the triangle."},{"Start":"12:11.830 ","End":"12:16.180","Text":"So it\u0027s equals square to the triangle, like here,"},{"Start":"12:16.180 ","End":"12:22.000","Text":"times the derivative of this,"},{"Start":"12:22.000 ","End":"12:24.280","Text":"where here I put the triangle,"},{"Start":"12:24.280 ","End":"12:26.275","Text":"natural log of square."},{"Start":"12:26.275 ","End":"12:28.960","Text":"Then you go ahead and differentiate it as usual,"},{"Start":"12:28.960 ","End":"12:29.980","Text":"and everything\u0027s the same,"},{"Start":"12:29.980 ","End":"12:32.410","Text":"it\u0027s just that the mnemonic uses shapes rather than"},{"Start":"12:32.410 ","End":"12:37.370","Text":"colors and it might be easier. All right."},{"Start":"12:38.820 ","End":"12:46.990","Text":"Now I\u0027d like to show you 2 other ways of differentiating"},{"Start":"12:46.990 ","End":"12:50.335","Text":"f of x to the power of g of x without using the formula."},{"Start":"12:50.335 ","End":"12:53.905","Text":"This section is not mandatory and you could skip it"},{"Start":"12:53.905 ","End":"12:57.550","Text":"and still be able to solve this kind of exercise."},{"Start":"12:57.550 ","End":"12:59.140","Text":"It\u0027s only for those who want it,"},{"Start":"12:59.140 ","End":"13:04.330","Text":"who might find an alternative method better and for the sake of completeness."},{"Start":"13:04.330 ","End":"13:11.385","Text":"So we\u0027re going to continue working on the same exercise as the 1 before,"},{"Start":"13:11.385 ","End":"13:14.745","Text":"except that we\u0027ll do it in different ways."},{"Start":"13:14.745 ","End":"13:16.800","Text":"Before I get started,"},{"Start":"13:16.800 ","End":"13:20.160","Text":"I want to write down the formulas I\u0027m going to need here,"},{"Start":"13:20.160 ","End":"13:22.415","Text":"in case you forgotten them."},{"Start":"13:22.415 ","End":"13:27.160","Text":"The rules I need is 1 from algebra,"},{"Start":"13:27.160 ","End":"13:29.815","Text":"which says that a to the power of b,"},{"Start":"13:29.815 ","End":"13:32.050","Text":"assuming certain conditions on a and b,"},{"Start":"13:32.050 ","End":"13:33.955","Text":"like a is positive and so on,"},{"Start":"13:33.955 ","End":"13:38.185","Text":"then a to the power of b is equal, algebraically,"},{"Start":"13:38.185 ","End":"13:40.270","Text":"to e to the power of b,"},{"Start":"13:40.270 ","End":"13:44.740","Text":"natural log of a. I\u0027m just bringing it from algebra,"},{"Start":"13:44.740 ","End":"13:48.595","Text":"I\u0027m not going to prove it again or anything, so that\u0027s there."},{"Start":"13:48.595 ","End":"13:55.420","Text":"The other rule is the template form of the e to the power of differentiation rule,"},{"Start":"13:55.420 ","End":"14:01.855","Text":"where the derivative of e to the power of box is also e to the power of box,"},{"Start":"14:01.855 ","End":"14:05.980","Text":"but times that it\u0027s internal derivative,"},{"Start":"14:05.980 ","End":"14:07.270","Text":"which we keep talking about."},{"Start":"14:07.270 ","End":"14:09.010","Text":"So it\u0027s e to the box, box prime."},{"Start":"14:09.010 ","End":"14:11.800","Text":"I\u0027m leaving these formulas there for when we need them,"},{"Start":"14:11.800 ","End":"14:16.970","Text":"and now I\u0027m going to show you how to solve this without using the formula."},{"Start":"14:17.040 ","End":"14:20.590","Text":"The first thing I do here is just write"},{"Start":"14:20.590 ","End":"14:25.840","Text":"the function in a different form using this algebra."},{"Start":"14:25.840 ","End":"14:27.790","Text":"So I\u0027m not differentiating or anything,"},{"Start":"14:27.790 ","End":"14:31.400","Text":"I\u0027m just going to write this using the rule,"},{"Start":"14:31.400 ","End":"14:36.340","Text":"a^b is e^b, natural log a."},{"Start":"14:36.340 ","End":"14:40.555","Text":"So this will be e to the power of, now,"},{"Start":"14:40.555 ","End":"14:45.205","Text":"the b is the original 2x"},{"Start":"14:45.205 ","End":"14:51.115","Text":"and natural log of the a is just the x,"},{"Start":"14:51.115 ","End":"14:53.800","Text":"so it\u0027s just natural log of x."},{"Start":"14:53.800 ","End":"14:57.130","Text":"Then I\u0027m going to start differentiating."},{"Start":"14:57.130 ","End":"14:59.590","Text":"So y prime equals,"},{"Start":"14:59.590 ","End":"15:05.680","Text":"and now I come to this rule for e to the power of something derivative."},{"Start":"15:05.680 ","End":"15:10.060","Text":"This is equal to the same thing as in the beginning,"},{"Start":"15:10.060 ","End":"15:15.370","Text":"which is e^2x,"},{"Start":"15:15.370 ","End":"15:19.180","Text":"natural log of x"},{"Start":"15:19.180 ","End":"15:27.145","Text":"times the derivative of 2x,"},{"Start":"15:27.145 ","End":"15:29.260","Text":"natural log x,"},{"Start":"15:29.260 ","End":"15:32.260","Text":"so all this derivative."},{"Start":"15:32.260 ","End":"15:35.770","Text":"Perhaps, I better put up the product rule again,"},{"Start":"15:35.770 ","End":"15:37.630","Text":"in case you\u0027ve forgotten it."},{"Start":"15:37.630 ","End":"15:39.280","Text":"So here it is again,"},{"Start":"15:39.280 ","End":"15:41.410","Text":"the product rule, we just had it a little while ago,"},{"Start":"15:41.410 ","End":"15:43.390","Text":"so I won\u0027t even go into explanations,"},{"Start":"15:43.390 ","End":"15:45.220","Text":"it\u0027ll just be a mnemonic."},{"Start":"15:45.220 ","End":"15:48.655","Text":"Back to here, we\u0027re going to continue expanding"},{"Start":"15:48.655 ","End":"15:53.200","Text":"the y prime because we still have some differentiating to do."},{"Start":"15:53.200 ","End":"15:54.955","Text":"So this is equal to."},{"Start":"15:54.955 ","End":"15:57.745","Text":"Now, I\u0027m going to copy this, but not exactly."},{"Start":"15:57.745 ","End":"15:59.635","Text":"I got this from this,"},{"Start":"15:59.635 ","End":"16:00.805","Text":"there\u0027s the equal sign."},{"Start":"16:00.805 ","End":"16:03.730","Text":"Let me go back to the way it was originally."},{"Start":"16:03.730 ","End":"16:08.950","Text":"So this thing was originally x^2x and we\u0027ll leave it like that,"},{"Start":"16:08.950 ","End":"16:12.625","Text":"and here we have the product rule, this prime."},{"Start":"16:12.625 ","End":"16:17.040","Text":"This is 2x times natural log of x,"},{"Start":"16:17.040 ","End":"16:19.815","Text":"so it\u0027s 2x derivative,"},{"Start":"16:19.815 ","End":"16:22.590","Text":"which is 2, and this 1, as is,"},{"Start":"16:22.590 ","End":"16:27.270","Text":"natural log of x plus this 1, as is,"},{"Start":"16:27.270 ","End":"16:37.200","Text":"2x times the derivative of natural log x, which is 1/x."},{"Start":"16:37.200 ","End":"16:41.350","Text":"That\u0027s the answer and I believe that this is what we had before,"},{"Start":"16:41.350 ","End":"16:43.000","Text":"except for simplification,"},{"Start":"16:43.000 ","End":"16:44.215","Text":"where like I said,"},{"Start":"16:44.215 ","End":"16:47.020","Text":"it\u0027s tempting to cancel the x with the x,"},{"Start":"16:47.020 ","End":"16:50.320","Text":"but that\u0027s not part of the differentiation process."},{"Start":"16:50.320 ","End":"16:53.945","Text":"This is the answer without using the formula,"},{"Start":"16:53.945 ","End":"16:58.185","Text":"but if you followed how I did the expansion and everything,"},{"Start":"16:58.185 ","End":"17:00.570","Text":"and if you tried it in general with f and g,"},{"Start":"17:00.570 ","End":"17:03.665","Text":"it gives you an indication of how I got to the formula."},{"Start":"17:03.665 ","End":"17:07.645","Text":"This method mimics the way I reached the formula."},{"Start":"17:07.645 ","End":"17:09.640","Text":"If you didn\u0027t understand that last part,"},{"Start":"17:09.640 ","End":"17:12.920","Text":"doesn\u0027t matter, not important."},{"Start":"17:15.450 ","End":"17:18.130","Text":"Let me call this method,"},{"Start":"17:18.130 ","End":"17:22.480","Text":"method II, Roman II."},{"Start":"17:22.480 ","End":"17:26.710","Text":"The very first 1 with the formula would be method I, lets say."},{"Start":"17:26.710 ","End":"17:29.815","Text":"Now I\u0027m going to show you a third 1."},{"Start":"17:29.815 ","End":"17:34.690","Text":"Method II was significantly similar to method I,"},{"Start":"17:34.690 ","End":"17:38.470","Text":"but III will be quite a bit different."},{"Start":"17:38.470 ","End":"17:44.660","Text":"Let me just scroll up a bit so I get some space here."},{"Start":"17:44.700 ","End":"17:47.230","Text":"For rule number 3,"},{"Start":"17:47.230 ","End":"17:49.705","Text":"I\u0027m just going to need a rule from algebra,"},{"Start":"17:49.705 ","End":"17:52.780","Text":"and that is the log to any base, for example,"},{"Start":"17:52.780 ","End":"17:56.860","Text":"the natural log of something to the power of something,"},{"Start":"17:56.860 ","End":"18:02.845","Text":"say a^b is equal to b times the natural log,"},{"Start":"18:02.845 ","End":"18:06.655","Text":"in this case, of a,"},{"Start":"18:06.655 ","End":"18:08.440","Text":"and actually works to any log."},{"Start":"18:08.440 ","End":"18:11.875","Text":"The log of a^b is b log a."},{"Start":"18:11.875 ","End":"18:13.330","Text":"That\u0027s all I\u0027ll need here,"},{"Start":"18:13.330 ","End":"18:18.670","Text":"and I\u0027m going to do the same exercise, that is,"},{"Start":"18:18.670 ","End":"18:23.935","Text":"that y equals x^2x,"},{"Start":"18:23.935 ","End":"18:27.970","Text":"and solve it using a third method."},{"Start":"18:27.970 ","End":"18:33.805","Text":"This method actually supposes that you remember how to do implicit differentiation,"},{"Start":"18:33.805 ","End":"18:35.590","Text":"so if you don\u0027t go brush up on it,"},{"Start":"18:35.590 ","End":"18:40.550","Text":"I\u0027ll just forget about number 3, implicit differentiation."},{"Start":"18:42.210 ","End":"18:47.380","Text":"We begin by taking the natural log of both sides,"},{"Start":"18:47.380 ","End":"18:55.135","Text":"so we get natural log of y equals natural log of this,"},{"Start":"18:55.135 ","End":"18:57.640","Text":"but the natural log of this,"},{"Start":"18:57.640 ","End":"19:00.190","Text":"if we use this formula,"},{"Start":"19:00.190 ","End":"19:06.040","Text":"will be b natural log of a,"},{"Start":"19:06.040 ","End":"19:11.005","Text":"which is 2x natural log,"},{"Start":"19:11.005 ","End":"19:13.850","Text":"and the a is just x."},{"Start":"19:13.860 ","End":"19:22.120","Text":"That\u0027s the first step. For the implicit differentiation,"},{"Start":"19:22.120 ","End":"19:24.070","Text":"which I hope you remember,"},{"Start":"19:24.070 ","End":"19:30.400","Text":"we differentiate natural log of y, which is, well,"},{"Start":"19:30.400 ","End":"19:35.065","Text":"I write it as y-prime/y,"},{"Start":"19:35.065 ","End":"19:39.070","Text":"but most people actually just write 1/y"},{"Start":"19:39.070 ","End":"19:42.640","Text":"and then they remember to multiply by the inner derivative,"},{"Start":"19:42.640 ","End":"19:44.035","Text":"which is y prime."},{"Start":"19:44.035 ","End":"19:46.330","Text":"So take your pick of 1 of these 2,"},{"Start":"19:46.330 ","End":"19:49.825","Text":"but preferably not both, like I did."},{"Start":"19:49.825 ","End":"19:52.390","Text":"Then, you know what?"},{"Start":"19:52.390 ","End":"19:54.310","Text":"I\u0027ll just go with the classic."},{"Start":"19:54.310 ","End":"19:59.230","Text":"So 1/y, which is what you would normally expect, times the internal,"},{"Start":"19:59.230 ","End":"20:01.705","Text":"which is what I call the box, the box prime,"},{"Start":"20:01.705 ","End":"20:03.730","Text":"is equal to, now here we have the product,"},{"Start":"20:03.730 ","End":"20:06.235","Text":"we still have the product rule up."},{"Start":"20:06.235 ","End":"20:12.970","Text":"In fact, we even have the answer here for this bit of differentiation,"},{"Start":"20:12.970 ","End":"20:17.530","Text":"which is 2 natural log of x,"},{"Start":"20:17.530 ","End":"20:24.850","Text":"I\u0027m just copying the work I did before, plus 2x 1/x."},{"Start":"20:24.850 ","End":"20:29.665","Text":"Still itching to cancel that x/x."},{"Start":"20:29.665 ","End":"20:31.355","Text":"Never mind, we\u0027ll leave it."},{"Start":"20:31.355 ","End":"20:36.085","Text":"Then what we do is what we need is y prime,"},{"Start":"20:36.085 ","End":"20:39.685","Text":"so we have to multiply both sides by y,"},{"Start":"20:39.685 ","End":"20:41.440","Text":"but you got to remember that"},{"Start":"20:41.440 ","End":"20:50.380","Text":"y equals"},{"Start":"20:50.380 ","End":"20:52.105","Text":"x times"},{"Start":"20:52.105 ","End":"20:53.515","Text":"2 to the x."},{"Start":"20:53.515 ","End":"20:57.310","Text":"So if I multiply both sides by y,"},{"Start":"20:57.310 ","End":"20:59.230","Text":"I know your algebra\u0027s good,"},{"Start":"20:59.230 ","End":"21:01.075","Text":"so I\u0027m doing 2 steps in 1,"},{"Start":"21:01.075 ","End":"21:03.760","Text":"so y prime equals putting y over here,"},{"Start":"21:03.760 ","End":"21:05.080","Text":"and as it moves over,"},{"Start":"21:05.080 ","End":"21:09.200","Text":"it transforms magically back into x^2x,"},{"Start":"21:09.510 ","End":"21:12.670","Text":"so this is this y from here,"},{"Start":"21:12.670 ","End":"21:19.825","Text":"times 2 natural log of x plus 2x 1/x,"},{"Start":"21:19.825 ","End":"21:23.600","Text":"which at the end will become just 2."},{"Start":"21:25.440 ","End":"21:30.490","Text":"That\u0027s the third method which uses implicit differentiation"},{"Start":"21:30.490 ","End":"21:39.560","Text":"and this rule of the natural log of a function which is not x."},{"Start":"21:41.420 ","End":"21:51.640","Text":"That was methods 2 and 3 and can get it all under 1 page. We\u0027re done."}],"ID":10449},{"Watched":false,"Name":"Exercise 1 - Parts 1-4","Duration":"22m 1s","ChapterTopicVideoID":10143,"CourseChapterTopicPlaylistID":8713,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.680 ","End":"00:06.105","Text":"Here we have really 12 exercises in 1, 1-12."},{"Start":"00:06.105 ","End":"00:09.100","Text":"They\u0027re all similar in the sense that they\u0027re"},{"Start":"00:09.100 ","End":"00:12.885","Text":"a function of x to the power of another function of x."},{"Start":"00:12.885 ","End":"00:19.330","Text":"We\u0027ll start with number 1 and you\u0027ll quickly catch on to what I\u0027m doing."},{"Start":"00:21.740 ","End":"00:31.390","Text":"Just a second. Number 1 is y equals x^2x."},{"Start":"00:34.190 ","End":"00:38.610","Text":"Now, I\u0027ll draw a little margin here."},{"Start":"00:38.610 ","End":"00:41.140","Text":"It\u0027s for our scratch work."},{"Start":"00:42.330 ","End":"00:46.060","Text":"The general trick, and it\u0027s an algebraic trick,"},{"Start":"00:46.060 ","End":"00:47.575","Text":"is when I have something,"},{"Start":"00:47.575 ","End":"00:51.295","Text":"say square to the power of something else,"},{"Start":"00:51.295 ","End":"00:55.600","Text":"triangle that this is equal to e to"},{"Start":"00:55.600 ","End":"01:02.900","Text":"the power of triangle times natural log of square."},{"Start":"01:03.050 ","End":"01:12.030","Text":"I\u0027ll just write that over here and we\u0027ll see how to proceed."},{"Start":"01:12.030 ","End":"01:16.545","Text":"This is equal to triangle,"},{"Start":"01:16.545 ","End":"01:21.090","Text":"and we\u0027ll use colors."},{"Start":"01:21.090 ","End":"01:26.940","Text":"We\u0027ll use colors that this will be green,"},{"Start":"01:26.940 ","End":"01:32.775","Text":"that will be the square and the square."},{"Start":"01:32.775 ","End":"01:38.355","Text":"I\u0027ll use the other color for the triangle,"},{"Start":"01:38.355 ","End":"01:41.735","Text":"the triangle, the triangle."},{"Start":"01:41.735 ","End":"01:46.855","Text":"Using this, I can write this thing"},{"Start":"01:46.855 ","End":"01:52.875","Text":"as e to the power of the triangle."},{"Start":"01:52.875 ","End":"01:55.990","Text":"The light blue is 2x,"},{"Start":"01:56.150 ","End":"02:01.815","Text":"natural log of the green,"},{"Start":"02:01.815 ","End":"02:04.090","Text":"which is the x."},{"Start":"02:13.610 ","End":"02:15.600","Text":"Just a second."},{"Start":"02:15.600 ","End":"02:28.680","Text":"The 2x and we have also the x which is here."},{"Start":"02:31.010 ","End":"02:34.850","Text":"Now you might ask, how does this help us?"},{"Start":"02:34.850 ","End":"02:37.715","Text":"Well, this helps us because there\u0027s another formula"},{"Start":"02:37.715 ","End":"02:41.150","Text":"that the derivative of e to, I don\u0027t know,"},{"Start":"02:41.150 ","End":"02:49.620","Text":"let\u0027s call it rectangle is e to the same thing,"},{"Start":"02:49.620 ","End":"02:52.090","Text":"let\u0027s called it rectangle,"},{"Start":"02:53.030 ","End":"03:01.150","Text":"I don\u0027t need the brackets really, times rectangle derived."},{"Start":"03:20.270 ","End":"03:23.499","Text":"I\u0027ve used up colors."},{"Start":"03:28.520 ","End":"03:31.840","Text":"I\u0027ll just underline it."},{"Start":"03:34.850 ","End":"03:37.440","Text":"That\u0027s going to be our rectangle."},{"Start":"03:37.440 ","End":"03:42.880","Text":"Y prime is going to be the same e to the power of the same rectangle,"},{"Start":"03:42.880 ","End":"03:50.395","Text":"2x natural log of x times this thing prime."},{"Start":"03:50.395 ","End":"03:52.405","Text":"Now, I won\u0027t actually differentiate,"},{"Start":"03:52.405 ","End":"03:54.715","Text":"yet I\u0027ll just write that I have to do it,"},{"Start":"03:54.715 ","End":"04:01.630","Text":"2x natural log of x prime."},{"Start":"04:02.540 ","End":"04:07.145","Text":"I have an idea of how to do this with color, hang on."},{"Start":"04:07.145 ","End":"04:10.540","Text":"It\u0027s my idea just to copy the same thing with it."},{"Start":"04:10.540 ","End":"04:13.945","Text":"Just put a yellow color instead of these colors."},{"Start":"04:13.945 ","End":"04:16.475","Text":"This formula says that if I derive it,"},{"Start":"04:16.475 ","End":"04:22.020","Text":"I get the same thing times the derivative of what was in the rectangle."},{"Start":"04:22.090 ","End":"04:25.130","Text":"Now, we\u0027ll do 2 things at once."},{"Start":"04:25.130 ","End":"04:26.510","Text":"On the 1 hand,"},{"Start":"04:26.510 ","End":"04:29.645","Text":"I can return this to the way it was before."},{"Start":"04:29.645 ","End":"04:31.040","Text":"This is the same as this,"},{"Start":"04:31.040 ","End":"04:33.635","Text":"but it came from here by algebra."},{"Start":"04:33.635 ","End":"04:35.210","Text":"This is the original."},{"Start":"04:35.210 ","End":"04:37.055","Text":"I prefer to stick with the original."},{"Start":"04:37.055 ","End":"04:40.745","Text":"So I\u0027ll rewrite it as x^2x."},{"Start":"04:40.745 ","End":"04:42.530","Text":"It\u0027s exactly the same; if it goes 1 way,"},{"Start":"04:42.530 ","End":"04:44.165","Text":"it goes backwards too."},{"Start":"04:44.165 ","End":"04:50.105","Text":"Now we have to just derive 2x natural log of x."},{"Start":"04:50.105 ","End":"04:53.465","Text":"Well, this is just a product,"},{"Start":"04:53.465 ","End":"04:56.280","Text":"2x times the natural log of x."},{"Start":"04:56.280 ","End":"05:03.750","Text":"Remember, fg prime is f prime g plus f g prime."},{"Start":"05:03.750 ","End":"05:09.060","Text":"In this case, let\u0027s keep the 2x as f."},{"Start":"05:09.060 ","End":"05:14.570","Text":"Say, this will be f and this will be g."},{"Start":"05:14.570 ","End":"05:20.540","Text":"It\u0027s f prime, which is just 2, times g,"},{"Start":"05:20.540 ","End":"05:22.750","Text":"which is natural log of x,"},{"Start":"05:22.750 ","End":"05:26.275","Text":"plus f, which is 2x,"},{"Start":"05:26.275 ","End":"05:30.390","Text":"times g prime, which is 1 over x,"},{"Start":"05:30.390 ","End":"05:34.345","Text":"natural log of x as the derivative of 1 over x."},{"Start":"05:34.345 ","End":"05:37.205","Text":"Then just keep simplifying."},{"Start":"05:37.205 ","End":"05:41.275","Text":"This is equal to x^2x."},{"Start":"05:41.275 ","End":"05:44.075","Text":"I can take the 2 outside the brackets,"},{"Start":"05:44.075 ","End":"05:51.295","Text":"2 times natural log of x plus,"},{"Start":"05:51.295 ","End":"05:54.905","Text":"and the 2s come out and the x over x cancels."},{"Start":"05:54.905 ","End":"05:57.395","Text":"Then we just get plus 1."},{"Start":"05:57.395 ","End":"06:00.780","Text":"That\u0027s the answer to number 1."},{"Start":"06:01.670 ","End":"06:06.375","Text":"We\u0027re going to use this same trick pretty much throughout."},{"Start":"06:06.375 ","End":"06:08.685","Text":"Let\u0027s go on to number 2."},{"Start":"06:08.685 ","End":"06:30.360","Text":"Number 2 is y equals x^natural log x, x^log x"},{"Start":"06:30.360 ","End":"06:33.430","Text":"Did I forget to put the y equals."},{"Start":"06:33.430 ","End":"06:38.750","Text":"I think I can squeeze it in here, y equals."},{"Start":"06:39.140 ","End":"06:44.620","Text":"Now using the same formula we used above,"},{"Start":"06:44.620 ","End":"06:47.340","Text":"I\u0027ll just write it again."},{"Start":"06:47.340 ","End":"06:52.840","Text":"Something to the power of something is equal to e to the power"},{"Start":"06:52.840 ","End":"06:59.160","Text":"of the second something natural log of the first something."},{"Start":"06:59.160 ","End":"07:03.055","Text":"Here we get that this is equal to"},{"Start":"07:03.055 ","End":"07:11.120","Text":"e^natural log of x."},{"Start":"07:11.120 ","End":"07:21.500","Text":"Again, natural log of x because this is the square."},{"Start":"07:21.950 ","End":"07:25.715","Text":"Maybe I better use colors."},{"Start":"07:25.715 ","End":"07:28.520","Text":"Let\u0027s try them again."},{"Start":"07:39.290 ","End":"07:48.105","Text":"This is the x here. That\u0027s part of the thing."},{"Start":"07:48.105 ","End":"07:51.640","Text":"Then this function here was the triangle."},{"Start":"07:53.060 ","End":"07:58.570","Text":"That this and that\u0027s this."},{"Start":"07:59.390 ","End":"08:06.840","Text":"Eventually, you\u0027ll get used to this formula and you\u0027ll stop using colors all the time."},{"Start":"08:08.660 ","End":"08:13.095","Text":"Now I\u0027ve just transformed it algebraically."},{"Start":"08:13.095 ","End":"08:17.760","Text":"Now we need y prime."},{"Start":"08:17.760 ","End":"08:19.789","Text":"That\u0027s where we need the other formula,"},{"Start":"08:19.789 ","End":"08:25.865","Text":"which was that the derivative of e to the power of something is"},{"Start":"08:25.865 ","End":"08:34.035","Text":"that same e to the power of something times that something prime."},{"Start":"08:34.035 ","End":"08:38.610","Text":"In this case, our something is this whole exponent."},{"Start":"08:38.610 ","End":"08:42.095","Text":"It\u0027s going to equal, first of all, the thing itself,"},{"Start":"08:42.095 ","End":"08:49.400","Text":"which is e^natural log of x,"},{"Start":"08:49.400 ","End":"08:54.795","Text":"which actually we should write as this thing squared,"},{"Start":"08:54.795 ","End":"09:01.610","Text":"times the derivative of the top."},{"Start":"09:01.610 ","End":"09:04.415","Text":"Now what we\u0027ll have to do is differentiate that,"},{"Start":"09:04.415 ","End":"09:06.600","Text":"that\u0027s this part here."},{"Start":"09:08.410 ","End":"09:17.400","Text":"Which is equal to and copying the e^natural log of x squared,"},{"Start":"09:17.400 ","End":"09:19.185","Text":"another derivative of this."},{"Start":"09:19.185 ","End":"09:24.410","Text":"If the derivative of x squared is 2x,"},{"Start":"09:24.410 ","End":"09:32.240","Text":"the derivative of this is 2 times natural log of x times the internal derivative."},{"Start":"09:32.240 ","End":"09:39.770","Text":"The internal derivative is the derivative of natural log of x,"},{"Start":"09:39.770 ","End":"09:42.480","Text":"which is 1 over x."},{"Start":"09:42.490 ","End":"09:47.930","Text":"If you want me to show you schematically, for example,"},{"Start":"09:47.930 ","End":"09:55.040","Text":"if I have something squared and I want it derived,"},{"Start":"09:55.040 ","End":"09:57.110","Text":"it\u0027s going to be twice that something,"},{"Start":"09:57.110 ","End":"09:59.440","Text":"but times that something prime."},{"Start":"09:59.440 ","End":"10:03.350","Text":"The something here in our case,"},{"Start":"10:04.990 ","End":"10:11.490","Text":"the square represents natural log of x in our case."},{"Start":"10:12.650 ","End":"10:15.030","Text":"You\u0027ll get used to these things,"},{"Start":"10:15.030 ","End":"10:19.175","Text":"I won\u0027t spell them out each time."},{"Start":"10:19.175 ","End":"10:21.770","Text":"Now simplification."},{"Start":"10:21.770 ","End":"10:24.305","Text":"Is there anything to simplify?"},{"Start":"10:24.305 ","End":"10:27.890","Text":"Not really. Could leave this as the answer."},{"Start":"10:27.890 ","End":"10:36.500","Text":"Or I might just write it as e^natural log of x"},{"Start":"10:36.500 ","End":"10:45.950","Text":"squared times 2 times natural log of x all over x,"},{"Start":"10:45.950 ","End":"10:48.715","Text":"just as well to leave it like this."},{"Start":"10:48.715 ","End":"10:52.865","Text":"That\u0027s done with the number 2."},{"Start":"10:52.865 ","End":"11:00.084","Text":"We\u0027ll continue with number 3."},{"Start":"11:00.084 ","End":"11:08.655","Text":"Number 3 is natural log of x, so the power of x,"},{"Start":"11:08.655 ","End":"11:12.045","Text":"and that\u0027s y. y equals"},{"Start":"11:12.045 ","End":"11:20.520","Text":"natural log of x^x."},{"Start":"11:20.520 ","End":"11:34.430","Text":"This equals, again, using that same formula, e^x."},{"Start":"11:34.430 ","End":"11:40.360","Text":"The original exponent was times the log of the base."},{"Start":"11:40.360 ","End":"11:47.990","Text":"It\u0027s log of log x,"},{"Start":"11:49.020 ","End":"11:54.355","Text":"and then we differentiate using the other formula."},{"Start":"11:54.355 ","End":"11:59.210","Text":"The derivative is equal to the same thing,"},{"Start":"12:01.530 ","End":"12:12.655","Text":"log of log x times the derivative of what was at the top."},{"Start":"12:12.655 ","End":"12:19.840","Text":"It\u0027s the x log of log x,"},{"Start":"12:19.840 ","End":"12:32.650","Text":"all natural logs, and prime."},{"Start":"12:32.650 ","End":"12:41.400","Text":"This equals, the first bit we just copy e^x log,"},{"Start":"12:41.400 ","End":"12:45.675","Text":"log x, and now we need the derivative of this."},{"Start":"12:45.675 ","End":"12:48.870","Text":"I would recommend using the product rule,"},{"Start":"12:48.870 ","End":"12:52.320","Text":"it\u0027s quite obviously a product where we say that"},{"Start":"12:52.320 ","End":"13:01.005","Text":"fg prime is f prime g plus fg prime,"},{"Start":"13:01.005 ","End":"13:03.035","Text":"or in our case,"},{"Start":"13:03.035 ","End":"13:10.300","Text":"this is going to be the f and the second half from here to here will be the g."},{"Start":"13:10.300 ","End":"13:16.090","Text":"I still leave the brackets here."},{"Start":"13:16.090 ","End":"13:19.015","Text":"F prime is 1,"},{"Start":"13:19.015 ","End":"13:22.870","Text":"and then we need g itself,"},{"Start":"13:22.870 ","End":"13:27.010","Text":"which is log of log x,"},{"Start":"13:27.010 ","End":"13:32.275","Text":"I say log instead of natural log plus f,"},{"Start":"13:32.275 ","End":"13:40.960","Text":"which is just x times g prime and that g prime I\u0027ll do this separately here,"},{"Start":"13:40.960 ","End":"13:49.695","Text":"g prime is natural log of natural log of x, all this prime."},{"Start":"13:49.695 ","End":"13:52.140","Text":"This is a chain rule."},{"Start":"13:52.140 ","End":"13:55.050","Text":"We have the log of the log of x."},{"Start":"13:55.050 ","End":"14:03.060","Text":"This is going to equal the external derivative,"},{"Start":"14:03.060 ","End":"14:04.515","Text":"because it\u0027s natural log,"},{"Start":"14:04.515 ","End":"14:10.180","Text":"it\u0027s 1 over natural log of x,"},{"Start":"14:10.180 ","End":"14:14.364","Text":"but the natural log of x is the internal function."},{"Start":"14:14.364 ","End":"14:19.570","Text":"The inner function needs it\u0027s derivative also times the derivative of natural log of x,"},{"Start":"14:19.570 ","End":"14:22.820","Text":"which is times 1/x."},{"Start":"14:28.350 ","End":"14:33.220","Text":"This we got up to the f and now we have to add the g prime,"},{"Start":"14:33.220 ","End":"14:36.500","Text":"and the g prime is,"},{"Start":"14:36.750 ","End":"14:39.160","Text":"instead of 1 over,"},{"Start":"14:39.160 ","End":"14:41.545","Text":"we can just put this as a fraction."},{"Start":"14:41.545 ","End":"14:48.835","Text":"On the denominator, put natural log of x times x."},{"Start":"14:48.835 ","End":"14:53.725","Text":"Something cancels the x with the x."},{"Start":"14:53.725 ","End":"14:56.840","Text":"But that\u0027s about it."},{"Start":"14:57.240 ","End":"15:02.380","Text":"All I can really do with the answer is write it"},{"Start":"15:02.380 ","End":"15:25.450","Text":"as e^x log, log x times,"},{"Start":"15:25.450 ","End":"15:27.860","Text":"let\u0027s see what we have here,"},{"Start":"15:32.670 ","End":"15:41.810","Text":"log of log x plus 1/x is what we\u0027re left with here."},{"Start":"15:45.210 ","End":"15:48.280","Text":"That\u0027s about as far as I can go."},{"Start":"15:48.280 ","End":"15:55.460","Text":"We\u0027re done with Number 3 and so on to Number 4."},{"Start":"15:58.020 ","End":"16:06.410","Text":"Number 4 says, we have x squared plus 1^4x."},{"Start":"16:18.390 ","End":"16:22.555","Text":"Now I\u0027m not writing that formula again,"},{"Start":"16:22.555 ","End":"16:34.780","Text":"but this is the same as e^4x times the log of this."},{"Start":"16:35.810 ","End":"16:48.770","Text":"So it\u0027s e^4x log of x squared plus 1."},{"Start":"16:48.770 ","End":"16:51.325","Text":"Now when we differentiate,"},{"Start":"16:51.325 ","End":"16:53.650","Text":"we use that other rule that when we differentiate,"},{"Start":"16:53.650 ","End":"16:56.140","Text":"we take this e to the power of as it is,"},{"Start":"16:56.140 ","End":"16:59.410","Text":"times the derivative of the exponent,"},{"Start":"16:59.410 ","End":"17:01.480","Text":"so this is equal to"},{"Start":"17:01.480 ","End":"17:10.810","Text":"e^4x natural log of"},{"Start":"17:10.810 ","End":"17:13.460","Text":"x squared plus 1,"},{"Start":"17:24.060 ","End":"17:30.820","Text":"we multiply that by the derivative of what was on the top,"},{"Start":"17:30.820 ","End":"17:39.160","Text":"which is 4x natural log of x squared plus 1 derivative."},{"Start":"17:39.160 ","End":"17:41.335","Text":"You still have to do the derivative."},{"Start":"17:41.335 ","End":"17:43.315","Text":"Now this equals,"},{"Start":"17:43.315 ","End":"17:45.970","Text":"there is 2 things we\u0027re going to do here, compute the derivative,"},{"Start":"17:45.970 ","End":"17:52.915","Text":"but also we can put that in its original form because this thing here is equal to this."},{"Start":"17:52.915 ","End":"17:56.990","Text":"This will be x squared plus 1^4x,"},{"Start":"17:58.080 ","End":"18:01.000","Text":"we don\u0027t have to do this, but it\u0027s nicer,"},{"Start":"18:01.000 ","End":"18:02.815","Text":"and oops, do you know what?"},{"Start":"18:02.815 ","End":"18:05.260","Text":"I forgot to do that in the previous exercise,"},{"Start":"18:05.260 ","End":"18:07.990","Text":"so forgive me for that."},{"Start":"18:07.990 ","End":"18:11.080","Text":"Let me just at the very end,"},{"Start":"18:11.080 ","End":"18:14.540","Text":"I\u0027m going to change this back."},{"Start":"18:20.310 ","End":"18:25.915","Text":"Yes, since we got to this thing, from this thing,"},{"Start":"18:25.915 ","End":"18:29.125","Text":"I should\u0027ve, from some point on,"},{"Start":"18:29.125 ","End":"18:31.464","Text":"I can even at the very end, revert."},{"Start":"18:31.464 ","End":"18:34.360","Text":"Let me just say I wish instead of writing this,"},{"Start":"18:34.360 ","End":"18:37.405","Text":"I had written natural,"},{"Start":"18:37.405 ","End":"18:40.240","Text":"I\u0027ll write it in a different color,"},{"Start":"18:40.240 ","End":"18:46.900","Text":"natural log of x^x."},{"Start":"18:46.900 ","End":"18:49.930","Text":"What\u0027s written here is true, it\u0027s correct."},{"Start":"18:49.930 ","End":"18:55.000","Text":"But it\u0027s tidier if after you do this conversion, you convert back."},{"Start":"18:55.000 ","End":"18:58.345","Text":"I think I may have even made that mistake in the previous,"},{"Start":"18:58.345 ","End":"19:01.810","Text":"not a mistake, it\u0027s just not the best 1 can do."},{"Start":"19:01.810 ","End":"19:03.610","Text":"But over here,"},{"Start":"19:03.610 ","End":"19:06.730","Text":"in this exercise also,"},{"Start":"19:06.730 ","End":"19:12.850","Text":"we got from x^log x."},{"Start":"19:12.850 ","End":"19:16.090","Text":"We algebraically converted it to that,"},{"Start":"19:16.090 ","End":"19:18.475","Text":"so at some point like from here,"},{"Start":"19:18.475 ","End":"19:20.830","Text":"we should have converted it back."},{"Start":"19:20.830 ","End":"19:25.570","Text":"Let\u0027s take this 1 away, and instead of it,"},{"Start":"19:25.570 ","End":"19:31.900","Text":"write x^ natural log of x."},{"Start":"19:31.900 ","End":"19:37.825","Text":"Everything was correct, but the simplification of when you convert it to the e form,"},{"Start":"19:37.825 ","End":"19:40.840","Text":"you can convert it back and it\u0027s neater."},{"Start":"19:40.840 ","End":"19:42.820","Text":"Excuse me for that,"},{"Start":"19:42.820 ","End":"19:44.455","Text":"but nothing is incorrect."},{"Start":"19:44.455 ","End":"19:48.205","Text":"Just not as tidy as it could be."},{"Start":"19:48.205 ","End":"19:51.085","Text":"Here we won\u0027t forget to do that,"},{"Start":"19:51.085 ","End":"19:52.540","Text":"and 1 from this form,"},{"Start":"19:52.540 ","End":"19:54.385","Text":"since we convert it from this to this,"},{"Start":"19:54.385 ","End":"19:58.150","Text":"we\u0027re going to convert back and it looks tidier. That\u0027s 1 thing we do."},{"Start":"19:58.150 ","End":"20:02.710","Text":"Write in the original form and then differentiate what was in the exponent."},{"Start":"20:02.710 ","End":"20:06.805","Text":"This part also looks like a quotient rule,"},{"Start":"20:06.805 ","End":"20:09.130","Text":"and I\u0027ll keep writing it."},{"Start":"20:09.130 ","End":"20:12.590","Text":"Because people don\u0027t know it by heart,"},{"Start":"20:12.960 ","End":"20:18.385","Text":"f prime g plus fg prime,"},{"Start":"20:18.385 ","End":"20:25.270","Text":"and what we\u0027ll do is this part will be f and the rest of it"},{"Start":"20:25.270 ","End":"20:32.395","Text":"will be g. Yes,"},{"Start":"20:32.395 ","End":"20:36.055","Text":"it\u0027s times, so first of all, f prime,"},{"Start":"20:36.055 ","End":"20:42.310","Text":"which is 4 times g natural log"},{"Start":"20:42.310 ","End":"20:47.500","Text":"of x squared plus 1 plus the other way around,"},{"Start":"20:47.500 ","End":"20:49.780","Text":"4x as is,"},{"Start":"20:49.780 ","End":"20:51.925","Text":"and the derivative of this,"},{"Start":"20:51.925 ","End":"20:53.575","Text":"because it\u0027s natural log,"},{"Start":"20:53.575 ","End":"21:02.290","Text":"it\u0027s going to be 1/x squared plus 1 because natural log is 1 over,"},{"Start":"21:02.290 ","End":"21:04.840","Text":"but there\u0027s an inner function, the x squared plus 1,"},{"Start":"21:04.840 ","End":"21:10.160","Text":"and we have to multiply by the derivative, so that\u0027s 2x."},{"Start":"21:11.730 ","End":"21:18.355","Text":"This equals, is there anything we can do to simplify."},{"Start":"21:18.355 ","End":"21:26.455","Text":"Well, we can certainly take 4 outside the brackets."},{"Start":"21:26.455 ","End":"21:32.320","Text":"That\u0027s 4, x squared plus 1^4x."},{"Start":"21:32.320 ","End":"21:37.330","Text":"This 4 and this 4 have already been taken care of out here,"},{"Start":"21:37.330 ","End":"21:45.130","Text":"and then we can have times natural log of x squared plus 1"},{"Start":"21:45.130 ","End":"21:53.680","Text":"plus x times 2x is 2x squared over x squared plus 1,"},{"Start":"21:53.680 ","End":"21:56.890","Text":"and I think that\u0027s about as much simplifications we can do."},{"Start":"21:56.890 ","End":"22:00.740","Text":"That\u0027s the end of Number 4."}],"ID":10450},{"Watched":false,"Name":"Exercise 1 - Parts 5-8","Duration":"25m 53s","ChapterTopicVideoID":10144,"CourseChapterTopicPlaylistID":8713,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.220","Text":"We just finished number 4 and I scrolled back up to see what\u0027s next in store for us."},{"Start":"00:05.220 ","End":"00:10.680","Text":"It\u0027s number 5, y equals x^x squared plus 1."},{"Start":"00:10.680 ","End":"00:17.100","Text":"As usual, we\u0027ll have to use some formulae."},{"Start":"00:17.100 ","End":"00:20.520","Text":"The main formula that we\u0027re going to use is how to"},{"Start":"00:20.520 ","End":"00:25.770","Text":"convert 1 function to the power of another in terms of e to the power of."},{"Start":"00:25.770 ","End":"00:28.545","Text":"Write it symbolically as that, say,"},{"Start":"00:28.545 ","End":"00:37.955","Text":"square to the power of triangle is equal to e to the power of triangle,"},{"Start":"00:37.955 ","End":"00:41.280","Text":"natural log of square."},{"Start":"00:41.280 ","End":"00:45.890","Text":"The other formula that we\u0027ll need that goes along with"},{"Start":"00:45.890 ","End":"00:50.330","Text":"this is that the derivative of e to the power of something,"},{"Start":"00:50.330 ","End":"00:51.680","Text":"it\u0027s going to be this whole thing,"},{"Start":"00:51.680 ","End":"00:54.350","Text":"let\u0027s just call it rectangle."},{"Start":"00:54.350 ","End":"00:58.820","Text":"Its derivative is just the thing itself that we had in"},{"Start":"00:58.820 ","End":"01:06.065","Text":"the beginning but multiplied by the derivative of rectangle,"},{"Start":"01:06.065 ","End":"01:09.130","Text":"whatever was in the exponent."},{"Start":"01:09.130 ","End":"01:13.275","Text":"Well, having said that, we can rewrite"},{"Start":"01:13.275 ","End":"01:21.285","Text":"our exercise as y equals e to the power of,"},{"Start":"01:21.285 ","End":"01:25.025","Text":"now the triangle here is the x squared plus 1,"},{"Start":"01:25.025 ","End":"01:33.300","Text":"so it\u0027s x squared plus 1 times the natural log,"},{"Start":"01:33.700 ","End":"01:36.380","Text":"I\u0027ll have to put brackets here,"},{"Start":"01:36.380 ","End":"01:41.730","Text":"natural log of the square, which is x."},{"Start":"01:42.680 ","End":"01:48.075","Text":"After that, we start differentiating y prime,"},{"Start":"01:48.075 ","End":"01:50.225","Text":"then we use the e to the power of rule,"},{"Start":"01:50.225 ","End":"01:53.015","Text":"which is just a copy the thing first of all,"},{"Start":"01:53.015 ","End":"02:01.355","Text":"and that\u0027s x squared plus 1 times natural log of x."},{"Start":"02:01.355 ","End":"02:08.220","Text":"But now, I have to multiply by the derivative of this thing,"},{"Start":"02:08.680 ","End":"02:17.090","Text":"which is a product and so I\u0027ll need the product rule"},{"Start":"02:17.090 ","End":"02:25.160","Text":"that fg prime is f prime g plus fg prime,"},{"Start":"02:25.160 ","End":"02:28.250","Text":"where this first bit will be my f,"},{"Start":"02:28.250 ","End":"02:33.680","Text":"and the second bit will be the g. Now we\u0027re still at the point"},{"Start":"02:33.680 ","End":"02:40.500","Text":"where we\u0027ve got the e to the rectangle and rectangle prime."},{"Start":"02:40.690 ","End":"02:44.660","Text":"You know what? I\u0027ll just write it in as, first of all,"},{"Start":"02:44.660 ","End":"02:46.280","Text":"just to show what we\u0027re doing."},{"Start":"02:46.280 ","End":"02:54.270","Text":"It\u0027s x squared plus 1 natural log of x prime."},{"Start":"02:54.270 ","End":"03:00.520","Text":"Now, I\u0027ll apply this rule."},{"Start":"03:00.860 ","End":"03:03.840","Text":"Now, the first part is okay,"},{"Start":"03:03.840 ","End":"03:05.385","Text":"it\u0027s e to the,"},{"Start":"03:05.385 ","End":"03:07.110","Text":"just copying it again,"},{"Start":"03:07.110 ","End":"03:10.770","Text":"x squared plus 1 natural log of"},{"Start":"03:10.770 ","End":"03:18.655","Text":"x times, hang on."},{"Start":"03:18.655 ","End":"03:24.965","Text":"I think we should not have this f here and the g shouldn\u0027t be there but instead,"},{"Start":"03:24.965 ","End":"03:31.240","Text":"I should write f above this and g above this, that looks better."},{"Start":"03:31.240 ","End":"03:37.370","Text":"F prime, x squared prime is 2x times g,"},{"Start":"03:37.370 ","End":"03:43.445","Text":"as is natural log of x plus f as is,"},{"Start":"03:43.445 ","End":"03:48.110","Text":"which is x squared plus 1 times g prime,"},{"Start":"03:48.110 ","End":"03:59.370","Text":"which is 1/x and"},{"Start":"03:59.370 ","End":"04:02.369","Text":"that\u0027s really the answer,"},{"Start":"04:02.369 ","End":"04:08.260","Text":"except that usually, when we get to this point,"},{"Start":"04:08.260 ","End":"04:12.800","Text":"we convert this e thing back to the original form."},{"Start":"04:13.650 ","End":"04:23.500","Text":"This is equal to x to the power of x squared plus 1 times this thing."},{"Start":"04:23.500 ","End":"04:27.380","Text":"Now, I\u0027m wondering, isn\u0027t there anything we can simplify here?"},{"Start":"04:29.360 ","End":"04:31.695","Text":"It doesn\u0027t look like it,"},{"Start":"04:31.695 ","End":"04:33.960","Text":"so we just leave it like that,"},{"Start":"04:33.960 ","End":"04:38.385","Text":"2x natural log of x plus,"},{"Start":"04:38.385 ","End":"04:42.600","Text":"I\u0027ll put the 1/x in front, 1/x,"},{"Start":"04:42.600 ","End":"04:46.240","Text":"x squared plus 1,"},{"Start":"04:48.080 ","End":"04:52.660","Text":"and that\u0027s the answer."},{"Start":"04:56.300 ","End":"05:04.440","Text":"That was number 5, and after number 5, I guess, comes number 6."},{"Start":"05:04.440 ","End":"05:14.775","Text":"We\u0027ll start with number 6 and number 6 is"},{"Start":"05:14.775 ","End":"05:23.010","Text":"y equals the square root of"},{"Start":"05:23.010 ","End":"05:30.055","Text":"x to the power"},{"Start":"05:30.055 ","End":"05:35.090","Text":"of the square root of 2x."},{"Start":"05:38.510 ","End":"05:44.080","Text":"I\u0027m not going to recopy those formulae,"},{"Start":"05:44.080 ","End":"05:49.975","Text":"I\u0027ll just take a quick peek above and see that we have this thing"},{"Start":"05:49.975 ","End":"05:56.440","Text":"of a power being converted to the e to the power form and the standard rule,"},{"Start":"05:56.440 ","End":"05:58.885","Text":"chain rule for e to the power of,"},{"Start":"05:58.885 ","End":"06:05.450","Text":"and that\u0027s where we are now."},{"Start":"06:06.500 ","End":"06:09.990","Text":"If we do that with the e thing,"},{"Start":"06:09.990 ","End":"06:17.900","Text":"we get y equals e to the power of the square root of"},{"Start":"06:17.900 ","End":"06:24.865","Text":"2x times the natural log"},{"Start":"06:24.865 ","End":"06:29.500","Text":"of square root of x."},{"Start":"06:32.000 ","End":"06:36.430","Text":"Now, when we differentiate,"},{"Start":"06:36.820 ","End":"06:41.660","Text":"we get e to the power of the same thing,"},{"Start":"06:41.660 ","End":"06:49.790","Text":"as just the square root of 2x times natural log of square root of"},{"Start":"06:49.790 ","End":"07:00.630","Text":"x and"},{"Start":"07:04.610 ","End":"07:06.650","Text":"it\u0027s the same thing"},{"Start":"07:06.650 ","End":"07:10.240","Text":"except we need the prime of what was above."},{"Start":"07:10.240 ","End":"07:17.134","Text":"What we\u0027re missing still is this thing prime,"},{"Start":"07:17.134 ","End":"07:21.050","Text":"but I\u0027ll also throw in a little shortcut."},{"Start":"07:21.050 ","End":"07:30.080","Text":"I just noticed because natural log of square root of x is natural log"},{"Start":"07:30.080 ","End":"07:34.670","Text":"of x^1/2 and natural log"},{"Start":"07:34.670 ","End":"07:41.345","Text":"of x^1/2 the exponent comes in front is 1/2 natural log of x."},{"Start":"07:41.345 ","End":"07:43.625","Text":"Whether this will shorten anything or not,"},{"Start":"07:43.625 ","End":"07:46.115","Text":"I\u0027m not sure, but looks good."},{"Start":"07:46.115 ","End":"07:50.720","Text":"What we have here is e^square root"},{"Start":"07:50.720 ","End":"07:58.295","Text":"of 2x times 1/2."},{"Start":"07:58.295 ","End":"08:02.960","Text":"I\u0027ll put the half in front,"},{"Start":"08:04.320 ","End":"08:08.725","Text":"times natural log of x."},{"Start":"08:08.725 ","End":"08:14.800","Text":"Basically just took the half out of the derivative,"},{"Start":"08:14.800 ","End":"08:18.050","Text":"we still haven\u0027t actually differentiated it."},{"Start":"08:19.020 ","End":"08:28.390","Text":"This thing, no,"},{"Start":"08:28.390 ","End":"08:33.280","Text":"oops, I\u0027m terribly sorry."},{"Start":"08:33.280 ","End":"08:36.835","Text":"It should be just the prime of what\u0027s above here,"},{"Start":"08:36.835 ","End":"08:39.370","Text":"I\u0027ll fix that in an instant,"},{"Start":"08:39.370 ","End":"08:42.710","Text":"there. Sorry about that."},{"Start":"08:42.750 ","End":"08:49.750","Text":"Now, this equals e to the power of square root of"},{"Start":"08:49.750 ","End":"08:56.870","Text":"2x times natural log"},{"Start":"08:59.340 ","End":"09:06.115","Text":"of square root of x times let\u0027s see how we differentiate that."},{"Start":"09:06.115 ","End":"09:13.240","Text":"Remember that fg prime is equal to f prime"},{"Start":"09:13.240 ","End":"09:18.410","Text":"g plus"},{"Start":"09:18.410 ","End":"09:23.470","Text":"fg prime,"},{"Start":"09:23.470 ","End":"09:30.820","Text":"the half can just stay and we\u0027ll call this bit"},{"Start":"09:30.820 ","End":"09:39.954","Text":"f and this bit will be g and so we\u0027ll get times,"},{"Start":"09:39.954 ","End":"09:50.064","Text":"the half can just come out of the derivative, so we have f prime."},{"Start":"09:50.064 ","End":"09:57.055","Text":"Now, if we had the square root of x,"},{"Start":"09:57.055 ","End":"09:59.965","Text":"then its derivative, it\u0027s well known,"},{"Start":"09:59.965 ","End":"10:01.870","Text":"you could compute it."},{"Start":"10:01.870 ","End":"10:06.130","Text":"But it\u0027s not well-known to be 1 over twice the square root of x."},{"Start":"10:06.130 ","End":"10:07.660","Text":"Now, here, I don\u0027t have x,"},{"Start":"10:07.660 ","End":"10:10.315","Text":"I have 2x so I\u0027ll do the same thing,"},{"Start":"10:10.315 ","End":"10:12.085","Text":"but using the chain rule,"},{"Start":"10:12.085 ","End":"10:19.045","Text":"I need a bracket here because I\u0027ll need the f prime g and then the fg prime."},{"Start":"10:19.045 ","End":"10:23.170","Text":"First, this is f. The f prime is going to be"},{"Start":"10:23.170 ","End":"10:32.140","Text":"1 over twice the square root of 2x from this rule,"},{"Start":"10:32.140 ","End":"10:41.005","Text":"which could be generalized to say that the square root"},{"Start":"10:41.005 ","End":"10:51.205","Text":"of something prime is 1 over twice the square root of that something."},{"Start":"10:51.205 ","End":"10:58.045","Text":"But I have to also multiply by the internal and my case that something is just 2x."},{"Start":"10:58.045 ","End":"10:59.680","Text":"So I have to multiply that by"},{"Start":"10:59.680 ","End":"11:05.740","Text":"2 to the derivative of 2x but instead of multiplying, putting a 2 here,"},{"Start":"11:05.740 ","End":"11:11.125","Text":"I\u0027ll notice that I can put it on the top"},{"Start":"11:11.125 ","End":"11:20.665","Text":"and instead of the 1, I can put the 2 here and then the 2 cancels with the 2,"},{"Start":"11:20.665 ","End":"11:23.470","Text":"so what we\u0027re left for the f prime,"},{"Start":"11:23.470 ","End":"11:27.655","Text":"that is just 1 over square root of 2x."},{"Start":"11:27.655 ","End":"11:30.760","Text":"Now, we come to the g part,"},{"Start":"11:30.760 ","End":"11:37.600","Text":"which is natural log of x and then f,"},{"Start":"11:37.600 ","End":"11:47.170","Text":"which is square root of 2x and then,"},{"Start":"11:47.170 ","End":"11:49.315","Text":"we need the g prime,"},{"Start":"11:49.315 ","End":"11:52.330","Text":"which is just the derivative of natural log of x,"},{"Start":"11:52.330 ","End":"11:57.290","Text":"which is 1 over x. I think this is right."},{"Start":"11:57.540 ","End":"12:01.525","Text":"Another thing that you should do along the way somewhere is to convert"},{"Start":"12:01.525 ","End":"12:05.425","Text":"this form back to the original."},{"Start":"12:05.425 ","End":"12:07.315","Text":"So I\u0027ll do it at this point,"},{"Start":"12:07.315 ","End":"12:09.250","Text":"it doesn\u0027t really matter at what point."},{"Start":"12:09.250 ","End":"12:11.470","Text":"Whenever you remember."},{"Start":"12:11.470 ","End":"12:19.045","Text":"Square root of x to the power of square root of 2x,"},{"Start":"12:19.045 ","End":"12:21.400","Text":"that\u0027s just converting from the e form to"},{"Start":"12:21.400 ","End":"12:25.225","Text":"the original form like we did from here to here only backwards,"},{"Start":"12:25.225 ","End":"12:30.460","Text":"times, all this,"},{"Start":"12:30.460 ","End":"12:33.385","Text":"does it simplify in any way?"},{"Start":"12:33.385 ","End":"12:38.245","Text":"Maybe it does, but I think we could just leave it as it is,"},{"Start":"12:38.245 ","End":"12:48.425","Text":"times 1/2 times 1 over square root."},{"Start":"12:48.425 ","End":"12:50.650","Text":"Well, if it\u0027s times natural log of x,"},{"Start":"12:50.650 ","End":"12:52.180","Text":"I can put that on the numerator,"},{"Start":"12:52.180 ","End":"12:57.895","Text":"natural log of x over square root of 2x,"},{"Start":"12:57.895 ","End":"13:04.465","Text":"plus square root of 2x over x."},{"Start":"13:04.465 ","End":"13:07.960","Text":"This, as I say, you could probably mess around with it and simplify it,"},{"Start":"13:07.960 ","End":"13:12.250","Text":"but I don\u0027t think there\u0027s any purpose in that, after all,"},{"Start":"13:12.250 ","End":"13:20.410","Text":"we\u0027re just studying how to do derivatives and I\u0027ll leave that as the answer to number 6,"},{"Start":"13:22.070 ","End":"13:26.920","Text":"moving on is number 7."},{"Start":"13:27.600 ","End":"13:30.520","Text":"Where in number 7,"},{"Start":"13:30.520 ","End":"13:39.560","Text":"y equals x to the power of e to the x,"},{"Start":"13:43.860 ","End":"13:50.020","Text":"stroke down a bit and once again,"},{"Start":"13:50.020 ","End":"13:54.535","Text":"we\u0027re going to use that formula converts it into the e part,"},{"Start":"13:54.535 ","End":"13:59.140","Text":"into the e form which is y equals e to the power"},{"Start":"13:59.140 ","End":"14:08.320","Text":"of e to the x times natural log of x."},{"Start":"14:08.320 ","End":"14:11.900","Text":"Extend this line here."},{"Start":"14:14.250 ","End":"14:20.185","Text":"What we need is now the product rule again."},{"Start":"14:20.185 ","End":"14:23.900","Text":"It\u0027s quick enough to write."},{"Start":"14:25.500 ","End":"14:31.190","Text":"That\u0027s what it is. Now, the derivative"},{"Start":"14:33.510 ","End":"14:39.145","Text":"is equal to derivative of e to the something is the same,"},{"Start":"14:39.145 ","End":"14:41.510","Text":"e to the something,"},{"Start":"14:43.110 ","End":"14:47.725","Text":"e to the x, natural log of x."},{"Start":"14:47.725 ","End":"14:52.480","Text":"But we have to multiply by the derivative of the exponent,"},{"Start":"14:52.480 ","End":"14:55.569","Text":"I have done this several times before,"},{"Start":"14:55.569 ","End":"15:00.530","Text":"to the x natural log of x derivative."},{"Start":"15:00.690 ","End":"15:03.265","Text":"This is equal to."},{"Start":"15:03.265 ","End":"15:06.640","Text":"Now, I remembered that at some point,"},{"Start":"15:06.640 ","End":"15:10.765","Text":"we should convert back just because it\u0027s nicer."},{"Start":"15:10.765 ","End":"15:15.715","Text":"This thing is just the original x to the power of e to the x,"},{"Start":"15:15.715 ","End":"15:23.110","Text":"x to the power of e to the x and here we can use the product where this part will be the"},{"Start":"15:23.110 ","End":"15:31.285","Text":"f and this part will be the g. What we get is f prime,"},{"Start":"15:31.285 ","End":"15:35.905","Text":"that\u0027s e to the x times g,"},{"Start":"15:35.905 ","End":"15:38.035","Text":"which is natural log of x,"},{"Start":"15:38.035 ","End":"15:41.560","Text":"plus f, which is also e to the x,"},{"Start":"15:41.560 ","End":"15:47.719","Text":"f and f prime are the same and g prime is 1 over x."},{"Start":"15:47.910 ","End":"15:56.665","Text":"Natural logs derivative and that\u0027s the answer except the simplification."},{"Start":"15:56.665 ","End":"15:58.430","Text":"I see e to the x twice,"},{"Start":"15:58.430 ","End":"16:00.244","Text":"I would take it out the brackets."},{"Start":"16:00.244 ","End":"16:06.980","Text":"So it\u0027s x e to the x times e to"},{"Start":"16:06.980 ","End":"16:16.940","Text":"the x times natural log of x plus 1 over x and that\u0027s the answer for number 7"},{"Start":"16:16.940 ","End":"16:27.325","Text":"and next after 7 is 8, and 8"},{"Start":"16:27.325 ","End":"16:36.994","Text":"is that y equals x to the power of x to the power of x."},{"Start":"16:36.994 ","End":"16:43.475","Text":"I want to emphasize that when you have something like"},{"Start":"16:43.475 ","End":"16:50.430","Text":"a to the power of b to the power of c, in general, and if you don\u0027t write any brackets,"},{"Start":"16:51.190 ","End":"16:57.350","Text":"it means it is defined as a to the power"},{"Start":"16:57.350 ","End":"17:04.910","Text":"of b to the power of c. That in another color,"},{"Start":"17:04.910 ","End":"17:12.775","Text":"it is not a to the power of b, all that,"},{"Start":"17:12.775 ","End":"17:17.720","Text":"to the power of c. Just pay attention to that,"},{"Start":"17:17.720 ","End":"17:21.410","Text":"that it\u0027s as if the top 2 were in parentheses."},{"Start":"17:21.410 ","End":"17:24.665","Text":"So what I could really do is write"},{"Start":"17:24.665 ","End":"17:29.090","Text":"the parentheses here just so it\u0027s absolutely clear which way"},{"Start":"17:29.090 ","End":"17:33.930","Text":"round it is. Now,"},{"Start":"17:33.970 ","End":"17:41.390","Text":"you really should know these formulae by heart by now."},{"Start":"17:41.390 ","End":"17:46.880","Text":"We put it into the e form, so y equals e to the power,"},{"Start":"17:46.880 ","End":"17:48.770","Text":"it\u0027s a square to the triangle."},{"Start":"17:48.770 ","End":"17:51.230","Text":"So it\u0027s triangle log square."},{"Start":"17:51.230 ","End":"17:54.320","Text":"This is e to the x,"},{"Start":"17:54.320 ","End":"17:56.135","Text":"to the power of x,"},{"Start":"17:56.135 ","End":"18:02.750","Text":"natural log of x. Now, we use"},{"Start":"18:02.750 ","End":"18:10.685","Text":"the chain rule with the e to the power of to say that this is equal to the thing itself,"},{"Start":"18:10.685 ","End":"18:16.085","Text":"e to the x to the x times natural log of x,"},{"Start":"18:16.085 ","End":"18:21.230","Text":"but times the derivative of that exponent, x to"},{"Start":"18:21.230 ","End":"18:28.950","Text":"the x natural log of x derivative."},{"Start":"18:32.730 ","End":"18:36.940","Text":"Now, at some point, when you remember, you convert back"},{"Start":"18:36.940 ","End":"18:40.315","Text":"from this e form to the original form."},{"Start":"18:40.315 ","End":"18:42.430","Text":"This is just algebraically equal."},{"Start":"18:42.430 ","End":"18:49.705","Text":"It\u0027s x^x^x times, now,"},{"Start":"18:49.705 ","End":"18:52.435","Text":"here we can use the chain rule,"},{"Start":"18:52.435 ","End":"18:59.440","Text":"and the chain rule, just memorize it."},{"Start":"18:59.440 ","End":"19:06.700","Text":"Very useful. f prime g plus f g prime."},{"Start":"19:06.700 ","End":"19:10.550","Text":"What I\u0027ll get is,"},{"Start":"19:11.790 ","End":"19:19.090","Text":"I\u0027m already stuck at the first part because I don\u0027t know what the derivative of x^x is."},{"Start":"19:19.090 ","End":"19:20.530","Text":"Let\u0027s just for the moment,"},{"Start":"19:20.530 ","End":"19:27.970","Text":"write it as x^x derivative times natural log of x"},{"Start":"19:27.970 ","End":"19:35.380","Text":"plus x^x times the derivative,"},{"Start":"19:35.380 ","End":"19:43.515","Text":"which is 1 over x. I think we can do this x^x at the side here."},{"Start":"19:43.515 ","End":"19:48.010","Text":"Let\u0027s take this part here and do it over here."},{"Start":"19:48.350 ","End":"19:50.790","Text":"Let\u0027s call it some letter."},{"Start":"19:50.790 ","End":"19:58.090","Text":"Let\u0027s say z is x^x,"},{"Start":"19:58.090 ","End":"20:00.490","Text":"and then I\u0027ll put z back in there."},{"Start":"20:00.490 ","End":"20:02.290","Text":"When I find the derivative of z,"},{"Start":"20:02.290 ","End":"20:04.090","Text":"that\u0027s what I want,"},{"Start":"20:04.090 ","End":"20:07.315","Text":"and then I\u0027ll put it back in."},{"Start":"20:07.315 ","End":"20:09.040","Text":"Now, before we do that,"},{"Start":"20:09.040 ","End":"20:13.990","Text":"let\u0027s just use that funny rule we had with the triangle and the square."},{"Start":"20:13.990 ","End":"20:24.310","Text":"This is e^x natural log of x."},{"Start":"20:24.310 ","End":"20:33.500","Text":"z prime is equal to this thing itself,"},{"Start":"20:35.190 ","End":"20:41.630","Text":"e^x natural log of x times the derivative,"},{"Start":"20:50.670 ","End":"20:53.260","Text":"it\u0027s without the e,"},{"Start":"20:53.260 ","End":"21:01.040","Text":"the derivative is just of the x natural log of x derivative."},{"Start":"21:01.500 ","End":"21:06.260","Text":"It\u0027s time for a second use of the chain rule."},{"Start":"21:08.430 ","End":"21:15.280","Text":"Now, also, put this back into the original form because this is equal to this."},{"Start":"21:15.280 ","End":"21:18.625","Text":"Let\u0027s put this as x^x and"},{"Start":"21:18.625 ","End":"21:27.160","Text":"use the product rule for when you multiply 2 things."},{"Start":"21:27.160 ","End":"21:28.870","Text":"That\u0027s going to be the f,"},{"Start":"21:28.870 ","End":"21:30.130","Text":"that\u0027s going to be the g,"},{"Start":"21:30.130 ","End":"21:33.430","Text":"so it\u0027s going to be f prime,"},{"Start":"21:33.430 ","End":"21:35.620","Text":"which is 1 times g,"},{"Start":"21:35.620 ","End":"21:41.050","Text":"which is natural log of x plus f in itself,"},{"Start":"21:41.050 ","End":"21:44.840","Text":"and g prime is 1 over x."},{"Start":"21:45.300 ","End":"21:47.545","Text":"What we\u0027re left with"},{"Start":"21:47.545 ","End":"21:55.045","Text":"is x^x"},{"Start":"21:55.045 ","End":"21:58.990","Text":"times natural log"},{"Start":"21:58.990 ","End":"22:01.795","Text":"of x plus 1,"},{"Start":"22:01.795 ","End":"22:04.670","Text":"and all this is z prime."},{"Start":"22:05.130 ","End":"22:08.275","Text":"That\u0027s exactly what we need here."},{"Start":"22:08.275 ","End":"22:10.315","Text":"We\u0027ll just plug it in there."},{"Start":"22:10.315 ","End":"22:18.190","Text":"We have x^x^x and inside the brackets,"},{"Start":"22:18.190 ","End":"22:21.505","Text":"we have z prime,"},{"Start":"22:21.505 ","End":"22:23.440","Text":"which is x^x,"},{"Start":"22:23.440 ","End":"22:28.540","Text":"z is actually my f from here."},{"Start":"22:28.540 ","End":"22:35.720","Text":"It\u0027s"},{"Start":"22:37.230 ","End":"22:40.900","Text":"x^x"},{"Start":"22:40.900 ","End":"22:44.740","Text":"times natural log"},{"Start":"22:44.740 ","End":"22:47.830","Text":"of x plus 1,"},{"Start":"22:47.830 ","End":"22:49.930","Text":"and that\u0027s just this part,"},{"Start":"22:49.930 ","End":"22:52.279","Text":"so I need another bracket,"},{"Start":"22:54.870 ","End":"22:57.790","Text":"natural log of x,"},{"Start":"22:57.790 ","End":"22:59.860","Text":"that\u0027s the g,"},{"Start":"22:59.860 ","End":"23:01.970","Text":"that\u0027s the f prime."},{"Start":"23:13.610 ","End":"23:18.165","Text":"We don\u0027t need to refer to that because we have it all written here."},{"Start":"23:18.165 ","End":"23:22.200","Text":"We\u0027re just simplifying this now after we\u0027ve got the derivative of that."},{"Start":"23:22.200 ","End":"23:28.960","Text":"Times natural log of x plus x^x."},{"Start":"23:31.050 ","End":"23:36.740","Text":"But look, 1 over x is x to the minus 1."},{"Start":"23:37.590 ","End":"23:44.155","Text":"I\u0027m using that rule, I can just write x^x, x minus 1."},{"Start":"23:44.155 ","End":"23:46.225","Text":"On the other hand,"},{"Start":"23:46.225 ","End":"23:47.890","Text":"I might want to factor it,"},{"Start":"23:47.890 ","End":"23:49.840","Text":"so you know what, I\u0027m going to just leave it as it is."},{"Start":"23:49.840 ","End":"23:52.150","Text":"That was just worth mentioning that if it was just this,"},{"Start":"23:52.150 ","End":"23:54.490","Text":"I would simplify it by subtracting 1 there,"},{"Start":"23:54.490 ","End":"23:56.005","Text":"but now, we\u0027ll leave it as it is,"},{"Start":"23:56.005 ","End":"23:59.290","Text":"because then we can simplify it more easily."},{"Start":"23:59.290 ","End":"24:03.320","Text":"Here we are with that."},{"Start":"24:04.170 ","End":"24:08.920","Text":"That is the answer except that I just want to simplify a bit."},{"Start":"24:08.920 ","End":"24:12.805","Text":"I\u0027ll take x^x outside the brackets here."},{"Start":"24:12.805 ","End":"24:19.280","Text":"It\u0027s x^x^x times"},{"Start":"24:27.060 ","End":"24:29.740","Text":"x^x."},{"Start":"24:29.740 ","End":"24:31.315","Text":"We\u0027re left with this bit,"},{"Start":"24:31.315 ","End":"24:33.175","Text":"which I can multiply out,"},{"Start":"24:33.175 ","End":"24:39.820","Text":"which is natural log of x all squared or we can just put the 2 here,"},{"Start":"24:39.820 ","End":"24:42.260","Text":"that\u0027s from this times this."},{"Start":"24:43.140 ","End":"24:47.230","Text":"This part has come out as this."},{"Start":"24:47.230 ","End":"24:54.775","Text":"Now, plus natural log of x, that\u0027s this,"},{"Start":"24:54.775 ","End":"25:02.170","Text":"and here we can say that x^x times 1 over x,"},{"Start":"25:02.170 ","End":"25:07.555","Text":"maybe here is where I\u0027ll simplify it and say it\u0027s x^x minus 1."},{"Start":"25:07.555 ","End":"25:12.830","Text":"Just like I said, this is x to the minus 1 and we take x plus minus 1."},{"Start":"25:15.660 ","End":"25:23.815","Text":"This is basically it. That\u0027s the answer."},{"Start":"25:23.815 ","End":"25:25.945","Text":"We could simplify these 2,"},{"Start":"25:25.945 ","End":"25:31.435","Text":"I don\u0027t know if it is simpler or not by saying that this is x^x"},{"Start":"25:31.435 ","End":"25:38.200","Text":"to the x plus x. I could add these 2,"},{"Start":"25:38.200 ","End":"25:42.680","Text":"I can add this to this and the rest of it is just the same."},{"Start":"25:44.250 ","End":"25:48.980","Text":"Either leave the answer here or simplify a bit more, up to you."},{"Start":"25:49.260 ","End":"25:52.760","Text":"That\u0027s number 8 done."}],"ID":10451},{"Watched":false,"Name":"Exercise 1 - Parts 9-12","Duration":"19m 52s","ChapterTopicVideoID":10145,"CourseChapterTopicPlaylistID":8713,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.605","Text":"Scroll back to the top just to see where we are."},{"Start":"00:04.605 ","End":"00:09.045","Text":"We just finished number 8 and so the next one is number 9."},{"Start":"00:09.045 ","End":"00:18.990","Text":"Y equals sine X^x."},{"Start":"00:18.990 ","End":"00:24.780","Text":"There we are. As usual,"},{"Start":"00:24.780 ","End":"00:31.725","Text":"we\u0027re going to use the same formulae."},{"Start":"00:31.725 ","End":"00:36.040","Text":"Just a bit of a margin here,"},{"Start":"00:36.500 ","End":"00:38.700","Text":"I\u0027ll remind you again,"},{"Start":"00:38.700 ","End":"00:46.850","Text":"when we have something like square to the power of triangle,"},{"Start":"00:46.850 ","End":"00:57.165","Text":"this is the same thing as e to the power of triangle log of square natural log."},{"Start":"00:57.165 ","End":"01:00.530","Text":"We also have the other rule that we use a lot,"},{"Start":"01:00.530 ","End":"01:07.040","Text":"is that when we want to differentiate e to the power of something,"},{"Start":"01:07.040 ","End":"01:09.785","Text":"that we get the same,"},{"Start":"01:09.785 ","End":"01:18.115","Text":"e to the power of something times the derivative of that something."},{"Start":"01:18.115 ","End":"01:22.720","Text":"So these 2 rules together will help us here."},{"Start":"01:22.720 ","End":"01:25.430","Text":"So using the first rule,"},{"Start":"01:25.430 ","End":"01:32.105","Text":"we get that this is the same thing as e to the power of"},{"Start":"01:32.105 ","End":"01:43.199","Text":"x times natural log of sine x."},{"Start":"01:45.430 ","End":"01:50.105","Text":"Because here, the square is"},{"Start":"01:50.105 ","End":"01:59.420","Text":"sine x and so it\u0027s log of sine x and the triangle is the x, so that\u0027s that."},{"Start":"01:59.420 ","End":"02:05.855","Text":"Now, differentiating this whole exponent is the rectangle."},{"Start":"02:05.855 ","End":"02:14.770","Text":"We get e^x times log sine x,"},{"Start":"02:14.770 ","End":"02:17.390","Text":"I think we have to put a bracket here,"},{"Start":"02:17.390 ","End":"02:22.925","Text":"times the derivative of"},{"Start":"02:22.925 ","End":"02:28.640","Text":"x times log of sine x."},{"Start":"02:28.640 ","End":"02:31.085","Text":"So this is the part I still have to do,"},{"Start":"02:31.085 ","End":"02:34.055","Text":"the differentiation of what\u0027s in the brackets."},{"Start":"02:34.055 ","End":"02:37.160","Text":"Let\u0027s do that."},{"Start":"02:37.160 ","End":"02:40.655","Text":"But using the product rule,"},{"Start":"02:40.655 ","End":"02:46.264","Text":"which is that fg prime is f prime,"},{"Start":"02:46.264 ","End":"02:52.530","Text":"g plus fg prime."},{"Start":"02:52.530 ","End":"02:58.985","Text":"The other thing we do is that this part here is the same as the original."},{"Start":"02:58.985 ","End":"03:03.050","Text":"Just to convert back from the e form to the original form."},{"Start":"03:03.050 ","End":"03:12.710","Text":"This is sine c^x and now we\u0027ll take this x to be the f,"},{"Start":"03:12.710 ","End":"03:22.580","Text":"and the rest of it, this part will be the g. We need to keep the brackets here."},{"Start":"03:22.580 ","End":"03:24.410","Text":"We need f prime,"},{"Start":"03:24.410 ","End":"03:27.575","Text":"which is 1 times g,"},{"Start":"03:27.575 ","End":"03:36.140","Text":"which is natural log of sine x plus f,"},{"Start":"03:36.140 ","End":"03:41.220","Text":"which is x times g prime."},{"Start":"03:41.480 ","End":"03:43.755","Text":"Here\u0027s the difficulty."},{"Start":"03:43.755 ","End":"03:45.350","Text":"Not difficulties."},{"Start":"03:45.350 ","End":"03:47.300","Text":"Let\u0027s do it at the side."},{"Start":"03:47.300 ","End":"03:54.485","Text":"What is the derivative of log of sine x?"},{"Start":"03:54.485 ","End":"03:56.990","Text":"How do I differentiate that?"},{"Start":"03:56.990 ","End":"04:05.580","Text":"Well, it\u0027s chain rule because it\u0027s natural log that makes it 1 over its argument,"},{"Start":"04:05.580 ","End":"04:08.060","Text":"1 over sine x."},{"Start":"04:08.060 ","End":"04:10.400","Text":"But because it was sine x and not x,"},{"Start":"04:10.400 ","End":"04:11.530","Text":"that\u0027s the inner function,"},{"Start":"04:11.530 ","End":"04:15.350","Text":"we need the derivative of the inner function, and if you"},{"Start":"04:15.350 ","End":"04:20.285","Text":"remember, the derivative of sine x is cosine x."},{"Start":"04:20.285 ","End":"04:27.065","Text":"This is equal to cosine x over sine x."},{"Start":"04:27.065 ","End":"04:29.510","Text":"That\u0027s just fine,"},{"Start":"04:29.510 ","End":"04:35.990","Text":"except that some people would prefer to call it cotangent."},{"Start":"04:35.990 ","End":"04:42.680","Text":"It\u0027s up to you if what we put here is cosine x over sine x,"},{"Start":"04:42.680 ","End":"04:47.500","Text":"or I\u0027ll just do it as cotangent x."},{"Start":"04:47.500 ","End":"04:52.970","Text":"But it\u0027s fine if you leave it like that and that\u0027s"},{"Start":"04:52.970 ","End":"04:58.490","Text":"basically the answer for possible simplification,"},{"Start":"04:58.490 ","End":"05:00.920","Text":"I don\u0027t even see where we can simplify,"},{"Start":"05:00.920 ","End":"05:04.850","Text":"except to remove the 1 so I\u0027ll just leave it like this."},{"Start":"05:04.850 ","End":"05:07.990","Text":"I don\u0027t want to write a new line with the 1 absent."},{"Start":"05:07.990 ","End":"05:17.770","Text":"But I could cheat and I have a little eraser and I could just cheat."},{"Start":"05:17.780 ","End":"05:21.250","Text":"That\u0027s how I\u0027ll leave my answer."},{"Start":"05:21.250 ","End":"05:24.285","Text":"That\u0027s enough."},{"Start":"05:24.285 ","End":"05:28.950","Text":"We\u0027re done with number 9 and we\u0027re going to move on to number 10."},{"Start":"05:29.800 ","End":"05:36.560","Text":"Number 10 is y equals x to"},{"Start":"05:36.560 ","End":"05:46.530","Text":"the power of cosine 2x."},{"Start":"05:46.530 ","End":"05:48.825","Text":"Still have the rules written here."},{"Start":"05:48.825 ","End":"05:59.025","Text":"This thing is equal to e to the power of cosine"},{"Start":"05:59.025 ","End":"06:06.000","Text":"2x times natural log"},{"Start":"06:06.000 ","End":"06:15.340","Text":"of just x. y prime is equal to,"},{"Start":"06:17.780 ","End":"06:21.225","Text":"we use this e rule,"},{"Start":"06:21.225 ","End":"06:25.250","Text":"and so we have just e to the power of"},{"Start":"06:25.250 ","End":"06:32.225","Text":"cosine 2x natural log of x times the derivative of this thing,"},{"Start":"06:32.225 ","End":"06:42.645","Text":"times cosine 2x times natural log of x derivative."},{"Start":"06:42.645 ","End":"06:45.585","Text":"We have to do the differentiation."},{"Start":"06:45.585 ","End":"06:50.165","Text":"As is customary, we also convert back,"},{"Start":"06:50.165 ","End":"06:58.355","Text":"the e foam was just so we could use this rule and we just go back to the original form,"},{"Start":"06:58.355 ","End":"07:05.260","Text":"which is x to the power of cosine 2x."},{"Start":"07:05.260 ","End":"07:08.750","Text":"But now we really do have to differentiate this using the product rule"},{"Start":"07:08.750 ","End":"07:12.080","Text":"where the cosine 2x will be the f and"},{"Start":"07:12.080 ","End":"07:19.130","Text":"the natural log of the x will be the g. What we have here is f prime."},{"Start":"07:19.130 ","End":"07:22.355","Text":"Now, the derivative of cosine,"},{"Start":"07:22.355 ","End":"07:25.714","Text":"if it was just cosine x,"},{"Start":"07:25.714 ","End":"07:28.805","Text":"derivative would be minus sine x."},{"Start":"07:28.805 ","End":"07:33.335","Text":"But it\u0027s not exactly cosine x because cosine is 2x."},{"Start":"07:33.335 ","End":"07:42.480","Text":"It wouldn\u0027t be minus sign of 2x, we\u0027d have to multiply it by the internal derivative,"},{"Start":"07:42.880 ","End":"07:47.905","Text":"which is 2, that from the 2x,"},{"Start":"07:47.905 ","End":"07:50.720","Text":"and that\u0027s just the f prime."},{"Start":"07:50.720 ","End":"07:52.640","Text":"Now, we need the g,"},{"Start":"07:52.640 ","End":"07:56.060","Text":"which is natural log of x."},{"Start":"07:56.060 ","End":"07:57.889","Text":"Now, the other way around,"},{"Start":"07:57.889 ","End":"07:59.960","Text":"which is fg prime,"},{"Start":"07:59.960 ","End":"08:07.900","Text":"so it\u0027s cosine 2x and g prime is 1 over x."},{"Start":"08:13.400 ","End":"08:18.060","Text":"Let\u0027s see, can we simplify anything here?"},{"Start":"08:21.320 ","End":"08:24.270","Text":"Not really."},{"Start":"08:24.270 ","End":"08:28.365","Text":"Maybe put the 2 and minus 2."},{"Start":"08:28.365 ","End":"08:29.850","Text":"I just leave it like that."},{"Start":"08:29.850 ","End":"08:31.710","Text":"If you want to simplify it,"},{"Start":"08:31.710 ","End":"08:41.410","Text":"go ahead and that\u0027s the answer to number 10, and we can move on to number 11."},{"Start":"08:42.080 ","End":"08:52.780","Text":"Number 11 is that y equals tangent x to the power of 2x."},{"Start":"08:58.760 ","End":"09:03.420","Text":"After I\u0027ve scrolled up, all my rules have disappeared,"},{"Start":"09:03.420 ","End":"09:09.705","Text":"but hopefully, they\u0027re still fresh in memory."},{"Start":"09:09.705 ","End":"09:14.080","Text":"That was a terrible line. Never mind."},{"Start":"09:17.240 ","End":"09:20.580","Text":"We write it in the e form."},{"Start":"09:20.580 ","End":"09:25.829","Text":"It\u0027s e to the power of whatever was in the power before,"},{"Start":"09:25.829 ","End":"09:29.880","Text":"together with the natural log of the base,"},{"Start":"09:29.880 ","End":"09:35.440","Text":"which is natural log of tangent x."},{"Start":"09:37.190 ","End":"09:42.840","Text":"Now, we differentiate the derivative of e to the power of something."},{"Start":"09:42.840 ","End":"09:44.460","Text":"It\u0027s just that same thing,"},{"Start":"09:44.460 ","End":"09:52.800","Text":"it\u0027s e to the 2x log of tangent x times"},{"Start":"09:52.800 ","End":"10:02.460","Text":"the derivative of 2x log"},{"Start":"10:02.460 ","End":"10:09.070","Text":"of tangent x derivative."},{"Start":"10:09.620 ","End":"10:13.890","Text":"Now, there\u0027s 2 things we can do here."},{"Start":"10:13.890 ","End":"10:19.020","Text":"At any point, we should convert back to the original form."},{"Start":"10:19.020 ","End":"10:20.745","Text":"Let\u0027s do it already,"},{"Start":"10:20.745 ","End":"10:28.050","Text":"put it as tangent x to the power of 2x."},{"Start":"10:28.050 ","End":"10:33.565","Text":"This derivative, just quickly write the product rule again,"},{"Start":"10:33.565 ","End":"10:39.070","Text":"f prime g plus fg prime."},{"Start":"10:41.260 ","End":"10:46.385","Text":"I\u0027ll take the 2 outside the brackets and this x will be the"},{"Start":"10:46.385 ","End":"10:51.780","Text":"f and this natural log of tangent x,"},{"Start":"10:51.780 ","End":"10:54.610","Text":"all this will be g,"},{"Start":"10:55.910 ","End":"10:58.485","Text":"and the 2 will just stay here."},{"Start":"10:58.485 ","End":"11:01.480","Text":"We have f prime g,"},{"Start":"11:01.520 ","End":"11:04.665","Text":"I\u0027ll leave it in the brackets,"},{"Start":"11:04.665 ","End":"11:08.490","Text":"f prime is 1 times g,"},{"Start":"11:08.490 ","End":"11:15.520","Text":"which is natural log of tangent x,"},{"Start":"11:15.950 ","End":"11:19.410","Text":"plus the other way around,"},{"Start":"11:19.410 ","End":"11:21.495","Text":"which is f as is,"},{"Start":"11:21.495 ","End":"11:26.110","Text":"which is x and g prime."},{"Start":"11:26.510 ","End":"11:30.270","Text":"This is where it might be a little bit difficult,"},{"Start":"11:30.270 ","End":"11:33.435","Text":"so I\u0027ll just put it in brackets,"},{"Start":"11:33.435 ","End":"11:44.940","Text":"natural log of tangent x prime."},{"Start":"11:44.940 ","End":"11:49.330","Text":"Then going to do this at the side."},{"Start":"11:49.820 ","End":"11:55.210","Text":"Let me just straighten this line out a bit. Hang on."},{"Start":"11:56.030 ","End":"11:59.640","Text":"I don\u0027t know if it\u0027s much better, slightly, anyway,"},{"Start":"11:59.640 ","End":"12:08.370","Text":"I want to do this as a side exercise to figure out what is the derivative of this thing,"},{"Start":"12:08.370 ","End":"12:10.725","Text":"Let\u0027s use some letter,"},{"Start":"12:10.725 ","End":"12:13.800","Text":"used up x and y, let\u0027s go for z."},{"Start":"12:13.800 ","End":"12:23.760","Text":"Let\u0027s say that z is like the g,"},{"Start":"12:23.760 ","End":"12:28.950","Text":"it\u0027s the natural log of"},{"Start":"12:28.950 ","End":"12:36.250","Text":"tangent x. I need to know what is z prime and then plug it in here."},{"Start":"12:38.150 ","End":"12:42.405","Text":"So z prime is going to be,"},{"Start":"12:42.405 ","End":"12:43.815","Text":"use the chain rule,"},{"Start":"12:43.815 ","End":"12:46.575","Text":"because it\u0027s natural log of something,"},{"Start":"12:46.575 ","End":"12:50.080","Text":"we need 1 over that something."},{"Start":"12:50.810 ","End":"12:55.990","Text":"But because that something,"},{"Start":"12:56.840 ","End":"12:59.070","Text":"it\u0027s an internal function,"},{"Start":"12:59.070 ","End":"13:00.675","Text":"it\u0027s not x, it\u0027s tangent x,"},{"Start":"13:00.675 ","End":"13:07.020","Text":"we need to multiply by tangent x derivative."},{"Start":"13:07.020 ","End":"13:10.845","Text":"Who remembers the derivative of tangent x?"},{"Start":"13:10.845 ","End":"13:12.720","Text":"You go and look it up."},{"Start":"13:12.720 ","End":"13:17.205","Text":"We could use sine over cosine and use the quotient rule,"},{"Start":"13:17.205 ","End":"13:19.810","Text":"but let\u0027s just go and look it up."},{"Start":"13:21.680 ","End":"13:23.760","Text":"Well, I just looked it up,"},{"Start":"13:23.760 ","End":"13:31.260","Text":"and the derivative of tangent of x is sometimes written as secant squared x,"},{"Start":"13:31.260 ","End":"13:34.365","Text":"but it\u0027s also written as 1 over,"},{"Start":"13:34.365 ","End":"13:40.575","Text":"which is the same thing as 1 over cosine squared of x."},{"Start":"13:40.575 ","End":"13:46.770","Text":"I prefer to use the cosine squared,"},{"Start":"13:46.770 ","End":"13:55.890","Text":"so if we put this equals 1"},{"Start":"13:55.890 ","End":"14:03.915","Text":"over tangent x times"},{"Start":"14:03.915 ","End":"14:08.265","Text":"1 over cosine squared x,"},{"Start":"14:08.265 ","End":"14:10.815","Text":"it might be able to simplify."},{"Start":"14:10.815 ","End":"14:14.865","Text":"For one thing, tangent of x is sine over cosine,"},{"Start":"14:14.865 ","End":"14:19.390","Text":"so this is equal to sine x"},{"Start":"14:24.860 ","End":"14:27.795","Text":"over cosine x."},{"Start":"14:27.795 ","End":"14:34.870","Text":"Actually, we get cosine cubed x, that\u0027s a 3."},{"Start":"14:38.240 ","End":"14:43.260","Text":"That\u0027s the answer to this part here,"},{"Start":"14:43.260 ","End":"14:45.690","Text":"so I\u0027ll just write it in."},{"Start":"14:45.690 ","End":"14:52.560","Text":"We have tangent x to the power of 2x,"},{"Start":"14:52.560 ","End":"15:02.535","Text":"times 2 times natural log"},{"Start":"15:02.535 ","End":"15:09.180","Text":"of tangent x plus x times this mass,"},{"Start":"15:09.180 ","End":"15:13.935","Text":"which is x times"},{"Start":"15:13.935 ","End":"15:20.385","Text":"sine x over cosine"},{"Start":"15:20.385 ","End":"15:26.460","Text":"squared x. I think that\u0027s about as far as we can go."},{"Start":"15:26.460 ","End":"15:27.945","Text":"You could put the 2 in front,"},{"Start":"15:27.945 ","End":"15:29.220","Text":"might look a bit nicer,"},{"Start":"15:29.220 ","End":"15:30.450","Text":"but that\u0027s basically it."},{"Start":"15:30.450 ","End":"15:32.565","Text":"We\u0027re done with number 11,"},{"Start":"15:32.565 ","End":"15:37.690","Text":"so it\u0027s time to move on to number 12."},{"Start":"15:38.360 ","End":"15:46.230","Text":"Number 12 is y equals sine x to the power"},{"Start":"15:46.230 ","End":"16:00.690","Text":"of natural log of x. I forgot the y,"},{"Start":"16:00.690 ","End":"16:03.550","Text":"so hang on a second."},{"Start":"16:05.360 ","End":"16:12.900","Text":"Got the y back in and we\u0027re going to use the same formula we\u0027ve always been using."},{"Start":"16:12.900 ","End":"16:16.305","Text":"That is that we bring it first of all in to the e form,"},{"Start":"16:16.305 ","End":"16:23.790","Text":"which is e to the power of whatever exponent was up there in the first place,"},{"Start":"16:23.790 ","End":"16:27.000","Text":"times natural log of the base,"},{"Start":"16:27.000 ","End":"16:32.580","Text":"times natural log of sine x."},{"Start":"16:32.580 ","End":"16:35.790","Text":"When we differentiate,"},{"Start":"16:35.790 ","End":"16:39.570","Text":"and I better scroll up a bit,"},{"Start":"16:39.570 ","End":"16:44.630","Text":"then we get this thing itself,"},{"Start":"16:44.630 ","End":"16:49.745","Text":"which is e to the power of natural log of"},{"Start":"16:49.745 ","End":"16:57.820","Text":"x times natural log of sine x,"},{"Start":"16:57.820 ","End":"17:04.040","Text":"times the derivative of that exponent."},{"Start":"17:04.040 ","End":"17:13.880","Text":"Just copy it. Log of sine x, all this derivative."},{"Start":"17:18.410 ","End":"17:20.950","Text":"This is equal to,"},{"Start":"17:20.950 ","End":"17:22.900","Text":"and I\u0027m also, at this point,"},{"Start":"17:22.900 ","End":"17:28.345","Text":"now that I remember that we should go back from the e form to the original form,"},{"Start":"17:28.345 ","End":"17:32.690","Text":"because it\u0027s more aesthetic that way that we kept."},{"Start":"17:32.990 ","End":"17:37.230","Text":"That\u0027s the way it\u0027s done, preferably."},{"Start":"17:37.230 ","End":"17:39.115","Text":"Not wrong to leave it like that,"},{"Start":"17:39.115 ","End":"17:41.920","Text":"but we should stay close to the original,"},{"Start":"17:41.920 ","End":"17:48.725","Text":"sine x to the power of natural log of x."},{"Start":"17:48.725 ","End":"17:53.970","Text":"Now, I\u0027m going to use the product rule."},{"Start":"17:53.970 ","End":"17:59.110","Text":"The first part will be f and the second part will be a g."},{"Start":"17:59.110 ","End":"18:06.930","Text":"The fg prime is f prime g plus fg prime."},{"Start":"18:06.930 ","End":"18:10.905","Text":"This part will be the f part,"},{"Start":"18:10.905 ","End":"18:14.980","Text":"and this whole thing from the natural log will be the g. What"},{"Start":"18:14.980 ","End":"18:19.140","Text":"we\u0027ll get will be, first of all,"},{"Start":"18:19.140 ","End":"18:24.810","Text":"f prime, that\u0027s 1 over x times g as is,"},{"Start":"18:24.810 ","End":"18:29.560","Text":"natural log of sine x."},{"Start":"18:30.920 ","End":"18:36.425","Text":"Plus the other way around, which is f,"},{"Start":"18:36.425 ","End":"18:43.640","Text":"which is natural log of x times g prime."},{"Start":"18:43.640 ","End":"18:49.315","Text":"Now, g prime, it\u0027s the chain rule,"},{"Start":"18:49.315 ","End":"18:51.230","Text":"because it\u0027s not the natural log of x,"},{"Start":"18:51.230 ","End":"18:53.450","Text":"it\u0027s natural log of the inner function."},{"Start":"18:53.450 ","End":"18:55.325","Text":"What we do is,"},{"Start":"18:55.325 ","End":"18:59.715","Text":"because it\u0027s natural log, it\u0027s 1 over x."},{"Start":"18:59.715 ","End":"19:02.010","Text":"If it was natural log of x, it would be 1 over x,"},{"Start":"19:02.010 ","End":"19:11.910","Text":"so here, it would be 1 over sine x. I\u0027m just not putting the 1 there yet,"},{"Start":"19:11.910 ","End":"19:13.410","Text":"imagine there\u0027s a 1 there,"},{"Start":"19:13.410 ","End":"19:18.485","Text":"but I have to multiply by the inner derivative,"},{"Start":"19:18.485 ","End":"19:20.420","Text":"meaning the derivative of the inner function,"},{"Start":"19:20.420 ","End":"19:21.620","Text":"the derivative of sine x,"},{"Start":"19:21.620 ","End":"19:25.550","Text":"which is cosine x. I won\u0027t even put the 1 here."},{"Start":"19:25.550 ","End":"19:29.760","Text":"I\u0027ll just put the cosine x here."},{"Start":"19:30.890 ","End":"19:34.400","Text":"That\u0027s basically it."},{"Start":"19:34.400 ","End":"19:39.050","Text":"Unless you would like to write the cosine x over sine x,"},{"Start":"19:39.050 ","End":"19:44.225","Text":"this part here, you could write it as cotangent x or not."},{"Start":"19:44.225 ","End":"19:47.810","Text":"I think this will be sufficient as a solution to number"},{"Start":"19:47.810 ","End":"19:52.080","Text":"12 and therefore to the whole series."}],"ID":10452}],"Thumbnail":null,"ID":8713},{"Name":"Implicit Differentiation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 - Parts 1-4","Duration":"13m 3s","ChapterTopicVideoID":10166,"CourseChapterTopicPlaylistID":8714,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.095","Text":"This exercise is really 12 exercises in 1."},{"Start":"00:04.095 ","End":"00:06.870","Text":"Here in this clip, we\u0027ll do the first 4."},{"Start":"00:06.870 ","End":"00:09.570","Text":"What we have to do is find y prime,"},{"Start":"00:09.570 ","End":"00:18.510","Text":"but the thing is in all these we have to do implicit differentiation."},{"Start":"00:18.510 ","End":"00:20.775","Text":"Let\u0027s start with the first one,"},{"Start":"00:20.775 ","End":"00:25.470","Text":"and I copied it and now we start differentiating with respect to x."},{"Start":"00:25.470 ","End":"00:29.130","Text":"Derivative of x squared is 2x"},{"Start":"00:29.130 ","End":"00:34.260","Text":"but the derivative of y squared because it\u0027s an expression in y,"},{"Start":"00:34.260 ","End":"00:39.135","Text":"we differentiate it as if it was x and say 2y,"},{"Start":"00:39.135 ","End":"00:42.200","Text":"but we have to remember that when it\u0027s an expression in y,"},{"Start":"00:42.200 ","End":"00:45.440","Text":"we have to also multiply by y prime."},{"Start":"00:45.440 ","End":"00:47.135","Text":"That\u0027s the big difference."},{"Start":"00:47.135 ","End":"00:50.695","Text":"On the right-hand side, it\u0027s equal to 0."},{"Start":"00:50.695 ","End":"00:54.690","Text":"The 2s cancel on both sides."},{"Start":"00:54.690 ","End":"00:57.380","Text":"All we need now is a bit of algebra."},{"Start":"00:57.380 ","End":"00:59.300","Text":"Bring the x to the other side,"},{"Start":"00:59.300 ","End":"01:03.650","Text":"get minus x and divide by y. I\u0027ll do it in 1 step."},{"Start":"01:03.650 ","End":"01:07.100","Text":"We\u0027ll write that y prime is minus x, from here,"},{"Start":"01:07.100 ","End":"01:09.265","Text":"and then divide by y,"},{"Start":"01:09.265 ","End":"01:12.225","Text":"and that\u0027s the answer to number 1."},{"Start":"01:12.225 ","End":"01:17.420","Text":"Now let\u0027s do number 2. I copied it."},{"Start":"01:17.420 ","End":"01:20.210","Text":"Now we\u0027ll do the differentiation."},{"Start":"01:20.210 ","End":"01:23.000","Text":"Notice that here I have a product."},{"Start":"01:23.000 ","End":"01:25.570","Text":"I\u0027ll remind you of the product rule."},{"Start":"01:25.570 ","End":"01:28.650","Text":"Here it is in condensed form."},{"Start":"01:28.650 ","End":"01:35.725","Text":"Let this be f and this will be g in the formula here."},{"Start":"01:35.725 ","End":"01:42.540","Text":"What we get is f prime is just 2x and"},{"Start":"01:42.540 ","End":"01:49.350","Text":"then G is y cubed plus fg prime,"},{"Start":"01:49.350 ","End":"01:53.025","Text":"f is x squared and g prime,"},{"Start":"01:53.025 ","End":"01:56.070","Text":"notice it\u0027s a function of y, it\u0027s y cubed."},{"Start":"01:56.070 ","End":"01:59.420","Text":"We start off with 3y-squared as if it was x,"},{"Start":"01:59.420 ","End":"02:02.524","Text":"and then we multiply by y prime."},{"Start":"02:02.524 ","End":"02:04.910","Text":"That\u0027s the implicit part."},{"Start":"02:04.910 ","End":"02:07.805","Text":"On the right-hand side, it\u0027s more straightforward."},{"Start":"02:07.805 ","End":"02:14.000","Text":"X just gives us 1 and y squared gives us not just 2y,"},{"Start":"02:14.000 ","End":"02:18.420","Text":"but the extra y prime, not to forget."},{"Start":"02:18.430 ","End":"02:21.410","Text":"Remember what we\u0027re looking for is y prime."},{"Start":"02:21.410 ","End":"02:24.440","Text":"I suggest the algebra is to bring"},{"Start":"02:24.440 ","End":"02:28.285","Text":"all the y prime stuff to the left and the rest on the right."},{"Start":"02:28.285 ","End":"02:31.260","Text":"I\u0027ll keep this x squared,"},{"Start":"02:31.260 ","End":"02:33.465","Text":"I\u0027ll put the 3 in front though,"},{"Start":"02:33.465 ","End":"02:41.645","Text":"3x squared times y squared and then y prime."},{"Start":"02:41.645 ","End":"02:48.795","Text":"Now from here, bring to the other side minus 2yy prime."},{"Start":"02:48.795 ","End":"02:50.300","Text":"Then everything else to the right,"},{"Start":"02:50.300 ","End":"02:52.160","Text":"the 1 was already on the right,"},{"Start":"02:52.160 ","End":"02:54.395","Text":"and then this one bring over,"},{"Start":"02:54.395 ","End":"02:58.590","Text":"2xy cubed with a minus."},{"Start":"02:58.730 ","End":"03:01.110","Text":"Then I\u0027ll do 2 steps in 1."},{"Start":"03:01.110 ","End":"03:05.825","Text":"We can take y prime outside the brackets and then divide by it."},{"Start":"03:05.825 ","End":"03:11.295","Text":"We get that y prime is equal to,"},{"Start":"03:11.295 ","End":"03:12.555","Text":"start off with this,"},{"Start":"03:12.555 ","End":"03:16.865","Text":"1 minus 2xy cubed divided by,"},{"Start":"03:16.865 ","End":"03:22.295","Text":"now, we divide by what we would have got alongside the y prime."},{"Start":"03:22.295 ","End":"03:25.790","Text":"It\u0027s 3x squared y squared,"},{"Start":"03:25.790 ","End":"03:30.095","Text":"from here, minus 2y."},{"Start":"03:30.095 ","End":"03:34.220","Text":"If you\u0027re not sure, do it in 2 steps take y prime outside the brackets,"},{"Start":"03:34.220 ","End":"03:40.010","Text":"you get y prime times this and then divide by it. That\u0027s number 2."},{"Start":"03:40.010 ","End":"03:41.870","Text":"Now onto 3."},{"Start":"03:41.870 ","End":"03:45.570","Text":"I scroll down a bit."},{"Start":"03:46.010 ","End":"03:50.260","Text":"I\u0027ll just copy number 3 first."},{"Start":"03:51.230 ","End":"04:00.705","Text":"Y squared plus x over y cubed minus 4x equals 1."},{"Start":"04:00.705 ","End":"04:04.640","Text":"You know what, before this whole scrolls out of sight,"},{"Start":"04:04.640 ","End":"04:08.285","Text":"let me also copy number 4."},{"Start":"04:08.285 ","End":"04:10.130","Text":"Well, I still see it."},{"Start":"04:10.130 ","End":"04:14.450","Text":"Square root of y plus square root of x equals 1,"},{"Start":"04:14.450 ","End":"04:17.360","Text":"but we\u0027re going to work on number 3."},{"Start":"04:17.360 ","End":"04:23.040","Text":"I don\u0027t need the original anymore, I have them copied."},{"Start":"04:23.040 ","End":"04:27.125","Text":"Here, we need the quotient rule not the product rule."},{"Start":"04:27.125 ","End":"04:30.295","Text":"Let me write it at the side here."},{"Start":"04:30.295 ","End":"04:34.655","Text":"If we have 1 function of x over another function of x,"},{"Start":"04:34.655 ","End":"04:39.229","Text":"and we differentiate, what we get is a quotient."},{"Start":"04:39.229 ","End":"04:41.675","Text":"The denominator is g squared,"},{"Start":"04:41.675 ","End":"04:46.340","Text":"the numerator is the derivative of f times g"},{"Start":"04:46.340 ","End":"04:52.880","Text":"minus f times the derivative of g. The numerator here looks pretty much like this,"},{"Start":"04:52.880 ","End":"04:54.310","Text":"but with a minus."},{"Start":"04:54.310 ","End":"05:00.885","Text":"Then there\u0027s this. This is f and this is g on the bottom."},{"Start":"05:00.885 ","End":"05:03.965","Text":"Let\u0027s differentiate both sides."},{"Start":"05:03.965 ","End":"05:08.045","Text":"On the left, we have the quotient here and from the formula,"},{"Start":"05:08.045 ","End":"05:11.500","Text":"we get the derivative of the numerator."},{"Start":"05:11.500 ","End":"05:14.400","Text":"Y squared gives us 2y,"},{"Start":"05:14.400 ","End":"05:17.400","Text":"but that\u0027s not all, y prime."},{"Start":"05:17.400 ","End":"05:20.265","Text":"X gives us 1."},{"Start":"05:20.265 ","End":"05:23.585","Text":"That\u0027s the f prime part."},{"Start":"05:23.585 ","End":"05:29.040","Text":"Now we need g as is y cubed minus 4x."},{"Start":"05:29.040 ","End":"05:31.380","Text":"Now we\u0027re up to the minus,"},{"Start":"05:31.380 ","End":"05:36.555","Text":"minus f, which is y squared plus x."},{"Start":"05:36.555 ","End":"05:44.120","Text":"Then g prime y cubed gives us 3y squared for starters,"},{"Start":"05:44.120 ","End":"05:46.535","Text":"but don\u0027t forget the y prime,"},{"Start":"05:46.535 ","End":"05:49.760","Text":"and then minus 4 from here."},{"Start":"05:49.760 ","End":"05:54.994","Text":"All this over the denominator squared,"},{"Start":"05:54.994 ","End":"06:01.785","Text":"which is y cubed minus 4x all squared."},{"Start":"06:01.785 ","End":"06:03.875","Text":"This is equal to,"},{"Start":"06:03.875 ","End":"06:08.045","Text":"now the derivative of the right-hand side, is just 0."},{"Start":"06:08.045 ","End":"06:10.730","Text":"Now, here we have a fraction."},{"Start":"06:10.730 ","End":"06:12.380","Text":"If a fraction is 0,"},{"Start":"06:12.380 ","End":"06:15.200","Text":"it must be that the numerator is 0."},{"Start":"06:15.200 ","End":"06:17.990","Text":"I could just multiply both sides by the denominator."},{"Start":"06:17.990 ","End":"06:22.410","Text":"Effectively, I can just ignore this denominator."},{"Start":"06:22.610 ","End":"06:24.945","Text":"The numerator is 0,"},{"Start":"06:24.945 ","End":"06:27.640","Text":"let\u0027s expand the numerator."},{"Start":"06:28.220 ","End":"06:32.600","Text":"We have to multiply each thing in here by each thing in here."},{"Start":"06:32.600 ","End":"06:41.355","Text":"Let\u0027s start with the 2yy prime times y cubed."},{"Start":"06:41.355 ","End":"06:50.565","Text":"Then minus 2yy prime times 4x."},{"Start":"06:50.565 ","End":"06:53.100","Text":"Then plus 1,"},{"Start":"06:53.100 ","End":"06:59.460","Text":"so it\u0027s plus y cubed and then minus 4x."},{"Start":"06:59.460 ","End":"07:02.530","Text":"Now the second part,"},{"Start":"07:02.810 ","End":"07:06.490","Text":"remember there\u0027s a minus here."},{"Start":"07:07.450 ","End":"07:11.630","Text":"I think I\u0027ll just leave the minus there and then put"},{"Start":"07:11.630 ","End":"07:16.700","Text":"a square bracket and then I will have to worry about the minus right away. Let\u0027s see."},{"Start":"07:16.700 ","End":"07:19.565","Text":"I\u0027ll take y squared with each of these."},{"Start":"07:19.565 ","End":"07:25.365","Text":"I\u0027ve got y squared times 3y squared,"},{"Start":"07:25.365 ","End":"07:29.490","Text":"y prime minus 4y squared."},{"Start":"07:29.490 ","End":"07:31.780","Text":"Then the x,"},{"Start":"07:31.880 ","End":"07:38.005","Text":"plus x times 3y squared,"},{"Start":"07:38.005 ","End":"07:42.800","Text":"y prime minus x times 4x."},{"Start":"07:43.530 ","End":"07:49.520","Text":"All this is equal to 0."},{"Start":"07:50.100 ","End":"07:58.939","Text":"Now, I want to collect terms with y prime and leave them on the left."},{"Start":"07:59.300 ","End":"08:04.395","Text":"I have y prime times,"},{"Start":"08:04.395 ","End":"08:11.890","Text":"now from here I get 2y times y cubed."},{"Start":"08:13.430 ","End":"08:18.540","Text":"From here I get minus,"},{"Start":"08:18.540 ","End":"08:22.930","Text":"I\u0027ll tell you what, I\u0027ll also simplify as I go along."},{"Start":"08:22.930 ","End":"08:29.930","Text":"I won\u0027t write yy cubed, I\u0027ll write y^4th."},{"Start":"08:29.930 ","End":"08:32.470","Text":"From here, if I take y prime out,"},{"Start":"08:32.470 ","End":"08:34.405","Text":"I\u0027ll get 2 times 4 is 8,"},{"Start":"08:34.405 ","End":"08:37.780","Text":"I\u0027ll write it as 8 and then the x,"},{"Start":"08:37.780 ","End":"08:40.910","Text":"and then the y."},{"Start":"08:40.910 ","End":"08:43.350","Text":"Here I don\u0027t have y prime,"},{"Start":"08:43.350 ","End":"08:45.600","Text":"here I have it with a minus,"},{"Start":"08:45.600 ","End":"08:47.595","Text":"so I\u0027ll have put a minus."},{"Start":"08:47.595 ","End":"08:50.590","Text":"Y squared times y squared is y to the 4th,"},{"Start":"08:50.590 ","End":"08:53.780","Text":"so it\u0027s 3y to the 4th."},{"Start":"08:54.260 ","End":"08:56.715","Text":"Where else do I have a y prime?"},{"Start":"08:56.715 ","End":"09:00.725","Text":"This term here, that\u0027s also going to be with a minus."},{"Start":"09:00.725 ","End":"09:04.240","Text":"I\u0027ve got what, 3xy squared."},{"Start":"09:04.240 ","End":"09:08.105","Text":"So minus 3xy squared."},{"Start":"09:08.105 ","End":"09:13.110","Text":"Now, what do I have besides the y prime?"},{"Start":"09:13.110 ","End":"09:16.470","Text":"The other terms are, and you know what,"},{"Start":"09:16.470 ","End":"09:24.140","Text":"I\u0027ll throw them over to the right-hand side already."},{"Start":"09:24.140 ","End":"09:34.485","Text":"This becomes minus y cubed and then plus 4x."},{"Start":"09:34.485 ","End":"09:37.715","Text":"Where else don\u0027t have y prime here."},{"Start":"09:37.715 ","End":"09:39.530","Text":"This is minus minus,"},{"Start":"09:39.530 ","End":"09:40.790","Text":"which is plus,"},{"Start":"09:40.790 ","End":"09:44.105","Text":"so on the right-hand side it becomes a minus again,"},{"Start":"09:44.105 ","End":"09:47.269","Text":"so minus 4y squared."},{"Start":"09:47.269 ","End":"09:48.560","Text":"The last one is this."},{"Start":"09:48.560 ","End":"09:51.080","Text":"Again, a minus, a minus, which is a plus,"},{"Start":"09:51.080 ","End":"09:52.505","Text":"but on the other side,"},{"Start":"09:52.505 ","End":"09:56.000","Text":"it becomes again minus 4x."},{"Start":"09:56.000 ","End":"10:03.130","Text":"Now all I have to do is extract y prime by dividing."},{"Start":"10:04.220 ","End":"10:08.110","Text":"Let\u0027s start with the big dividing line."},{"Start":"10:08.270 ","End":"10:12.695","Text":"Now what I want is the right-hand side over this."},{"Start":"10:12.695 ","End":"10:16.460","Text":"But look also the 4x cancels with 4x."},{"Start":"10:16.540 ","End":"10:24.890","Text":"I get minus y cubed minus 4y squared over this."},{"Start":"10:24.890 ","End":"10:27.740","Text":"Here also, I can collect like terms,"},{"Start":"10:27.740 ","End":"10:34.025","Text":"2y to the 4th minus 3y to the 4th is just minus y to the 4th."},{"Start":"10:34.025 ","End":"10:36.950","Text":"Then minus"},{"Start":"10:36.950 ","End":"10:45.295","Text":"8xy and minus 3xy squared."},{"Start":"10:45.295 ","End":"10:48.620","Text":"That\u0027s the answer, but I noticed that everything\u0027s negative."},{"Start":"10:48.620 ","End":"10:52.130","Text":"Why don\u0027t I just multiply top and bottom by minus 1."},{"Start":"10:52.130 ","End":"10:53.300","Text":"It look nicer."},{"Start":"10:53.300 ","End":"10:59.790","Text":"I\u0027ll write it as y cubed plus 4y squared over y to"},{"Start":"10:59.790 ","End":"11:06.825","Text":"the 4th plus 8xy plus 3xy squared."},{"Start":"11:06.825 ","End":"11:09.030","Text":"That settles number 3."},{"Start":"11:09.030 ","End":"11:11.485","Text":"Now on to number 4."},{"Start":"11:11.485 ","End":"11:15.440","Text":"I\u0027ll copy it from here to here. Here it is."},{"Start":"11:15.440 ","End":"11:22.745","Text":"I\u0027d like to remind you that the derivative of the square root of x,"},{"Start":"11:22.745 ","End":"11:24.815","Text":"we\u0027ve done this many times,"},{"Start":"11:24.815 ","End":"11:28.745","Text":"is 1 over twice the square root of x."},{"Start":"11:28.745 ","End":"11:35.510","Text":"Using this here, what I get is for the square root of y,"},{"Start":"11:35.510 ","End":"11:38.225","Text":"I start off with something similar,"},{"Start":"11:38.225 ","End":"11:44.605","Text":"I start off with 1 over 2 square root of y."},{"Start":"11:44.605 ","End":"11:47.470","Text":"But because it\u0027s not x,"},{"Start":"11:47.470 ","End":"11:48.760","Text":"it\u0027s y, remember,"},{"Start":"11:48.760 ","End":"11:51.595","Text":"we have to multiply by the y prime."},{"Start":"11:51.595 ","End":"11:53.960","Text":"Here we can proceed just like here,"},{"Start":"11:53.960 ","End":"12:01.850","Text":"1 over twice square root of x and derivative of the right-hand side is 0."},{"Start":"12:03.590 ","End":"12:08.815","Text":"Now all I have to do is put stuff to the other side,"},{"Start":"12:08.815 ","End":"12:11.600","Text":"I want to extract y prime."},{"Start":"12:12.330 ","End":"12:18.760","Text":"We get that y prime is equal to."},{"Start":"12:18.760 ","End":"12:22.135","Text":"Now, first of all, throw this over to the other side,"},{"Start":"12:22.135 ","End":"12:26.300","Text":"1 over twice the square root of x,"},{"Start":"12:26.300 ","End":"12:28.300","Text":"but with a minus."},{"Start":"12:28.300 ","End":"12:30.950","Text":"Then I have to divide by this."},{"Start":"12:30.950 ","End":"12:36.605","Text":"But dividing by a fraction is like multiplying by the reciprocal."},{"Start":"12:36.605 ","End":"12:38.135","Text":"When I bring this over,"},{"Start":"12:38.135 ","End":"12:43.139","Text":"it becomes times 2 square root of y."},{"Start":"12:43.160 ","End":"12:46.290","Text":"The 2s cancel,"},{"Start":"12:46.290 ","End":"12:48.795","Text":"and I\u0027m just left with,"},{"Start":"12:48.795 ","End":"12:57.095","Text":"this is equal to minus the square root of y over the square root of x."},{"Start":"12:57.095 ","End":"12:59.105","Text":"That\u0027s the answer to number 4."},{"Start":"12:59.105 ","End":"13:03.030","Text":"We\u0027ll continue number 5 on the next clip."}],"ID":10471},{"Watched":false,"Name":"Exercise 1 - Parts 5-8","Duration":"18m 42s","ChapterTopicVideoID":10167,"CourseChapterTopicPlaylistID":8714,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.650","Text":"Just run out of space,"},{"Start":"00:01.650 ","End":"00:10.635","Text":"so just wiped clean everything we did before and we finished 1 through 4."},{"Start":"00:10.635 ","End":"00:14.350","Text":"Now we\u0027re about to start number 5."},{"Start":"00:15.440 ","End":"00:18.820","Text":"I\u0027ve written it already."},{"Start":"00:20.150 ","End":"00:23.440","Text":"We\u0027ll take it from there."},{"Start":"00:24.530 ","End":"00:28.470","Text":"Implicit differentiation is where we are."},{"Start":"00:28.470 ","End":"00:34.060","Text":"What we do is let\u0027s differentiate both sides of the equation."},{"Start":"00:34.060 ","End":"00:39.780","Text":"Now we have 3 or rather something cubed."},{"Start":"00:39.780 ","End":"00:47.105","Text":"We have to say that the derivative is 3 times whatever this"},{"Start":"00:47.105 ","End":"00:55.820","Text":"is squared times the derivative of what\u0027s inside the inner function,"},{"Start":"00:55.820 ","End":"00:57.935","Text":"which is y plus 2."},{"Start":"00:57.935 ","End":"01:01.205","Text":"We have to write the derivative of y plus 2."},{"Start":"01:01.205 ","End":"01:06.460","Text":"The y is just y prime and the 2."},{"Start":"01:07.700 ","End":"01:12.100","Text":"That\u0027s where the y prime gets in this way."},{"Start":"01:12.830 ","End":"01:18.090","Text":"On the other side of the equation we have a product."},{"Start":"01:18.530 ","End":"01:22.340","Text":"I guess I really do need to write down"},{"Start":"01:22.340 ","End":"01:25.805","Text":"the formula each time because someone might have forgotten."},{"Start":"01:25.805 ","End":"01:31.820","Text":"If we have f times g and we want to differentiate it, the product,"},{"Start":"01:31.820 ","End":"01:35.870","Text":"we take the derivative of the first and then times the"},{"Start":"01:35.870 ","End":"01:41.750","Text":"second plus the other way around the first and the derivative of the second."},{"Start":"01:41.750 ","End":"01:47.490","Text":"What we get here is x."},{"Start":"01:47.490 ","End":"01:50.220","Text":"The x prime is just 1,"},{"Start":"01:50.220 ","End":"01:59.150","Text":"and the other 1 is y plus x times the derivative of y,"},{"Start":"01:59.150 ","End":"02:04.245","Text":"which you could just say it\u0027s 1 and then remember to add the y prime,"},{"Start":"02:04.245 ","End":"02:08.400","Text":"or you could just straight away say that derivative of y is y prime."},{"Start":"02:14.330 ","End":"02:21.960","Text":"What we have to do now is isolate the places where we have y,"},{"Start":"02:22.010 ","End":"02:24.390","Text":"what we have is,"},{"Start":"02:24.390 ","End":"02:26.910","Text":"just a second there we just color it,"},{"Start":"02:26.910 ","End":"02:30.270","Text":"there\u0027s a y prime and there\u0027s y prime."},{"Start":"02:30.270 ","End":"02:34.480","Text":"We want to get those collected together."},{"Start":"02:35.030 ","End":"02:37.965","Text":"This taken outside the bracket."},{"Start":"02:37.965 ","End":"02:48.240","Text":"Let\u0027s see, I suggest moving this piece to this side so we get 3 y plus"},{"Start":"02:48.240 ","End":"02:57.195","Text":"2 squared times y prime minus xy prime"},{"Start":"02:57.195 ","End":"03:00.475","Text":"is just equal to y."},{"Start":"03:00.475 ","End":"03:04.835","Text":"Now if we take y prime outside the brackets,"},{"Start":"03:04.835 ","End":"03:09.630","Text":"we\u0027ll have this 3."},{"Start":"03:18.010 ","End":"03:23.190","Text":"Something gets fishy here."},{"Start":"03:24.050 ","End":"03:26.190","Text":"I\u0027m just terribly sorry."},{"Start":"03:26.190 ","End":"03:30.829","Text":"I wrote the 2 inside the bracket instead of outside."},{"Start":"03:30.829 ","End":"03:37.060","Text":"Give me a second. That\u0027s a small thing, but it\u0027s important."},{"Start":"03:37.190 ","End":"03:46.479","Text":"We\u0027ll take all this 3 y plus 2"},{"Start":"03:50.090 ","End":"03:56.620","Text":"squared and minus x"},{"Start":"03:58.010 ","End":"04:08.970","Text":"y prime is equal to y. y prime is going to equal this over this."},{"Start":"04:08.970 ","End":"04:14.110","Text":"It\u0027s going to equal to y over"},{"Start":"04:14.120 ","End":"04:22.005","Text":"3 y plus 2 squared minus x."},{"Start":"04:22.005 ","End":"04:27.420","Text":"That\u0027s the answer except that there is a domain."},{"Start":"04:27.550 ","End":"04:31.370","Text":"I can\u0027t say exactly,"},{"Start":"04:31.370 ","End":"04:33.800","Text":"but I can say that this mustn\u0027t be 0."},{"Start":"04:33.800 ","End":"04:36.230","Text":"All I have to do is just to say this"},{"Start":"04:36.230 ","End":"04:39.500","Text":"holds true provided this is not 0, or in other words,"},{"Start":"04:39.500 ","End":"04:50.464","Text":"3 y plus 2 squared cannot equal x."},{"Start":"04:50.464 ","End":"04:53.690","Text":"You could take it as a restriction on x. x can be"},{"Start":"04:53.690 ","End":"04:57.515","Text":"anything except 3 times y plus 2 squared."},{"Start":"04:57.515 ","End":"05:00.920","Text":"That\u0027s just so we don\u0027t divide by 0,"},{"Start":"05:00.920 ","End":"05:03.990","Text":"we should properly state that."},{"Start":"05:04.240 ","End":"05:12.380","Text":"That\u0027s it for number 5."},{"Start":"05:12.380 ","End":"05:15.570","Text":"Next 1 will be number 6."},{"Start":"05:19.210 ","End":"05:28.685","Text":"Number 6 is e^x plus e^y equals 1,"},{"Start":"05:28.685 ","End":"05:36.850","Text":"it reminds me of the 1 we had with the square root."},{"Start":"05:38.010 ","End":"05:44.545","Text":"Differentiate both sides using implicit differentiation."},{"Start":"05:44.545 ","End":"05:47.080","Text":"e^x is just e^x,"},{"Start":"05:47.080 ","End":"05:51.235","Text":"and e^y is just e^y, or is it,"},{"Start":"05:51.235 ","End":"05:58.060","Text":"remember we have to add the y prime and that equals 0 because derivative of 1 is 0."},{"Start":"05:58.060 ","End":"06:04.440","Text":"Now we have to isolate the y prime,"},{"Start":"06:04.440 ","End":"06:06.830","Text":"and in order to do that,"},{"Start":"06:06.830 ","End":"06:11.705","Text":"all I have to do is throw e^x to the other side and divide by e^y."},{"Start":"06:11.705 ","End":"06:14.510","Text":"We have the y prime equals"},{"Start":"06:14.510 ","End":"06:24.005","Text":"minus e^x over e^y."},{"Start":"06:24.005 ","End":"06:30.335","Text":"This will always be defined because e to the something is always positive."},{"Start":"06:30.335 ","End":"06:36.665","Text":"But some people might like to rewrite it."},{"Start":"06:36.665 ","End":"06:42.605","Text":"This is fine, but using the laws of exponents and powers,"},{"Start":"06:42.605 ","End":"06:44.770","Text":"this will be minus e,"},{"Start":"06:44.770 ","End":"06:47.905","Text":"we could say to the x minus y,"},{"Start":"06:47.905 ","End":"06:51.605","Text":"an option, but certainly this would be the correct answer."},{"Start":"06:51.605 ","End":"06:55.760","Text":"That\u0027s number 6. After 6,"},{"Start":"06:55.760 ","End":"07:06.395","Text":"we have and 7 is natural log of x plus natural log of y is equal to y."},{"Start":"07:06.395 ","End":"07:08.345","Text":"Let me write that down."},{"Start":"07:08.345 ","End":"07:17.945","Text":"Natural log of x plus natural log of y is equal to y."},{"Start":"07:17.945 ","End":"07:21.460","Text":"Let\u0027s get differentiating."},{"Start":"07:21.460 ","End":"07:26.550","Text":"This thing gives us 1 over x and it\u0027s a plus."},{"Start":"07:26.550 ","End":"07:27.810","Text":"The plus stays plus,"},{"Start":"07:27.810 ","End":"07:35.505","Text":"and this thing gives us 1 over y. I hope someone was going to say wait,"},{"Start":"07:35.505 ","End":"07:37.455","Text":"where\u0027s the y prime?"},{"Start":"07:37.455 ","End":"07:41.115","Text":"Well, there\u0027s the y prime because it\u0027s a function of y."},{"Start":"07:41.115 ","End":"07:42.740","Text":"On the other side also,"},{"Start":"07:42.740 ","End":"07:44.150","Text":"we don\u0027t say it\u0027s 1,"},{"Start":"07:44.150 ","End":"07:47.625","Text":"it\u0027s actually derivative of y is y prime."},{"Start":"07:47.625 ","End":"07:53.595","Text":"That means that we have a y prime here and a y prime here."},{"Start":"07:53.595 ","End":"07:58.980","Text":"We want to get it on 1 side and everything else on the other side."},{"Start":"07:59.260 ","End":"08:02.645","Text":"How would I do that?"},{"Start":"08:02.645 ","End":"08:12.740","Text":"Well, let\u0027s say, how about we throw this over to this side and say that 1 over"},{"Start":"08:12.740 ","End":"08:17.965","Text":"x is equal to"},{"Start":"08:17.965 ","End":"08:24.430","Text":"y prime minus 1 over y, y prime."},{"Start":"08:33.720 ","End":"08:37.495","Text":"If we take y prime outside the brackets,"},{"Start":"08:37.495 ","End":"08:39.160","Text":"and I\u0027ll also switch sides,"},{"Start":"08:39.160 ","End":"08:46.180","Text":"so it\u0027s y prime times 1 minus 1 over"},{"Start":"08:46.180 ","End":"08:53.815","Text":"y is equal to 1 over x."},{"Start":"08:53.815 ","End":"09:02.170","Text":"Then all I have to do is divide by 1 minus 1 over y."},{"Start":"09:02.170 ","End":"09:07.750","Text":"Y prime is equal to 1 over"},{"Start":"09:07.750 ","End":"09:15.535","Text":"x divided by 1 minus 1 over y."},{"Start":"09:15.535 ","End":"09:18.595","Text":"But I don\u0027t really like it like that,"},{"Start":"09:18.595 ","End":"09:24.200","Text":"I think we should simplify it a bit."},{"Start":"09:24.420 ","End":"09:29.540","Text":"What I\u0027m going to do is a bit of work at the side here."},{"Start":"09:32.400 ","End":"09:38.289","Text":"1 minus 1 over y,"},{"Start":"09:38.289 ","End":"09:40.390","Text":"if I give it a common denominator,"},{"Start":"09:40.390 ","End":"09:42.640","Text":"will be something over y,"},{"Start":"09:42.640 ","End":"09:52.465","Text":"will be y minus 1 over y. I want to also remind you that when we divide fractions,"},{"Start":"09:52.465 ","End":"09:55.520","Text":"let\u0027s say if we have a over b,"},{"Start":"09:59.820 ","End":"10:02.650","Text":"some people write division in this way and"},{"Start":"10:02.650 ","End":"10:12.430","Text":"some people at least they do it in England where I grew up they put divide like this,"},{"Start":"10:12.430 ","End":"10:14.500","Text":"divided by c over d,"},{"Start":"10:14.500 ","End":"10:21.129","Text":"we learn division of fractions is what we do is we take a over b and multiply"},{"Start":"10:21.129 ","End":"10:27.730","Text":"by the inverse fraction by d over c. Now why am I saying that?"},{"Start":"10:27.730 ","End":"10:31.405","Text":"Because over here we have 1 over x"},{"Start":"10:31.405 ","End":"10:38.060","Text":"divided by this thing which we\u0027ve already shown is y minus 1 over y."},{"Start":"10:40.320 ","End":"10:42.865","Text":"All I have to do is take this,"},{"Start":"10:42.865 ","End":"10:46.135","Text":"multiply it by the inverse of this."},{"Start":"10:46.135 ","End":"10:54.880","Text":"This is equal to 1 over x times the opposite fraction,"},{"Start":"10:54.880 ","End":"10:59.090","Text":"y over y minus 1."},{"Start":"10:59.850 ","End":"11:06.205","Text":"Perhaps we don\u0027t even need this 1 because we could just write it as"},{"Start":"11:06.205 ","End":"11:13.495","Text":"y divided by x times y minus 1."},{"Start":"11:13.495 ","End":"11:16.390","Text":"What restrictions do we have?"},{"Start":"11:16.390 ","End":"11:18.835","Text":"A denominator can\u0027t be 0."},{"Start":"11:18.835 ","End":"11:20.680","Text":"If a product is 0,"},{"Start":"11:20.680 ","End":"11:22.810","Text":"it must mean at least 1 of them is 0."},{"Start":"11:22.810 ","End":"11:26.350","Text":"In other words, neither these can be 0."},{"Start":"11:26.350 ","End":"11:30.850","Text":"What we have to have is x must not be 0."},{"Start":"11:30.850 ","End":"11:32.290","Text":"If this won\u0027t be 0,"},{"Start":"11:32.290 ","End":"11:36.710","Text":"we have to say y must not equal to 1."},{"Start":"11:36.720 ","End":"11:44.020","Text":"That\u0027s the answer for y prime in exercise number 7."},{"Start":"11:44.020 ","End":"11:49.040","Text":"We\u0027re going to go on to number 8."},{"Start":"11:53.580 ","End":"12:00.010","Text":"Number 8, let\u0027s see what that is."},{"Start":"12:00.010 ","End":"12:02.110","Text":"Doesn\u0027t look too bad."},{"Start":"12:02.110 ","End":"12:06.820","Text":"Natural log of y all squared,"},{"Start":"12:06.820 ","End":"12:09.620","Text":"but we could have put the 2 here,"},{"Start":"12:11.130 ","End":"12:20.335","Text":"plus y natural log of x is equal to 1."},{"Start":"12:20.335 ","End":"12:24.475","Text":"We differentiate using the chain rule."},{"Start":"12:24.475 ","End":"12:31.940","Text":"It\u0027s twice natural log of y."},{"Start":"12:32.940 ","End":"12:35.665","Text":"But first of all,"},{"Start":"12:35.665 ","End":"12:39.190","Text":"we need the inner function."},{"Start":"12:39.190 ","End":"12:44.620","Text":"We need to use the chain rule and multiply the derivative of natural log,"},{"Start":"12:44.620 ","End":"12:47.170","Text":"which is 1 over,"},{"Start":"12:47.170 ","End":"12:55.280","Text":"but that\u0027s still not it because it\u0027s a function of y we also have to throw in a y prime."},{"Start":"12:56.070 ","End":"12:59.650","Text":"Chain rule says it\u0027s twice this times"},{"Start":"12:59.650 ","End":"13:04.975","Text":"the derivative of the inside and then because it\u0027s a function of y, this also."},{"Start":"13:04.975 ","End":"13:09.670","Text":"The second piece is the chain rule,"},{"Start":"13:09.670 ","End":"13:14.060","Text":"and I\u0027m happy to write the chain rule again."},{"Start":"13:16.170 ","End":"13:21.160","Text":"Once again, the product rule by popular demand,"},{"Start":"13:21.160 ","End":"13:29.270","Text":"the fg prime equals f prime g plus fg prime."},{"Start":"13:29.670 ","End":"13:34.630","Text":"In our case, this is going to be the product and this one\u0027s going to be the f,"},{"Start":"13:34.630 ","End":"13:40.090","Text":"and this one\u0027s going to be the G. We just finished with this part,"},{"Start":"13:40.090 ","End":"13:42.565","Text":"now we\u0027re doing this part using the formula."},{"Start":"13:42.565 ","End":"13:46.155","Text":"We have f prime,"},{"Start":"13:46.155 ","End":"13:51.670","Text":"which you might think is just 1 but no,"},{"Start":"13:51.670 ","End":"13:55.345","Text":"it\u0027s 1 times y prime because it\u0027s a function of y,"},{"Start":"13:55.345 ","End":"13:59.860","Text":"or you can just simply say if f is f prime and y is y prime,"},{"Start":"13:59.860 ","End":"14:06.250","Text":"times g as is natural log of"},{"Start":"14:06.250 ","End":"14:16.420","Text":"x plus y as is and natural log of x derived,"},{"Start":"14:16.420 ","End":"14:19.940","Text":"which is just 1 over x."},{"Start":"14:20.220 ","End":"14:25.610","Text":"All this equals 0 because a constant gives 0."},{"Start":"14:27.960 ","End":"14:33.100","Text":"What I suggest is collect together the terms with y prime,"},{"Start":"14:33.100 ","End":"14:37.075","Text":"which is here and here,"},{"Start":"14:37.075 ","End":"14:46.940","Text":"these 2, and throw the other one to the other side and then divide."},{"Start":"14:47.100 ","End":"14:51.800","Text":"Yeah, that looks like the way to do it."},{"Start":"14:52.380 ","End":"14:57.610","Text":"We\u0027ll take y prime out of the brackets,"},{"Start":"14:57.610 ","End":"15:00.520","Text":"and what we have is this whole thing."},{"Start":"15:00.520 ","End":"15:08.515","Text":"I will write it as 2 natural log of y over y,"},{"Start":"15:08.515 ","End":"15:10.660","Text":"rather than separately,"},{"Start":"15:10.660 ","End":"15:16.075","Text":"times y prime plus natural log of"},{"Start":"15:16.075 ","End":"15:23.960","Text":"x is equal to minus y over x."},{"Start":"15:27.060 ","End":"15:33.740","Text":"Of course, here\u0027s my y prime. This is where I got to."},{"Start":"15:33.960 ","End":"15:38.500","Text":"Now we just have to divide by this,"},{"Start":"15:38.500 ","End":"15:42.560","Text":"but I would suggest putting a common denominator first."},{"Start":"15:44.010 ","End":"15:51.145","Text":"Let\u0027s put a common denominator here and make it all over y."},{"Start":"15:51.145 ","End":"15:54.250","Text":"For the first thing, I don\u0027t need anything."},{"Start":"15:54.250 ","End":"16:00.145","Text":"The first term is 2 natural log of y."},{"Start":"16:00.145 ","End":"16:02.800","Text":"My y didn\u0027t come out so great."},{"Start":"16:02.800 ","End":"16:07.495","Text":"Here, I have to multiply top and bottom by y"},{"Start":"16:07.495 ","End":"16:14.365","Text":"so I have y natural log of x,"},{"Start":"16:14.365 ","End":"16:23.180","Text":"and all this times y prime equals minus y over x."},{"Start":"16:23.970 ","End":"16:29.455","Text":"Now, in one of the earlier exercises,"},{"Start":"16:29.455 ","End":"16:32.455","Text":"I mentioned that when we divide fractions,"},{"Start":"16:32.455 ","End":"16:35.935","Text":"we multiply by the inverse and I can do this right away."},{"Start":"16:35.935 ","End":"16:38.140","Text":"I want to isolate y primes."},{"Start":"16:38.140 ","End":"16:40.240","Text":"I want to divide by this fraction."},{"Start":"16:40.240 ","End":"16:42.445","Text":"When I throw it over the other side,"},{"Start":"16:42.445 ","End":"16:44.680","Text":"it becomes the inverse fraction."},{"Start":"16:44.680 ","End":"16:51.970","Text":"We get that y prime is minus y over"},{"Start":"16:51.970 ","End":"17:01.930","Text":"x divided"},{"Start":"17:01.930 ","End":"17:03.490","Text":"by this thing"},{"Start":"17:03.490 ","End":"17:05.815","Text":"which is times its opposite."},{"Start":"17:05.815 ","End":"17:12.130","Text":"The opposite is reciprocal inverse whatever,"},{"Start":"17:12.130 ","End":"17:17.860","Text":"is y over 2 natural log of"},{"Start":"17:17.860 ","End":"17:27.300","Text":"y plus y natural log of x."},{"Start":"17:30.580 ","End":"17:35.675","Text":"There\u0027s not really much to do for simplification,"},{"Start":"17:35.675 ","End":"17:40.295","Text":"I suppose it might be nicer to write minus at the side,"},{"Start":"17:40.295 ","End":"17:49.445","Text":"y squared from the y times y at the top over x times"},{"Start":"17:49.445 ","End":"17:59.805","Text":"this thing to natural log y plus y, natural log x."},{"Start":"17:59.805 ","End":"18:03.620","Text":"What restrictions do we have here on the domain?"},{"Start":"18:03.620 ","End":"18:06.870","Text":"Well, certainly x shouldn\u0027t be 0."},{"Start":"18:07.210 ","End":"18:13.945","Text":"That\u0027s one exclusion where x cannot be 0."},{"Start":"18:13.945 ","End":"18:18.770","Text":"Also, this thing should not come out to be 0."},{"Start":"18:18.770 ","End":"18:26.700","Text":"That 2 natural log y plus y natural log x"},{"Start":"18:26.700 ","End":"18:29.120","Text":"should also not be allowed to be 0."},{"Start":"18:29.120 ","End":"18:30.980","Text":"We can\u0027t substitute x or y,"},{"Start":"18:30.980 ","End":"18:34.050","Text":"that will make one of these things 0."},{"Start":"18:35.370 ","End":"18:39.710","Text":"That looks like it for number 8."}],"ID":10472},{"Watched":false,"Name":"Exercise 1 - Parts 9-12","Duration":"38m 19s","ChapterTopicVideoID":27370,"CourseChapterTopicPlaylistID":8714,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:02.565","Text":"We just did number 8."},{"Start":"00:02.565 ","End":"00:04.650","Text":"Next is number 9,"},{"Start":"00:04.650 ","End":"00:07.080","Text":"and here we are,"},{"Start":"00:07.080 ","End":"00:12.105","Text":"a sin y plus cosine x equals y squared."},{"Start":"00:12.105 ","End":"00:15.390","Text":"Let\u0027s do the implicit differentiation"},{"Start":"00:15.390 ","End":"00:19.320","Text":"and not forget to put the y prime where it\u0027s supposed to go."},{"Start":"00:19.320 ","End":"00:25.560","Text":"Derivative of sine is cosine so cosine y,"},{"Start":"00:25.560 ","End":"00:28.155","Text":"but because it\u0027s y not x,"},{"Start":"00:28.155 ","End":"00:30.579","Text":"we put y prime."},{"Start":"00:30.579 ","End":"00:35.205","Text":"Derivative of cosine is minus sine."},{"Start":"00:35.205 ","End":"00:39.990","Text":"It\u0027s plus minus sine x,"},{"Start":"00:39.990 ","End":"00:44.310","Text":"just written that as I could have just put a minus there never mind,"},{"Start":"00:45.470 ","End":"00:52.655","Text":"equals derivative of y squared is 2y."},{"Start":"00:52.655 ","End":"00:53.735","Text":"That\u0027s just the start."},{"Start":"00:53.735 ","End":"00:56.880","Text":"We have to remember to put the y prime."},{"Start":"00:58.060 ","End":"01:03.685","Text":"Now we need to isolate."},{"Start":"01:03.685 ","End":"01:11.550","Text":"What we really want is y prime and we have a y prime here and we have a y prime here."},{"Start":"01:12.010 ","End":"01:15.470","Text":"What we\u0027ll do is, let\u0027s say,"},{"Start":"01:15.470 ","End":"01:17.690","Text":"we\u0027ll put this over this side and this over"},{"Start":"01:17.690 ","End":"01:21.710","Text":"this side and take y prime outside the brackets all in 1."},{"Start":"01:21.710 ","End":"01:23.435","Text":"I think we can do that."},{"Start":"01:23.435 ","End":"01:27.720","Text":"We have cosine y,"},{"Start":"01:30.980 ","End":"01:35.170","Text":"and then minus 2y."},{"Start":"01:35.270 ","End":"01:43.680","Text":"All this y prime equals this on the other side is sine x."},{"Start":"01:44.420 ","End":"01:49.295","Text":"That just leaves us with a little bit more to do,"},{"Start":"01:49.295 ","End":"01:55.355","Text":"which is that y prime is equal to this over this,"},{"Start":"01:55.355 ","End":"02:00.185","Text":"which is sine x"},{"Start":"02:00.185 ","End":"02:07.570","Text":"over cosine y minus 2y."},{"Start":"02:08.960 ","End":"02:14.385","Text":"The only thing I would do is just because it has to be defined,"},{"Start":"02:14.385 ","End":"02:15.440","Text":"it has to be a domain,"},{"Start":"02:15.440 ","End":"02:18.500","Text":"I can\u0027t say exactly what the solution to this is,"},{"Start":"02:18.500 ","End":"02:21.575","Text":"but I mean the anti solution, it should not be 0."},{"Start":"02:21.575 ","End":"02:24.020","Text":"What I\u0027m saying is denominator should not be 0."},{"Start":"02:24.020 ","End":"02:27.140","Text":"I\u0027ll just write it in words that this is not 0."},{"Start":"02:27.140 ","End":"02:29.060","Text":"I can write it another way,"},{"Start":"02:29.060 ","End":"02:33.910","Text":"cosine y not equal to 2y."},{"Start":"02:33.910 ","End":"02:40.070","Text":"I mean, there are possibly some values of y like this and there this thing won\u0027t work."},{"Start":"02:40.070 ","End":"02:44.270","Text":"Other than that, that\u0027s the solution to number 9."},{"Start":"02:44.270 ","End":"02:47.015","Text":"After 9, we have,"},{"Start":"02:47.015 ","End":"02:49.355","Text":"lets see, we have 10,"},{"Start":"02:49.355 ","End":"02:52.290","Text":"I\u0027ll just scroll up a bit,"},{"Start":"02:53.770 ","End":"02:58.260","Text":"and we have number 10."},{"Start":"03:01.480 ","End":"03:05.720","Text":"Start here, number 10,"},{"Start":"03:05.720 ","End":"03:10.780","Text":"x tangent y equals"},{"Start":"03:10.780 ","End":"03:17.890","Text":"the cube root of y."},{"Start":"03:23.300 ","End":"03:26.900","Text":"Let\u0027s just remember the product rule."},{"Start":"03:26.900 ","End":"03:28.145","Text":"We\u0027re going to need it here."},{"Start":"03:28.145 ","End":"03:30.395","Text":"It\u0027s like a product."},{"Start":"03:30.395 ","End":"03:33.035","Text":"I\u0027ll write it for you."},{"Start":"03:33.035 ","End":"03:35.530","Text":"I\u0027ll just do it in purple."},{"Start":"03:35.530 ","End":"03:39.840","Text":"That if we have f times g,"},{"Start":"03:39.840 ","End":"03:42.270","Text":"we want a derivative."},{"Start":"03:42.270 ","End":"03:45.105","Text":"It\u0027s the first one derived,"},{"Start":"03:45.105 ","End":"03:47.165","Text":"second one as is,"},{"Start":"03:47.165 ","End":"03:48.650","Text":"plus the other way around,"},{"Start":"03:48.650 ","End":"03:52.925","Text":"first one as is, second one derived."},{"Start":"03:52.925 ","End":"03:57.170","Text":"In our case, we\u0027re calling, let\u0027s say,"},{"Start":"03:57.170 ","End":"04:02.550","Text":"the first one is f and the second one is g, the tangent,"},{"Start":"04:02.550 ","End":"04:05.550","Text":"what we\u0027ll get is f prime,"},{"Start":"04:05.550 ","End":"04:11.220","Text":"which is 1 times g,"},{"Start":"04:11.220 ","End":"04:15.735","Text":"which is just tangent of y as is,"},{"Start":"04:15.735 ","End":"04:18.795","Text":"plus the other way around,"},{"Start":"04:18.795 ","End":"04:26.070","Text":"x as the one that stays as is and we need the tangent y derived."},{"Start":"04:29.300 ","End":"04:37.740","Text":"We have used this before and the tangent y, its derivative."},{"Start":"04:38.650 ","End":"04:41.720","Text":"Let me say this."},{"Start":"04:41.720 ","End":"04:45.890","Text":"If we just had tangent x and we derived it,"},{"Start":"04:45.890 ","End":"04:53.775","Text":"the answer is secant squared x or for those who don\u0027t like the secant,"},{"Start":"04:53.775 ","End":"05:00.225","Text":"secant is 1 over cosine squared x."},{"Start":"05:00.225 ","End":"05:05.570","Text":"It\u0027s not hard to show if you\u0027d use the quotient rule on sine over cosine,"},{"Start":"05:05.570 ","End":"05:09.060","Text":"that\u0027s what you would get."},{"Start":"05:10.370 ","End":"05:20.145","Text":"Here, we have tangent y not tangent of x."},{"Start":"05:20.145 ","End":"05:25.185","Text":"Its derivative, and this is a g prime part I\u0027m talking about,"},{"Start":"05:25.185 ","End":"05:26.800","Text":"its"},{"Start":"05:43.730 ","End":"05:45.885","Text":"1 over"},{"Start":"05:45.885 ","End":"05:48.760","Text":"cosine squared."},{"Start":"05:49.910 ","End":"05:56.055","Text":"Something\u0027s wrong with my pen. There it is."},{"Start":"05:56.055 ","End":"05:58.845","Text":"Cosine squared x,"},{"Start":"05:58.845 ","End":"06:02.830","Text":"oops, still not right."},{"Start":"06:03.160 ","End":"06:07.230","Text":"Just shows you how tricky all this is."},{"Start":"06:07.310 ","End":"06:12.680","Text":"Because the derivative of tangent x is 1 over cosine squared x,"},{"Start":"06:12.680 ","End":"06:17.735","Text":"the derivative of tangent y is 1 over cosine squared y."},{"Start":"06:17.735 ","End":"06:23.555","Text":"But with a big exception that here we have to throw in the y prime."},{"Start":"06:23.555 ","End":"06:25.950","Text":"That\u0027s the way it works."},{"Start":"06:28.030 ","End":"06:33.230","Text":"That\u0027s just the left-hand side."},{"Start":"06:33.230 ","End":"06:35.975","Text":"Now we also need the right-hand side,"},{"Start":"06:35.975 ","End":"06:40.530","Text":"and we need to know what is the derivative of the cube root."},{"Start":"06:42.820 ","End":"06:49.700","Text":"What I\u0027d like to do is to figure out the derivative of the cube root."},{"Start":"06:49.700 ","End":"06:51.665","Text":"I\u0027ll do it at the side."},{"Start":"06:51.665 ","End":"06:54.815","Text":"Let me choose a color."},{"Start":"06:54.815 ","End":"06:57.500","Text":"Let\u0027s just say we have in general,"},{"Start":"06:57.500 ","End":"07:01.865","Text":"f of x equals cube root of x."},{"Start":"07:01.865 ","End":"07:04.410","Text":"How do we differentiate that?"},{"Start":"07:04.730 ","End":"07:08.000","Text":"Simplest, in my opinion,"},{"Start":"07:08.000 ","End":"07:10.730","Text":"is just to write it as a fractional power."},{"Start":"07:10.730 ","End":"07:14.015","Text":"This is equal x to the power of 1/3,"},{"Start":"07:14.015 ","End":"07:16.805","Text":"and for exponents, we know what to do."},{"Start":"07:16.805 ","End":"07:23.625","Text":"F prime of x is equal to 1/3 x to the power of,"},{"Start":"07:23.625 ","End":"07:30.940","Text":"I\u0027ve to take-off 1 from this and 1/3 less 1 is minus 2/3."},{"Start":"07:31.430 ","End":"07:37.510","Text":"From there, all I need to do is just simplify a bit."},{"Start":"07:37.610 ","End":"07:40.740","Text":"This is equal to 1/3."},{"Start":"07:40.740 ","End":"07:42.615","Text":"Now instead of the minus,"},{"Start":"07:42.615 ","End":"07:50.010","Text":"I can put 1 over x to the 2/3,"},{"Start":"07:50.010 ","End":"07:57.734","Text":"and this will give me 1 over 3."},{"Start":"07:57.734 ","End":"08:00.195","Text":"Now x to the 2/3,"},{"Start":"08:00.195 ","End":"08:05.865","Text":"in general when you have a fraction power,"},{"Start":"08:05.865 ","End":"08:14.850","Text":"it\u0027s just x to the power of 2 cube root."},{"Start":"08:14.850 ","End":"08:17.105","Text":"Let me write this a bit bigger."},{"Start":"08:17.105 ","End":"08:23.460","Text":"It\u0027s 1 over 3"},{"Start":"08:23.460 ","End":"08:29.150","Text":"times the cube root"},{"Start":"08:29.150 ","End":"08:31.820","Text":"of x squared."},{"Start":"08:31.820 ","End":"08:36.020","Text":"Otherwise, when you have 2 over 3, in general,"},{"Start":"08:36.020 ","End":"08:42.950","Text":"you could say that x to the power of m over n is the nth root of x to the"},{"Start":"08:42.950 ","End":"08:52.455","Text":"m. Now if I make use of all this over here,"},{"Start":"08:52.455 ","End":"08:58.925","Text":"I need the derivative of the cube root of y."},{"Start":"08:58.925 ","End":"09:06.840","Text":"Now it\u0027s y not x. I start off like this with 1 over,"},{"Start":"09:06.840 ","End":"09:09.120","Text":"I\u0027ll leave the 1 a minute,"},{"Start":"09:09.120 ","End":"09:11.700","Text":"there\u0027s a 1 here over"},{"Start":"09:11.700 ","End":"09:22.010","Text":"the 3 times the cube root of y squared."},{"Start":"09:22.010 ","End":"09:25.460","Text":"But because it\u0027s y and not x,"},{"Start":"09:25.460 ","End":"09:28.280","Text":"we also have to throw in y prime,"},{"Start":"09:28.280 ","End":"09:32.430","Text":"which I can just put in the numerator where I would have put the 1."},{"Start":"09:33.040 ","End":"09:40.040","Text":"This is already the implicit differentiation done."},{"Start":"09:40.040 ","End":"09:44.120","Text":"But the question asks specifically to isolate y prime."},{"Start":"09:44.120 ","End":"09:46.640","Text":"What is y prime equal?"},{"Start":"09:49.700 ","End":"09:56.900","Text":"Let\u0027s first of all point out where y prime is."},{"Start":"09:56.900 ","End":"09:59.445","Text":"It\u0027s here, and it\u0027s here."},{"Start":"09:59.445 ","End":"10:02.095","Text":"How do we isolate it?"},{"Start":"10:02.095 ","End":"10:10.860","Text":"What I would do is transfer this expression to this side and this to the other side,"},{"Start":"10:11.890 ","End":"10:17.585","Text":"and then just take it outside the brackets and divide,"},{"Start":"10:17.585 ","End":"10:20.910","Text":"just the usual algebraic stuff."},{"Start":"10:23.890 ","End":"10:29.390","Text":"Here it goes. In fact,"},{"Start":"10:29.390 ","End":"10:34.205","Text":"maybe I\u0027ll just leave the tangent and take this the other way around."},{"Start":"10:34.205 ","End":"10:44.475","Text":"We have tangent of y is equal to,"},{"Start":"10:44.475 ","End":"10:47.220","Text":"and I\u0027ll take the y prime."},{"Start":"10:47.220 ","End":"10:50.380","Text":"It\u0027s going to be this minus this."},{"Start":"10:50.530 ","End":"10:59.490","Text":"Otherwise I\u0027m going to have something minus something."},{"Start":"11:00.020 ","End":"11:04.520","Text":"Then I\u0027m going to take the y prime outside the brackets."},{"Start":"11:04.520 ","End":"11:07.115","Text":"The first something is what\u0027s here."},{"Start":"11:07.115 ","End":"11:17.950","Text":"It\u0027s 1 over 3 times the cube root of y squared,"},{"Start":"11:23.690 ","End":"11:26.385","Text":"that just about got it,"},{"Start":"11:26.385 ","End":"11:35.265","Text":"less this thing which is x over cosine squared y."},{"Start":"11:35.265 ","End":"11:39.070","Text":"Now to get y prime,"},{"Start":"11:39.830 ","End":"11:46.790","Text":"all I need to do is take tangent of y and divide it by this expression."},{"Start":"11:47.810 ","End":"11:53.650","Text":"Tangent y divided by"},{"Start":"11:55.160 ","End":"12:03.960","Text":"3 times the cube root of y squared."},{"Start":"12:03.960 ","End":"12:06.135","Text":"I\u0027m sorry, it\u0027s 1 over."},{"Start":"12:06.135 ","End":"12:10.560","Text":"Hang on. Let\u0027s do it right."},{"Start":"12:10.560 ","End":"12:20.700","Text":"It\u0027s 1 over 3 times the cube root of"},{"Start":"12:20.700 ","End":"12:26.634","Text":"y squared minus x"},{"Start":"12:26.634 ","End":"12:37.310","Text":"over cosine squared y and that\u0027s y prime."},{"Start":"12:38.150 ","End":"12:40.485","Text":"That\u0027s basically the answer."},{"Start":"12:40.485 ","End":"12:43.155","Text":"There might be some simplification 1 could do."},{"Start":"12:43.155 ","End":"12:44.940","Text":"Common denominator."},{"Start":"12:44.940 ","End":"12:47.565","Text":"I don\u0027t think it\u0027ll get much better."},{"Start":"12:47.565 ","End":"12:51.645","Text":"I\u0027ll leave it like that if you want to algebraically go on, you can."},{"Start":"12:51.645 ","End":"12:54.195","Text":"Also, I would just like to make a note"},{"Start":"12:54.195 ","End":"12:58.170","Text":"that I\u0027m not even going to write it in detail that whenever"},{"Start":"12:58.170 ","End":"13:03.140","Text":"this denominator is that should not be"},{"Start":"13:03.140 ","End":"13:08.900","Text":"0 or I could just say the denominator,"},{"Start":"13:08.900 ","End":"13:10.370","Text":"I\u0027m not going to copy the whole thing."},{"Start":"13:10.370 ","End":"13:16.725","Text":"The denominator, not to be equal to 0."},{"Start":"13:16.725 ","End":"13:18.840","Text":"If any x or y make it 0,"},{"Start":"13:18.840 ","End":"13:22.090","Text":"then it\u0027s not defined then, anyway."},{"Start":"13:22.460 ","End":"13:33.220","Text":"That\u0027s done with the number 10 and after 10 comes 11."},{"Start":"13:33.530 ","End":"13:42.090","Text":"Let\u0027s scroll a bit and we\u0027ll do number 11."},{"Start":"13:42.090 ","End":"13:48.130","Text":"Number 11 is x to the y plus y to the x equals 1."},{"Start":"13:48.500 ","End":"13:52.770","Text":"Here I write number 11,"},{"Start":"13:52.770 ","End":"14:02.620","Text":"x to the y plus y to the x is equal to 1."},{"Start":"14:03.110 ","End":"14:14.800","Text":"Let\u0027s start. We have 1 of these functions to the power of a function."},{"Start":"14:14.900 ","End":"14:18.285","Text":"I\u0027m going to, I guess,"},{"Start":"14:18.285 ","End":"14:23.290","Text":"remind you we need some form really here."},{"Start":"14:25.580 ","End":"14:28.305","Text":"Remember we did this thing."},{"Start":"14:28.305 ","End":"14:31.530","Text":"Square to the power of triangle,"},{"Start":"14:31.530 ","End":"14:35.385","Text":"is e to the power of triangle,"},{"Start":"14:35.385 ","End":"14:37.530","Text":"natural log of square."},{"Start":"14:37.530 ","End":"14:42.015","Text":"That\u0027s 1 useful thing we had and the other thing we had"},{"Start":"14:42.015 ","End":"14:47.280","Text":"was that when we have e to the something,"},{"Start":"14:47.280 ","End":"14:49.635","Text":"let\u0027s say make it a rectangle,"},{"Start":"14:49.635 ","End":"14:57.105","Text":"and we want to differentiate that or we get the thing itself e to this rectangle,"},{"Start":"14:57.105 ","End":"15:03.100","Text":"but also the derivative of the rectangle."},{"Start":"15:04.640 ","End":"15:08.410","Text":"How are we going to use that here?"},{"Start":"15:09.320 ","End":"15:16.920","Text":"First time I\u0027m going to take this is x and this is y and what I"},{"Start":"15:16.920 ","End":"15:25.995","Text":"get is that I write in the form of e to the power of I get e to the power of y,"},{"Start":"15:25.995 ","End":"15:31.350","Text":"natural log of x plus."},{"Start":"15:31.350 ","End":"15:33.060","Text":"Here I\u0027ll have the opposite."},{"Start":"15:33.060 ","End":"15:38.265","Text":"I\u0027ll have e to the power of x,"},{"Start":"15:38.265 ","End":"15:46.810","Text":"natural log of y and this equals 1."},{"Start":"15:47.060 ","End":"15:51.525","Text":"Now to differentiate this,"},{"Start":"15:51.525 ","End":"15:58.740","Text":"the first 1 is e to the power of y natural log of x"},{"Start":"15:58.740 ","End":"16:06.600","Text":"times the derivative of y natural log x derivative."},{"Start":"16:06.600 ","End":"16:09.735","Text":"I\u0027m not going to derive it just now,"},{"Start":"16:09.735 ","End":"16:12.720","Text":"just note that I have to do it."},{"Start":"16:12.720 ","End":"16:18.360","Text":"Plus e to the power of x natural log of"},{"Start":"16:18.360 ","End":"16:27.990","Text":"y times the derivative of x natural log y prime."},{"Start":"16:27.990 ","End":"16:30.105","Text":"That\u0027s to that bit and for 1,"},{"Start":"16:30.105 ","End":"16:31.995","Text":"we just get 0."},{"Start":"16:31.995 ","End":"16:38.310","Text":"I have 2 exercises that I have to do at the side."},{"Start":"16:38.310 ","End":"16:42.284","Text":"Let\u0027s maybe give them colors."},{"Start":"16:42.284 ","End":"16:45.840","Text":"Let\u0027s say this exercise here,"},{"Start":"16:45.840 ","End":"16:55.495","Text":"I\u0027ll do it at the side in blue and the other 1 I\u0027ll do at the side and say orange."},{"Start":"16:55.495 ","End":"17:01.850","Text":"Let\u0027s start with the blue 1."},{"Start":"17:01.850 ","End":"17:09.875","Text":"Of course, I\u0027m going to need the product rule that it gets to be blue, fine."},{"Start":"17:09.875 ","End":"17:15.005","Text":"The orange, f g prime is"},{"Start":"17:15.005 ","End":"17:21.680","Text":"f prime times g plus f times g prime,"},{"Start":"17:21.680 ","End":"17:24.600","Text":"we\u0027ve seen this a lot."},{"Start":"17:26.050 ","End":"17:33.825","Text":"In this case, for the y natural log x,"},{"Start":"17:33.825 ","End":"17:40.680","Text":"we have that y natural log"},{"Start":"17:40.680 ","End":"17:45.960","Text":"x prime is I\u0027m just following the cookbook,"},{"Start":"17:45.960 ","End":"17:51.210","Text":"f prime, which is, if it was x,"},{"Start":"17:51.210 ","End":"17:52.350","Text":"it would just be 1,"},{"Start":"17:52.350 ","End":"17:58.200","Text":"but it\u0027s y so it\u0027s y prime times g as it is."},{"Start":"17:58.200 ","End":"18:02.850","Text":"Natural log of x plus f as it is,"},{"Start":"18:02.850 ","End":"18:05.565","Text":"which is y, and g prime,"},{"Start":"18:05.565 ","End":"18:08.560","Text":"which is 1 over x."},{"Start":"18:20.780 ","End":"18:23.370","Text":"Well, we\u0027ll put it back in a minute."},{"Start":"18:23.370 ","End":"18:27.190","Text":"Let\u0027s do the other 1."},{"Start":"18:27.950 ","End":"18:32.220","Text":"The other 1 is x natural log of y."},{"Start":"18:32.220 ","End":"18:38.970","Text":"Again, x natural log"},{"Start":"18:38.970 ","End":"18:45.015","Text":"y derivative."},{"Start":"18:45.015 ","End":"18:47.850","Text":"It\u0027s the same product rule only I wish I"},{"Start":"18:47.850 ","End":"18:56.025","Text":"had underlined it in orange to say that this time we\u0027re taking x as our f,"},{"Start":"18:56.025 ","End":"18:58.620","Text":"and natural log y as the"},{"Start":"18:58.620 ","End":"19:03.045","Text":"g. If you think of it"},{"Start":"19:03.045 ","End":"19:05.160","Text":"as derivative of the first times the"},{"Start":"19:05.160 ","End":"19:08.175","Text":"second plus the first times derivative of the second,"},{"Start":"19:08.175 ","End":"19:10.230","Text":"then you don\u0027t have to worry."},{"Start":"19:10.230 ","End":"19:20.080","Text":"Derivative of the first is 1 and the second as is,"},{"Start":"19:22.100 ","End":"19:33.730","Text":"which is natural log of x plus the first,"},{"Start":"19:35.570 ","End":"19:40.410","Text":"let\u0027s see if I got it right, yeah,"},{"Start":"19:40.410 ","End":"19:47.084","Text":"the first which is just y times the derivative of the second,"},{"Start":"19:47.084 ","End":"19:52.090","Text":"which is also 1 over x"},{"Start":"19:52.100 ","End":"20:01.755","Text":"and that should be okay, I\u0027m going to say."},{"Start":"20:01.755 ","End":"20:06.000","Text":"Now I think we\u0027re getting confused here with the blue and the orange."},{"Start":"20:06.000 ","End":"20:12.360","Text":"Tell you what, I\u0027m going to just erase that previous line,"},{"Start":"20:12.360 ","End":"20:20.790","Text":"take out the eraser and just erase this,"},{"Start":"20:20.790 ","End":"20:23.220","Text":"and then maybe I won\u0027t get confused and I want to erase."},{"Start":"20:23.220 ","End":"20:26.085","Text":"I\u0027ve seen I\u0027ve started to do it wrong."},{"Start":"20:26.085 ","End":"20:32.625","Text":"I\u0027m just going to look at this and here we go again."},{"Start":"20:32.625 ","End":"20:41.110","Text":"The first 1 is f. Just a second there."},{"Start":"20:43.550 ","End":"20:47.010","Text":"Here we go, f is x,"},{"Start":"20:47.010 ","End":"20:49.740","Text":"so f prime is 1,"},{"Start":"20:49.740 ","End":"20:53.475","Text":"and g is natural log of y,"},{"Start":"20:53.475 ","End":"20:56.595","Text":"just stays natural log of y."},{"Start":"20:56.595 ","End":"20:59.505","Text":"Then the other way around,"},{"Start":"20:59.505 ","End":"21:01.755","Text":"f stays as is,"},{"Start":"21:01.755 ","End":"21:03.180","Text":"which is x,"},{"Start":"21:03.180 ","End":"21:05.370","Text":"and then we need the g prime,"},{"Start":"21:05.370 ","End":"21:08.490","Text":"which is a derivative of natural log y."},{"Start":"21:08.490 ","End":"21:16.940","Text":"The derivative of natural log is first of all 1 over y because natural log is 1 over,"},{"Start":"21:16.940 ","End":"21:20.690","Text":"but because it\u0027s a y it contains only y,"},{"Start":"21:20.690 ","End":"21:23.975","Text":"we also have to put in this extra y prime."},{"Start":"21:23.975 ","End":"21:27.570","Text":"I do believe that this is correct"},{"Start":"21:32.910 ","End":"21:44.080","Text":"but I was a bit hasty in erasing the blue because I don\u0027t have it for putting in here."},{"Start":"21:44.080 ","End":"21:51.385","Text":"Let me quickly go back to blue and I will let"},{"Start":"21:51.385 ","End":"21:56.500","Text":"undo again the y natural log of"},{"Start":"21:56.500 ","End":"22:01.795","Text":"x derivative and it\u0027ll be a good sign if I get the same thing as I did before."},{"Start":"22:01.795 ","End":"22:04.240","Text":"Once again, it\u0027s the derivative of the first,"},{"Start":"22:04.240 ","End":"22:07.570","Text":"which was y prime times the second,"},{"Start":"22:07.570 ","End":"22:14.625","Text":"which is natural log of x plus the other way around y stays as is,"},{"Start":"22:14.625 ","End":"22:19.590","Text":"and log x gets derived and that\u0027s 1 over x."},{"Start":"22:19.590 ","End":"22:22.160","Text":"Now I can put these things,"},{"Start":"22:22.160 ","End":"22:26.530","Text":"the blue instead of the blue and the orange instead of the orange."},{"Start":"22:26.530 ","End":"22:32.620","Text":"I\u0027ll altogether change my pen back to black and everything should be fine."},{"Start":"22:32.620 ","End":"22:39.535","Text":"We now get e to the power of y,"},{"Start":"22:39.535 ","End":"22:48.110","Text":"natural log of x times,"},{"Start":"22:48.110 ","End":"22:52.390","Text":"this I\u0027ll take from the blue part,"},{"Start":"22:52.390 ","End":"23:00.880","Text":"which is y prime natural log"},{"Start":"23:00.880 ","End":"23:05.240","Text":"of x plus y over x."},{"Start":"23:07.860 ","End":"23:11.635","Text":"That\u0027s the blue bit."},{"Start":"23:11.635 ","End":"23:18.985","Text":"Now the orange bit is going to be e to the power of x,"},{"Start":"23:18.985 ","End":"23:24.770","Text":"natural log y times."},{"Start":"23:24.960 ","End":"23:35.109","Text":"No more derivatives we note this is already the answer just not algebraically complete."},{"Start":"23:35.109 ","End":"23:38.080","Text":"Anyway, I\u0027m copying the orange one,"},{"Start":"23:38.080 ","End":"23:46.660","Text":"which is natural log of y plus this thing which is x,"},{"Start":"23:46.660 ","End":"23:49.760","Text":"y prime over y."},{"Start":"23:53.990 ","End":"23:58.630","Text":"It\u0027s an equation that\u0027s equal to 0."},{"Start":"23:58.700 ","End":"24:03.690","Text":"The next step and this is something that\u0027s customarily done."},{"Start":"24:03.690 ","End":"24:07.090","Text":"When we went from here to here,"},{"Start":"24:08.330 ","End":"24:14.320","Text":"then we took the original form of a power something to the power of something,"},{"Start":"24:14.320 ","End":"24:19.015","Text":"and we made it e to the power of same thing as we went from here to here."},{"Start":"24:19.015 ","End":"24:22.015","Text":"Now this bit is equal to this bit,"},{"Start":"24:22.015 ","End":"24:23.710","Text":"and this bit is equal to this bit."},{"Start":"24:23.710 ","End":"24:28.525","Text":"But aesthetically, it\u0027s more correct to leave it in the original form."},{"Start":"24:28.525 ","End":"24:34.750","Text":"This bit here should go back to being x to the power of y,"},{"Start":"24:34.750 ","End":"24:41.210","Text":"and this bit here should go back to being y to the power of x."},{"Start":"24:41.970 ","End":"24:47.230","Text":"Here we have x to the power of y,"},{"Start":"24:47.230 ","End":"24:50.140","Text":"and I\u0027ll multiply it out times"},{"Start":"24:50.140 ","End":"24:59.289","Text":"y prime log x"},{"Start":"24:59.289 ","End":"25:06.490","Text":"plus x to the power of y."},{"Start":"25:06.490 ","End":"25:08.815","Text":"Because this is x to the power of y,"},{"Start":"25:08.815 ","End":"25:16.550","Text":"changed it back times y over x."},{"Start":"25:18.570 ","End":"25:20.800","Text":"That\u0027s this bit."},{"Start":"25:20.800 ","End":"25:22.180","Text":"Now next bit,"},{"Start":"25:22.180 ","End":"25:25.450","Text":"I change this back to y, to the x."},{"Start":"25:25.450 ","End":"25:34.855","Text":"First of all I multiply it by this."},{"Start":"25:34.855 ","End":"25:39.610","Text":"I get y to the x, natural log y,"},{"Start":"25:39.610 ","End":"25:44.920","Text":"and then plus y to the x,"},{"Start":"25:44.920 ","End":"25:49.150","Text":"instead of this thing, times x,"},{"Start":"25:49.150 ","End":"25:57.560","Text":"y prime over y and all this is equal to 0."},{"Start":"25:59.490 ","End":"26:03.235","Text":"It\u0027s beginning to look extremely nasty here,"},{"Start":"26:03.235 ","End":"26:06.100","Text":"but let\u0027s remember to keep the focus,"},{"Start":"26:06.100 ","End":"26:08.454","Text":"what we need is y prime."},{"Start":"26:08.454 ","End":"26:11.110","Text":"Let\u0027s look at where we can find y prime."},{"Start":"26:11.110 ","End":"26:12.805","Text":"We can find it here,"},{"Start":"26:12.805 ","End":"26:14.860","Text":"and we can find it,"},{"Start":"26:14.860 ","End":"26:18.385","Text":"it\u0027s got to be hiding somewhere. There it is."},{"Start":"26:18.385 ","End":"26:23.185","Text":"What we have to do is isolate, separate y prime."},{"Start":"26:23.185 ","End":"26:27.430","Text":"What I suggest is we have 4 terms here,"},{"Start":"26:27.430 ","End":"26:30.970","Text":"we\u0027ll keep the 2 with the y prime on this side of the equals."},{"Start":"26:30.970 ","End":"26:35.860","Text":"We put the other 2 remaining terms on the other side,"},{"Start":"26:35.860 ","End":"26:40.600","Text":"we\u0027ll take y prime outside the brackets and then divide by that."},{"Start":"26:40.600 ","End":"26:44.500","Text":"That\u0027s the general idea."},{"Start":"26:44.500 ","End":"26:47.660","Text":"That\u0027s the algebraic way."},{"Start":"26:51.210 ","End":"26:57.000","Text":"Just let me change my pen back, there we are."},{"Start":"26:57.000 ","End":"27:01.310","Text":"I\u0027ll take the y primes I\u0027ll leave them here."},{"Start":"27:01.310 ","End":"27:05.395","Text":"I\u0027ll go a bit left so I have more room."},{"Start":"27:05.395 ","End":"27:08.750","Text":"Why am I telling you all this?"},{"Start":"27:09.690 ","End":"27:16.150","Text":"X to the y times natural log of"},{"Start":"27:16.150 ","End":"27:22.540","Text":"x. I\u0027m not putting the y prime because I\u0027m going to put it outside the brackets."},{"Start":"27:22.540 ","End":"27:25.045","Text":"Then the other one,"},{"Start":"27:25.045 ","End":"27:30.200","Text":"y to the x times"},{"Start":"27:30.480 ","End":"27:36.950","Text":"x over y,"},{"Start":"27:39.270 ","End":"27:44.005","Text":"all this times y prime."},{"Start":"27:44.005 ","End":"27:46.015","Text":"These are these 2 terms."},{"Start":"27:46.015 ","End":"27:50.690","Text":"The other 2, I\u0027ll put on the other side."},{"Start":"27:51.720 ","End":"28:00.205","Text":"I actually prefer to just put them inside a minus rather than making each 1 minus."},{"Start":"28:00.205 ","End":"28:07.194","Text":"I have y to the x natural log of y."},{"Start":"28:07.194 ","End":"28:10.255","Text":"Then it\u0027s a plus because there\u0027s a minus here,"},{"Start":"28:10.255 ","End":"28:13.760","Text":"just as it is plus y to the x."},{"Start":"28:17.820 ","End":"28:23.710","Text":"After all this, it should be just straightforward now to write y prime."},{"Start":"28:23.710 ","End":"28:29.350","Text":"Y prime is going to equal minus"},{"Start":"28:29.350 ","End":"28:37.010","Text":"this thing which is y to the x, natural log y."},{"Start":"28:39.900 ","End":"28:43.030","Text":"But taking a look at these 2,"},{"Start":"28:43.030 ","End":"28:44.590","Text":"y^x is there in both."},{"Start":"28:44.590 ","End":"28:46.450","Text":"So before I do that,"},{"Start":"28:46.450 ","End":"28:53.425","Text":"I think I can just make it in time to put a bracket here and plus 1."},{"Start":"28:53.425 ","End":"28:58.060","Text":"See what I did? Just took y^x outside this bracket,"},{"Start":"28:58.060 ","End":"29:04.760","Text":"and then this will go over this thing here."},{"Start":"29:07.590 ","End":"29:10.165","Text":"Something must be wrong."},{"Start":"29:10.165 ","End":"29:17.450","Text":"How can I have 3 times y^x and only once x^y. It\u0027s possible."},{"Start":"29:19.920 ","End":"29:22.675","Text":"I\u0027ve done something wrong somewhere."},{"Start":"29:22.675 ","End":"29:25.300","Text":"Yes. If you\u0027ll forgive me,"},{"Start":"29:25.300 ","End":"29:30.730","Text":"I\u0027ll just change this to x^y."},{"Start":"29:30.730 ","End":"29:34.885","Text":"Yes. You see, I miscopied this x^y as y^x,"},{"Start":"29:34.885 ","End":"29:36.685","Text":"and now I fixed it."},{"Start":"29:36.685 ","End":"29:42.085","Text":"Which means that everything else is now okay."},{"Start":"29:42.085 ","End":"29:47.230","Text":"Except that this also has to be x^y for the moment."},{"Start":"29:47.230 ","End":"29:51.805","Text":"Yes. A little bit tricky here as x^y."},{"Start":"29:51.805 ","End":"29:55.840","Text":"Miscopying is very common form of mistake,"},{"Start":"29:55.840 ","End":"29:58.495","Text":"and just showing you what you shouldn\u0027t do."},{"Start":"29:58.495 ","End":"30:00.890","Text":"You should copy correctly."},{"Start":"30:01.170 ","End":"30:04.750","Text":"So the denominator. Now,"},{"Start":"30:04.750 ","End":"30:07.390","Text":"it gives us hope that these 2 are the same,"},{"Start":"30:07.390 ","End":"30:08.665","Text":"x^y, x^y,"},{"Start":"30:08.665 ","End":"30:11.050","Text":"and here also y^x,"},{"Start":"30:11.050 ","End":"30:14.120","Text":"y^x, or vice versa."},{"Start":"30:20.700 ","End":"30:23.710","Text":"I\u0027ve really messed this 1 up, haven\u0027t I?"},{"Start":"30:23.710 ","End":"30:26.215","Text":"I have miscopied the y^x."},{"Start":"30:26.215 ","End":"30:28.240","Text":"It was right after all."},{"Start":"30:28.240 ","End":"30:33.925","Text":"Forgive. We\u0027re back with y^x."},{"Start":"30:33.925 ","End":"30:35.395","Text":"I\u0027m sticking with that,"},{"Start":"30:35.395 ","End":"30:38.725","Text":"y^x it is. It\u0027s final."},{"Start":"30:38.725 ","End":"30:41.530","Text":"Now, in the denominator,"},{"Start":"30:41.530 ","End":"30:45.085","Text":"that\u0027s where I take, divide by this."},{"Start":"30:45.085 ","End":"30:48.115","Text":"Here, I can take out x^y."},{"Start":"30:48.115 ","End":"30:51.520","Text":"So here, it is definitely x^y."},{"Start":"30:51.520 ","End":"30:54.055","Text":"I\u0027m not going back on that."},{"Start":"30:54.055 ","End":"31:00.700","Text":"Times and natural log of"},{"Start":"31:00.700 ","End":"31:09.020","Text":"x plus x over y from here."},{"Start":"31:12.990 ","End":"31:19.765","Text":"That\u0027s because I\u0027ve isolated the y-prime by dividing both sides by this thing here."},{"Start":"31:19.765 ","End":"31:22.300","Text":"I don\u0027t think this can be simplified,"},{"Start":"31:22.300 ","End":"31:25.340","Text":"and it just looks fine to me."},{"Start":"31:26.100 ","End":"31:35.390","Text":"The only thing I could say is that we must get a 0 in the denominator."},{"Start":"31:36.240 ","End":"31:40.360","Text":"There are several things that 1 could say about the domain,"},{"Start":"31:40.360 ","End":"31:42.550","Text":"and I\u0027m not going to say too much,"},{"Start":"31:42.550 ","End":"31:48.680","Text":"but just acknowledge the fact that there is a domain."},{"Start":"31:50.190 ","End":"31:52.360","Text":"What can go wrong?"},{"Start":"31:52.360 ","End":"31:57.760","Text":"Well, in order to take exponents,"},{"Start":"31:57.760 ","End":"32:01.150","Text":"we need the base to be positive,"},{"Start":"32:01.150 ","End":"32:05.530","Text":"so we need x to be positive."},{"Start":"32:05.530 ","End":"32:16.690","Text":"We need y to be positive because the power of is defined for bigger or equal to 0."},{"Start":"32:16.690 ","End":"32:19.645","Text":"Anyway, I\u0027ll just write it as bigger than,"},{"Start":"32:19.645 ","End":"32:22.660","Text":"actually could be 0 to the power of something."},{"Start":"32:22.660 ","End":"32:26.215","Text":"But negative numbers like yeah,"},{"Start":"32:26.215 ","End":"32:29.169","Text":"what\u0027s minus 1 to the power of a half?"},{"Start":"32:29.169 ","End":"32:31.990","Text":"Square root has to be positive,"},{"Start":"32:31.990 ","End":"32:34.390","Text":"but I could allow it to be 0."},{"Start":"32:34.390 ","End":"32:36.520","Text":"What else do we need?"},{"Start":"32:36.520 ","End":"32:40.465","Text":"We also need that, again, x,"},{"Start":"32:40.465 ","End":"32:43.850","Text":"because of the natural log,"},{"Start":"32:44.640 ","End":"32:48.190","Text":"they have to be definitely bigger than 0."},{"Start":"32:48.190 ","End":"32:54.560","Text":"So let\u0027s see if I just thicken the writing."},{"Start":"32:56.100 ","End":"33:00.320","Text":"Yeah. Just put the eraser."},{"Start":"33:01.620 ","End":"33:05.305","Text":"Because of the natural log,"},{"Start":"33:05.305 ","End":"33:08.470","Text":"it has to be strictly positive."},{"Start":"33:08.470 ","End":"33:10.915","Text":"Now, what else could be?"},{"Start":"33:10.915 ","End":"33:17.125","Text":"Any other thing that could go wrong would be if the denominator was 0."},{"Start":"33:17.125 ","End":"33:24.475","Text":"Well, this is not going to be 0 because x is not 0, but this could be 0."},{"Start":"33:24.475 ","End":"33:32.185","Text":"So we have to have to require that the natural log of x should not be"},{"Start":"33:32.185 ","End":"33:43.280","Text":"plus x over y should not be equal to 0 also."},{"Start":"33:44.130 ","End":"33:47.425","Text":"Less important these restrictions,"},{"Start":"33:47.425 ","End":"33:49.315","Text":"most important are the techniques."},{"Start":"33:49.315 ","End":"33:53.590","Text":"Notice, it\u0027s not really difficult just using formulae,"},{"Start":"33:53.590 ","End":"33:55.180","Text":"but there\u0027s a lot of writing,"},{"Start":"33:55.180 ","End":"33:58.855","Text":"and don\u0027t use me as a negative example,"},{"Start":"33:58.855 ","End":"34:02.330","Text":"copy correctly and be careful."},{"Start":"34:02.400 ","End":"34:06.730","Text":"If this is number 11,"},{"Start":"34:06.730 ","End":"34:10.690","Text":"and it means that we only have 1 more to go in this series,"},{"Start":"34:10.690 ","End":"34:15.110","Text":"and that will be number 12."},{"Start":"34:16.830 ","End":"34:19.690","Text":"Scroll up a bit,"},{"Start":"34:19.690 ","End":"34:26.210","Text":"and let\u0027s write number 12."},{"Start":"34:27.780 ","End":"34:37.130","Text":"Number 12 is y to the power of natural log of x"},{"Start":"34:37.530 ","End":"34:43.300","Text":"plus x to the power of natural log of"},{"Start":"34:43.300 ","End":"34:49.640","Text":"y is going to equal 4."},{"Start":"34:50.970 ","End":"34:54.760","Text":"Now, I know that the 4 is not going to make any difference."},{"Start":"34:54.760 ","End":"34:56.350","Text":"It could have been 5, or 7,"},{"Start":"34:56.350 ","End":"34:59.755","Text":"or 100, but it happens to be 4."},{"Start":"34:59.755 ","End":"35:03.830","Text":"So let\u0027s differentiate each 1."},{"Start":"35:04.020 ","End":"35:07.030","Text":"Since it\u0027s rolled off the screen,"},{"Start":"35:07.030 ","End":"35:10.210","Text":"I\u0027ll just remind you,"},{"Start":"35:10.210 ","End":"35:15.280","Text":"just of the first formula that the square to the power of"},{"Start":"35:15.280 ","End":"35:22.015","Text":"triangle is equal e to the power of triangle,"},{"Start":"35:22.015 ","End":"35:24.205","Text":"natural log of square."},{"Start":"35:24.205 ","End":"35:27.925","Text":"The other 1 with the e to the power of derivative,"},{"Start":"35:27.925 ","End":"35:31.250","Text":"I\u0027ll assume you remember it."},{"Start":"35:32.940 ","End":"35:38.080","Text":"We rewrite this in the e form using the e,"},{"Start":"35:38.080 ","End":"35:48.410","Text":"so we have e to the power of natural log of x times natural log of y,"},{"Start":"35:49.500 ","End":"35:55.480","Text":"plus e to the power of natural log of"},{"Start":"35:55.480 ","End":"36:03.650","Text":"y times natural log of x."},{"Start":"36:05.130 ","End":"36:08.960","Text":"Well, this is equal to 4."},{"Start":"36:10.350 ","End":"36:13.270","Text":"Now, as far as I can remember,"},{"Start":"36:13.270 ","End":"36:16.210","Text":"the order of multiplication makes no difference,"},{"Start":"36:16.210 ","End":"36:17.620","Text":"so natural log of x,"},{"Start":"36:17.620 ","End":"36:18.715","Text":"natural log of y,"},{"Start":"36:18.715 ","End":"36:20.170","Text":"or y and then x,"},{"Start":"36:20.170 ","End":"36:21.970","Text":"it makes no difference,"},{"Start":"36:21.970 ","End":"36:27.850","Text":"so I\u0027ll just write it down as twice this."},{"Start":"36:27.850 ","End":"36:31.345","Text":"With your permission, I\u0027ll also divide by 2"},{"Start":"36:31.345 ","End":"36:34.510","Text":"because if this is the same as this and they add up together to 4,"},{"Start":"36:34.510 ","End":"36:36.070","Text":"this one\u0027s going to be 2."},{"Start":"36:36.070 ","End":"36:45.260","Text":"This gives us that e to the power of natural log of x."},{"Start":"36:45.260 ","End":"36:54.760","Text":"Natural log of y has got to equal 2."},{"Start":"36:54.760 ","End":"37:00.985","Text":"If I have something like a plus a equals 4, then a is 2."},{"Start":"37:00.985 ","End":"37:05.590","Text":"Finally going to get down to doing some differentiation."},{"Start":"37:05.590 ","End":"37:11.260","Text":"Now, the derivative of this is"},{"Start":"37:11.260 ","End":"37:18.520","Text":"just e to the power of natural log x,"},{"Start":"37:18.520 ","End":"37:25.720","Text":"natural log y times the derivative of the exponent,"},{"Start":"37:25.720 ","End":"37:29.155","Text":"which is natural log x,"},{"Start":"37:29.155 ","End":"37:34.900","Text":"natural log y derivative,"},{"Start":"37:34.900 ","End":"37:38.210","Text":"and the 2 gives us 0."},{"Start":"37:40.620 ","End":"37:43.660","Text":"I think, again,"},{"Start":"37:43.660 ","End":"37:48.805","Text":"we\u0027ll be using the product rule,"},{"Start":"37:48.805 ","End":"37:55.030","Text":"which is that f times g. Its"},{"Start":"37:55.030 ","End":"38:03.040","Text":"derivative is derivative of f times g plus f times the derivative of g,"},{"Start":"38:03.040 ","End":"38:05.930","Text":"which we\u0027ll need here."},{"Start":"38:08.910 ","End":"38:13.300","Text":"The other thing to remember is that at some point,"},{"Start":"38:13.300 ","End":"38:22.260","Text":"we should write things"},{"Start":"38:22.260 ","End":"38:24.465","Text":"back in their original form,"},{"Start":"38:24.465 ","End":"38:27.030","Text":"and it turns out actually,"},{"Start":"38:27.030 ","End":"38:29.205","Text":"and this is the strange thing,"},{"Start":"38:29.205 ","End":"38:39.535","Text":"that y to the natural log of x is actually equal to x to the power of natural log of y."},{"Start":"38:39.535 ","End":"38:44.440","Text":"This actually surprises me because they both give the same thing."},{"Start":"38:44.440 ","End":"38:48.830","Text":"What shall I use if I only have 1?"},{"Start":"38:51.990 ","End":"38:55.120","Text":"I\u0027ll go with the first one."},{"Start":"38:55.120 ","End":"39:00.760","Text":"We have y to the power of natural log of"},{"Start":"39:00.760 ","End":"39:10.165","Text":"x. I must admit it did surprise me that these 2 expressions are actually the same."},{"Start":"39:10.165 ","End":"39:12.715","Text":"When we put them both in the e-form, we could see it."},{"Start":"39:12.715 ","End":"39:14.155","Text":"I hadn\u0027t expected that."},{"Start":"39:14.155 ","End":"39:17.785","Text":"I\u0027ll go with this one just this first."},{"Start":"39:17.785 ","End":"39:25.075","Text":"Times, and now in here we\u0027ll do the product rule."},{"Start":"39:25.075 ","End":"39:31.135","Text":"It\u0027s the first one prime,"},{"Start":"39:31.135 ","End":"39:34.015","Text":"which is 1 over x,"},{"Start":"39:34.015 ","End":"39:36.910","Text":"and the second one as is,"},{"Start":"39:36.910 ","End":"39:41.950","Text":"natural log y plus the other way round."},{"Start":"39:41.950 ","End":"40:01.375","Text":"Then this one stays as is, natural log of x"},{"Start":"40:01.375 ","End":"40:04.345","Text":"as is times the derivative of this,"},{"Start":"40:04.345 ","End":"40:08.380","Text":"and the derivative of this is not exactly 1 over y."},{"Start":"40:08.380 ","End":"40:12.775","Text":"We have to remember to put the y prime in there,"},{"Start":"40:12.775 ","End":"40:16.460","Text":"and this is equal to 0."},{"Start":"40:19.410 ","End":"40:24.790","Text":"Next thing we need to do is to isolate y prime,"},{"Start":"40:24.790 ","End":"40:30.175","Text":"and y prime is hiding somewhere,"},{"Start":"40:30.175 ","End":"40:34.280","Text":"and it looks like it\u0027s hiding here."},{"Start":"40:37.830 ","End":"40:41.515","Text":"There\u0027s the y prime,"},{"Start":"40:41.515 ","End":"40:45.020","Text":"and we have to get this on one side."},{"Start":"40:46.590 ","End":"40:58.690","Text":"A better idea yet."},{"Start":"40:58.690 ","End":"41:06.830","Text":"If we have a product that is equal to 0,"},{"Start":"41:10.920 ","End":"41:15.580","Text":"then we can just divide by y to the power of natural log of"},{"Start":"41:15.580 ","End":"41:21.050","Text":"x because an exponent is never going to be 0."},{"Start":"41:21.540 ","End":"41:25.615","Text":"Because we have natural log of y, for one thing,"},{"Start":"41:25.615 ","End":"41:29.020","Text":"I know that y is positive,"},{"Start":"41:29.020 ","End":"41:31.400","Text":"and this will never be 0."},{"Start":"41:31.400 ","End":"41:36.940","Text":"Yeah, just completely throw it"},{"Start":"41:36.940 ","End":"41:47.499","Text":"out and write that 1 over x natural log y"},{"Start":"41:55.290 ","End":"42:05.470","Text":"plus natural log of x over"},{"Start":"42:05.470 ","End":"42:15.460","Text":"y times y prime is equal to 0."},{"Start":"42:15.460 ","End":"42:20.630","Text":"I\u0027m going slowly now because I\u0027m going to mix those x and y\u0027s up otherwise."},{"Start":"42:24.750 ","End":"42:29.530","Text":"If I bring this to the other side."},{"Start":"42:29.530 ","End":"42:38.800","Text":"I now get that y prime times natural log of x over"},{"Start":"42:38.800 ","End":"42:43.675","Text":"y equals natural log of y"},{"Start":"42:43.675 ","End":"42:50.590","Text":"over x with a minus in front of it because I took it from that side."},{"Start":"42:50.590 ","End":"42:53.035","Text":"Now all I have to do is divide by this,"},{"Start":"42:53.035 ","End":"42:56.950","Text":"and remember that dividing by a fraction is like multiplying by"},{"Start":"42:56.950 ","End":"43:02.110","Text":"the inverse fraction by the reciprocal."},{"Start":"43:02.110 ","End":"43:06.250","Text":"Y prime is equal to"},{"Start":"43:06.250 ","End":"43:14.860","Text":"minus natural log y over x times the opposite of this,"},{"Start":"43:14.860 ","End":"43:23.530","Text":"which is y over"},{"Start":"43:23.530 ","End":"43:27.595","Text":"natural log of x. [inaudible] I said it wrong,"},{"Start":"43:27.595 ","End":"43:30.380","Text":"is y\u0027s and x\u0027s."},{"Start":"43:32.520 ","End":"43:37.810","Text":"I don\u0027t think this could be really simplified anymore,"},{"Start":"43:37.810 ","End":"43:43.225","Text":"but I will just say that is the answer."},{"Start":"43:43.225 ","End":"43:46.975","Text":"We should at least point out that there is a domain."},{"Start":"43:46.975 ","End":"43:53.780","Text":"For example, x has to be bigger than 0 because of the natural log,"},{"Start":"43:55.020 ","End":"43:57.160","Text":"and then automatically,"},{"Start":"43:57.160 ","End":"43:59.275","Text":"x will not be 0 here,"},{"Start":"43:59.275 ","End":"44:02.830","Text":"and also because of the natural log,"},{"Start":"44:02.830 ","End":"44:05.860","Text":"we also have to have y bigger than 0."},{"Start":"44:05.860 ","End":"44:08.425","Text":"Here it\u0027s fairly simple,"},{"Start":"44:08.425 ","End":"44:12.050","Text":"x and y both have to be positive."},{"Start":"44:13.530 ","End":"44:16.390","Text":"We\u0027re actually done with number 12,"},{"Start":"44:16.390 ","End":"44:19.250","Text":"and therefore we\u0027re done with the whole series."}],"ID":28483},{"Watched":false,"Name":"Exercise 2 - Parts 1-2","Duration":"14m 11s","ChapterTopicVideoID":27326,"CourseChapterTopicPlaylistID":8714,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.540 ","End":"00:05.170","Text":"In this exercise, there are really 4 exercises and"},{"Start":"00:05.170 ","End":"00:09.040","Text":"it\u0027s similar to the previous exercise set."},{"Start":"00:09.040 ","End":"00:13.960","Text":"It involves implicit differentiation, implicit functions."},{"Start":"00:13.960 ","End":"00:17.784","Text":"The difference is that here we have to also find the second derivative."},{"Start":"00:17.784 ","End":"00:20.235","Text":"There we only had the first."},{"Start":"00:20.235 ","End":"00:23.110","Text":"Let\u0027s begin with the first 1."},{"Start":"00:23.110 ","End":"00:24.620","Text":"I\u0027ve already written it here."},{"Start":"00:24.620 ","End":"00:29.410","Text":"I want to remind you that the main thing here is that when we"},{"Start":"00:29.410 ","End":"00:34.165","Text":"differentiate this implicit function,"},{"Start":"00:34.165 ","End":"00:39.160","Text":"where every time we have an expression with y we differentiated as if it was x,"},{"Start":"00:39.160 ","End":"00:42.600","Text":"but we have to add this y prime in addition,"},{"Start":"00:42.600 ","End":"00:46.620","Text":"mustn\u0027t forget to multiply by y prime."},{"Start":"00:46.820 ","End":"00:53.780","Text":"The first 1. Just start differentiating x squared just behaves normal 2x."},{"Start":"00:53.780 ","End":"00:55.805","Text":"Now here\u0027s what I was talking about."},{"Start":"00:55.805 ","End":"00:57.065","Text":"If it was x cubed,"},{"Start":"00:57.065 ","End":"00:59.150","Text":"you would write 3x squared."},{"Start":"00:59.150 ","End":"01:02.074","Text":"Similarly you would think 3y squared,"},{"Start":"01:02.074 ","End":"01:04.175","Text":"but you have to add,"},{"Start":"01:04.175 ","End":"01:08.315","Text":"in a sense of multiply y prime."},{"Start":"01:08.315 ","End":"01:12.875","Text":"This is important, equals 1, so equals 0."},{"Start":"01:12.875 ","End":"01:19.110","Text":"At this point, we just have to take y prime and isolate it."},{"Start":"01:19.110 ","End":"01:20.450","Text":"Let\u0027s, I don\u0027t know,"},{"Start":"01:20.450 ","End":"01:27.185","Text":"let\u0027s color it in some nice color here."},{"Start":"01:27.185 ","End":"01:32.900","Text":"If we, take the 2x to the other side and divide by 3y squared,"},{"Start":"01:32.900 ","End":"01:38.200","Text":"we simply get that y prime is equal to minus"},{"Start":"01:38.200 ","End":"01:43.945","Text":"2x over 3y squared."},{"Start":"01:43.945 ","End":"01:49.240","Text":"That\u0027s all there is to the y-prime."},{"Start":"01:50.780 ","End":"01:55.005","Text":"Which again, I\u0027d like to highlight."},{"Start":"01:55.005 ","End":"01:59.370","Text":"Continuing, now, that\u0027s not what they asked"},{"Start":"01:59.370 ","End":"02:03.850","Text":"in the question they wanted first, and second derivatives."},{"Start":"02:05.510 ","End":"02:10.350","Text":"We have to also give them y double-prime."},{"Start":"02:11.740 ","End":"02:17.690","Text":"Hang on a second to change the pen."},{"Start":"02:17.690 ","End":"02:20.720","Text":"Y double-prime equals,"},{"Start":"02:20.720 ","End":"02:22.310","Text":"I have a quotient here,"},{"Start":"02:22.310 ","End":"02:24.080","Text":"had that often before."},{"Start":"02:24.080 ","End":"02:27.075","Text":"Maybe it\u0027s time to re-memorize the quotient rule."},{"Start":"02:27.075 ","End":"02:35.790","Text":"If we have f over g derivative, it\u0027s f prime,"},{"Start":"02:36.170 ","End":"02:43.560","Text":"g minus f and g prime all over g squared,"},{"Start":"02:43.560 ","End":"02:48.435","Text":"where f will be this and g would be this."},{"Start":"02:48.435 ","End":"02:51.615","Text":"Continuing doing that here,"},{"Start":"02:51.615 ","End":"02:57.960","Text":"we will get f prime is minus"},{"Start":"02:57.960 ","End":"03:04.575","Text":"2 and then g is 3y squared."},{"Start":"03:04.575 ","End":"03:08.460","Text":"Then just following the formula minus f,"},{"Start":"03:08.460 ","End":"03:11.910","Text":"which is minus 2x."},{"Start":"03:11.910 ","End":"03:14.900","Text":"Allow me to put a plus here because it\u0027s a minus with"},{"Start":"03:14.900 ","End":"03:20.855","Text":"a minus and it\u0027s 2x times the derivative of this."},{"Start":"03:20.855 ","End":"03:23.190","Text":"Now, normally if it was 3x squared,"},{"Start":"03:23.190 ","End":"03:26.380","Text":"you would say straight away 2 times 3 is,"},{"Start":"03:27.320 ","End":"03:29.640","Text":"put it in brackets here,"},{"Start":"03:29.640 ","End":"03:31.155","Text":"2 times 3 is 6."},{"Start":"03:31.155 ","End":"03:34.390","Text":"Lower the power by 1, so it\u0027s 6y."},{"Start":"03:34.390 ","End":"03:37.040","Text":"However, it\u0027s a function of y,"},{"Start":"03:37.040 ","End":"03:39.895","Text":"we need to add the y prime."},{"Start":"03:39.895 ","End":"03:47.420","Text":"Then the last thing is the denominator squared, that\u0027s 9y^4."},{"Start":"03:47.420 ","End":"03:51.655","Text":"In some sense, this would be the answer."},{"Start":"03:51.655 ","End":"03:57.710","Text":"I don\u0027t accept this as an answer because it"},{"Start":"03:57.710 ","End":"04:03.770","Text":"has a y prime in it and I would like y double prime in terms of just x and y."},{"Start":"04:03.770 ","End":"04:08.000","Text":"What I\u0027m going to do is just substitute."},{"Start":"04:08.000 ","End":"04:09.545","Text":"We had y prime here,"},{"Start":"04:09.545 ","End":"04:11.585","Text":"put that into here,"},{"Start":"04:11.585 ","End":"04:13.835","Text":"and then we will get,"},{"Start":"04:13.835 ","End":"04:15.650","Text":"let\u0027s scroll again,"},{"Start":"04:15.650 ","End":"04:24.190","Text":"y double-prime equals the first thing is minus 6y squared."},{"Start":"04:24.320 ","End":"04:33.810","Text":"Next, plus 12xy times"},{"Start":"04:33.810 ","End":"04:39.880","Text":"y prime is minus 2x over 3y squared,"},{"Start":"04:42.610 ","End":"04:45.920","Text":"which is what we had here."},{"Start":"04:45.920 ","End":"04:51.210","Text":"Then all over 9y^4."},{"Start":"04:55.220 ","End":"05:00.860","Text":"How I would simplify this would be to multiply top and"},{"Start":"05:00.860 ","End":"05:06.275","Text":"bottom of the main fraction by 3y squared."},{"Start":"05:06.275 ","End":"05:10.270","Text":"If we multiply top and bottom by that will get,"},{"Start":"05:10.270 ","End":"05:13.295","Text":"I\u0027ll just write, multiply it,"},{"Start":"05:13.295 ","End":"05:17.960","Text":"say by 3y squared over 3y squared, which is 1."},{"Start":"05:17.960 ","End":"05:21.335","Text":"That\u0027s will make it simpler."},{"Start":"05:21.335 ","End":"05:24.245","Text":"In the denominator,"},{"Start":"05:24.245 ","End":"05:32.660","Text":"we will have 27y^6."},{"Start":"05:32.660 ","End":"05:37.965","Text":"In the numerator in the second part,"},{"Start":"05:37.965 ","End":"05:42.480","Text":"we will have plus without the 3y-squared,"},{"Start":"05:42.480 ","End":"05:47.760","Text":"which will cancel with that we\u0027ll have not plus but minus."},{"Start":"05:47.760 ","End":"05:57.450","Text":"Hang on. I\u0027m just going to take my eraser out and just erase."},{"Start":"05:57.970 ","End":"06:09.290","Text":"We\u0027ll put it as minus. Let\u0027s see."},{"Start":"06:09.290 ","End":"06:12.950","Text":"Without this denominator we have 12x."},{"Start":"06:12.950 ","End":"06:14.750","Text":"It\u0027s 24 collect numbers."},{"Start":"06:14.750 ","End":"06:18.095","Text":"It\u0027s 24 in the number department,"},{"Start":"06:18.095 ","End":"06:19.640","Text":"and in the x department,"},{"Start":"06:19.640 ","End":"06:23.850","Text":"it\u0027s x times x is x squared."},{"Start":"06:24.850 ","End":"06:30.940","Text":"We still have a y here."},{"Start":"06:31.190 ","End":"06:36.949","Text":"Well, indeed needing to multiply it by 3y squared,"},{"Start":"06:36.949 ","End":"06:41.540","Text":"we could have actually settled for just 3y."},{"Start":"06:41.540 ","End":"06:47.525","Text":"Let\u0027s do that. I\u0027m going to take out the eraser and just erase the"},{"Start":"06:47.525 ","End":"06:53.240","Text":"2 from here and that will suffice."},{"Start":"06:53.240 ","End":"06:57.025","Text":"Because here already I have y."},{"Start":"06:57.025 ","End":"07:02.135","Text":"This y together with this 3y cancels out."},{"Start":"07:02.135 ","End":"07:11.435","Text":"In other words, this y with this 3y will cancel out with the 3y squared."},{"Start":"07:11.435 ","End":"07:17.330","Text":"What we\u0027re left with again is minus 24x squared dot parts."},{"Start":"07:17.330 ","End":"07:21.900","Text":"Here we have minus 6y squared."},{"Start":"07:23.000 ","End":"07:27.240","Text":"The same thing as we had down there before."},{"Start":"07:27.240 ","End":"07:29.850","Text":"That\u0027s basically the answer."},{"Start":"07:29.850 ","End":"07:35.485","Text":"Well, I\u0027ll go 1 more step to see if everything is divisible by 3."},{"Start":"07:35.485 ","End":"07:38.505","Text":"I can take the minus up in front."},{"Start":"07:38.505 ","End":"07:41.470","Text":"Let\u0027s do a little bit more."},{"Start":"07:42.650 ","End":"07:46.125","Text":"That is equal to minus,"},{"Start":"07:46.125 ","End":"07:53.445","Text":"now dividing everything by 3 it\u0027s 2y squared plus"},{"Start":"07:53.445 ","End":"08:02.430","Text":"8x squared because we took the minus outside there and here also by 3, that\u0027s 9y^6."},{"Start":"08:05.740 ","End":"08:13.415","Text":"That\u0027s the answer. That\u0027s done with number 1."},{"Start":"08:13.415 ","End":"08:16.970","Text":"I suppose the next thing to do would be number 2."},{"Start":"08:16.970 ","End":"08:19.005","Text":"But I\u0027ve forgotten what it is."},{"Start":"08:19.005 ","End":"08:22.365","Text":"Hang on. Let\u0027s just take a look up there."},{"Start":"08:22.365 ","End":"08:27.715","Text":"X squared y cubed equals x plus y. I think I can remember that,"},{"Start":"08:27.715 ","End":"08:30.130","Text":"x squared y cubed equals x plus y,"},{"Start":"08:30.130 ","End":"08:41.110","Text":"is that correct?"},{"Start":"08:41.110 ","End":"08:44.660","Text":"Yes, it is."},{"Start":"08:46.890 ","End":"08:50.880","Text":"Same tricks as usual."},{"Start":"08:50.880 ","End":"08:53.375","Text":"This time we have product."},{"Start":"08:53.375 ","End":"08:55.820","Text":"Let\u0027s write down the product rule."},{"Start":"08:55.820 ","End":"08:59.110","Text":"Because I know that people just forget it."},{"Start":"08:59.270 ","End":"09:01.460","Text":"Just write it each time."},{"Start":"09:01.460 ","End":"09:04.730","Text":"It\u0027s not very long to do, f prime,"},{"Start":"09:04.730 ","End":"09:08.615","Text":"g plus fg prime,"},{"Start":"09:08.615 ","End":"09:14.705","Text":"where of course f is the x squared bit and g is the y cubed bit."},{"Start":"09:14.705 ","End":"09:17.790","Text":"Just following it,"},{"Start":"09:19.480 ","End":"09:21.935","Text":"for this bit here,"},{"Start":"09:21.935 ","End":"09:29.150","Text":"f is f prime."},{"Start":"09:29.150 ","End":"09:31.610","Text":"They gave me that would just take my eraser,"},{"Start":"09:31.610 ","End":"09:35.600","Text":"erase that, and then back to x squared."},{"Start":"09:35.600 ","End":"09:38.440","Text":"The first 1 is not differentiated,"},{"Start":"09:38.440 ","End":"09:42.970","Text":"the second oh boy."},{"Start":"09:42.970 ","End":"09:44.660","Text":"There I go again."},{"Start":"09:44.660 ","End":"09:46.400","Text":"I\u0027m so sorry."},{"Start":"09:46.400 ","End":"09:47.945","Text":"It is 2x,"},{"Start":"09:47.945 ","End":"09:53.235","Text":"of course and then this 1 is untouched y cubed."},{"Start":"09:53.235 ","End":"09:55.800","Text":"Then plus that\u0027s this, plus f,"},{"Start":"09:55.800 ","End":"09:57.270","Text":"that\u0027s the untouched part,"},{"Start":"09:57.270 ","End":"09:59.720","Text":"x squared and the g prime."},{"Start":"09:59.720 ","End":"10:03.725","Text":"Very tempting to just write 3y-squared and leave it at that."},{"Start":"10:03.725 ","End":"10:06.845","Text":"But it\u0027s an expression with y."},{"Start":"10:06.845 ","End":"10:09.140","Text":"We have to add the y prime."},{"Start":"10:09.140 ","End":"10:10.910","Text":"That\u0027s the left-hand side."},{"Start":"10:10.910 ","End":"10:14.020","Text":"Right-hand side, derivative of x is 1,"},{"Start":"10:14.020 ","End":"10:16.070","Text":"derivative of y is not 1."},{"Start":"10:16.070 ","End":"10:18.730","Text":"It\u0027s 1 times y prime"},{"Start":"10:18.730 ","End":"10:23.370","Text":"or you could just say when you derive anything you put prime over it."},{"Start":"10:23.680 ","End":"10:26.750","Text":"What we want is y prime."},{"Start":"10:26.750 ","End":"10:32.720","Text":"Let me just highlight that we have a y prime here,"},{"Start":"10:32.720 ","End":"10:40.295","Text":"and we have a y prime here and we want to isolate it and see what it is on its own."},{"Start":"10:40.295 ","End":"10:43.920","Text":"Let\u0027s see, how shall we do that?"},{"Start":"10:47.340 ","End":"10:53.515","Text":"Let\u0027s say we throw the 1 over,"},{"Start":"10:53.515 ","End":"10:55.705","Text":"it doesn\u0027t really matter which way,"},{"Start":"10:55.705 ","End":"10:57.430","Text":"one of them we have to throw on 1 side,"},{"Start":"10:57.430 ","End":"10:59.185","Text":"and 1 on the other side."},{"Start":"10:59.185 ","End":"11:03.830","Text":"Let\u0027s say I throw y prime over,"},{"Start":"11:04.050 ","End":"11:07.960","Text":"this 1 will go over to this side of the equation,"},{"Start":"11:07.960 ","End":"11:11.020","Text":"and this bit will go over to that side,"},{"Start":"11:11.020 ","End":"11:14.540","Text":"and then we\u0027ll get all the y primes on the left,"},{"Start":"11:19.080 ","End":"11:22.210","Text":"but on the other side it\u0027s minus."},{"Start":"11:22.210 ","End":"11:25.880","Text":"We have x squared"},{"Start":"11:35.310 ","End":"11:42.955","Text":"times 3y squared, on this side,"},{"Start":"11:42.955 ","End":"11:45.610","Text":"I\u0027m leaving out the y prime just for a moment,"},{"Start":"11:45.610 ","End":"11:50.935","Text":"you\u0027ll see, minus y prime."},{"Start":"11:50.935 ","End":"11:55.780","Text":"Basically what I\u0027m saying is I\u0027m taking y prime outside the brackets,"},{"Start":"11:55.780 ","End":"11:57.700","Text":"so the y prime is just 1,"},{"Start":"11:57.700 ","End":"12:00.460","Text":"and there it goes from there."},{"Start":"12:00.460 ","End":"12:02.665","Text":"The rest of it is,"},{"Start":"12:02.665 ","End":"12:04.270","Text":"leave the 1 here,"},{"Start":"12:04.270 ","End":"12:05.920","Text":"subtract the other bit,"},{"Start":"12:05.920 ","End":"12:08.660","Text":"which is the 2xy cubed."},{"Start":"12:10.500 ","End":"12:13.600","Text":"We still want to isolate y prime."},{"Start":"12:13.600 ","End":"12:16.420","Text":"All we do now is divide by this part,"},{"Start":"12:16.420 ","End":"12:20.050","Text":"and get that y prime is equal to"},{"Start":"12:20.050 ","End":"12:28.360","Text":"1 minus 2xy cubed over x squared,"},{"Start":"12:28.360 ","End":"12:29.620","Text":"we\u0027ll put the 3 in front,"},{"Start":"12:29.620 ","End":"12:32.245","Text":"usually you put the constant in front,"},{"Start":"12:32.245 ","End":"12:38.665","Text":"3x squared, y squared minus 1."},{"Start":"12:38.665 ","End":"12:42.505","Text":"That\u0027s y prime and that\u0027s the first derivative,"},{"Start":"12:42.505 ","End":"12:46.075","Text":"but we still need the second derivative."},{"Start":"12:46.075 ","End":"12:50.560","Text":"I probably like to also have that highlighted there."},{"Start":"12:50.560 ","End":"12:53.420","Text":"That\u0027s our y prime."},{"Start":"12:55.260 ","End":"13:00.800","Text":"It\u0027s consistent that\u0027s per all our y primes."},{"Start":"13:00.930 ","End":"13:04.075","Text":"It\u0027s not nice with all those colors."},{"Start":"13:04.075 ","End":"13:08.165","Text":"Continuing, we need y double prime."},{"Start":"13:08.165 ","End":"13:11.700","Text":"This time, it\u0027s not the product rule we need,"},{"Start":"13:11.700 ","End":"13:14.924","Text":"it\u0027s the quotient rule because we have a fraction."},{"Start":"13:14.924 ","End":"13:18.020","Text":"I\u0027m going to write what it is again."},{"Start":"13:18.020 ","End":"13:22.585","Text":"F over g prime is"},{"Start":"13:22.585 ","End":"13:29.470","Text":"f prime g minus fg prime all over,"},{"Start":"13:29.470 ","End":"13:32.780","Text":"what was the original denominator squared?"},{"Start":"13:32.780 ","End":"13:35.830","Text":"If we do that,"},{"Start":"13:36.150 ","End":"13:39.235","Text":"we\u0027re obviously taking this as f,"},{"Start":"13:39.235 ","End":"13:40.465","Text":"this is g,"},{"Start":"13:40.465 ","End":"13:42.535","Text":"so we get f prime."},{"Start":"13:42.535 ","End":"13:44.860","Text":"Now what is f prime?"},{"Start":"13:44.860 ","End":"13:47.545","Text":"The 1 is 0."},{"Start":"13:47.545 ","End":"13:54.385","Text":"This primed can take a constant outside the bracket, it\u0027s minus twice."},{"Start":"13:54.385 ","End":"13:56.305","Text":"We have a product here."},{"Start":"13:56.305 ","End":"13:58.660","Text":"Let\u0027s just meanwhile not differentiate,"},{"Start":"13:58.660 ","End":"14:04.585","Text":"but just write that we have to differentiate xy cubed prime."},{"Start":"14:04.585 ","End":"14:06.700","Text":"That\u0027s the f prime,"},{"Start":"14:06.700 ","End":"14:09.025","Text":"and we still have to throw in the g,"},{"Start":"14:09.025 ","End":"14:15.850","Text":"which is 3x squared y squared minus 1,"},{"Start":"14:15.850 ","End":"14:19.735","Text":"that\u0027s the g. Now we\u0027re up to this minus here."},{"Start":"14:19.735 ","End":"14:28.280","Text":"This minus, and then f. 1 minus 2xy cubed,"},{"Start":"14:29.490 ","End":"14:32.049","Text":"and then g prime."},{"Start":"14:32.049 ","End":"14:34.285","Text":"Now similar to before,"},{"Start":"14:34.285 ","End":"14:39.190","Text":"you\u0027ve got to differentiate each piece separately."},{"Start":"14:39.190 ","End":"14:41.995","Text":"The minus 1 is not going to give us anything, but this will."},{"Start":"14:41.995 ","End":"14:44.665","Text":"3 comes out of the brackets"},{"Start":"14:44.665 ","End":"14:49.555","Text":"because constants can come out of the derivative or the prime,"},{"Start":"14:49.555 ","End":"14:59.060","Text":"and we\u0027re left with x squared y squared prime, and still over."},{"Start":"14:59.430 ","End":"15:06.420","Text":"Oh yeah, not still, but freshly over the original denominator squared,"},{"Start":"15:06.420 ","End":"15:15.380","Text":"the g squared which is 3x squared y squared minus 1, all squared."},{"Start":"15:15.380 ","End":"15:18.280","Text":"We still haven\u0027t finished differentiating because"},{"Start":"15:18.280 ","End":"15:21.260","Text":"we have a prime here, and a prime here."},{"Start":"15:23.130 ","End":"15:27.310","Text":"Now, we still see the product rule."},{"Start":"15:27.310 ","End":"15:28.585","Text":"I\u0027ll just follow that."},{"Start":"15:28.585 ","End":"15:34.150","Text":"I\u0027m not going to go into every tiny detail at this stage is sophisticated enough."},{"Start":"15:34.150 ","End":"15:36.610","Text":"Minus 2. Now,"},{"Start":"15:36.610 ","End":"15:38.395","Text":"I\u0027m just going to write all this,"},{"Start":"15:38.395 ","End":"15:40.610","Text":"say in a square bracket."},{"Start":"15:40.610 ","End":"15:44.150","Text":"Derived is 1,"},{"Start":"15:44.150 ","End":"15:50.230","Text":"and underived y cubed"},{"Start":"15:50.230 ","End":"15:59.245","Text":"plus underived x, underived 3y squared."},{"Start":"15:59.245 ","End":"16:02.200","Text":"Almost forgot to do it, but there it is."},{"Start":"16:02.200 ","End":"16:03.760","Text":"You need to add the y prime."},{"Start":"16:03.760 ","End":"16:05.305","Text":"Do not forget that,"},{"Start":"16:05.305 ","End":"16:08.810","Text":"very important, during the whole exercise."},{"Start":"16:10.130 ","End":"16:13.470","Text":"That\u0027s this part minus 2,"},{"Start":"16:13.470 ","End":"16:17.040","Text":"the derived part is here."},{"Start":"16:17.040 ","End":"16:19.725","Text":"Then we come to this bit,"},{"Start":"16:19.725 ","End":"16:23.410","Text":"which is 3x squared, y squared minus 1."},{"Start":"16:25.490 ","End":"16:28.590","Text":"Starting to look messy."},{"Start":"16:28.590 ","End":"16:39.610","Text":"Minus 1 minus 2xy cubed times 3."},{"Start":"16:39.610 ","End":"16:43.345","Text":"Now, make sure this is a 3."},{"Start":"16:43.345 ","End":"16:49.840","Text":"Now we come again to the product rule only this time this is f,"},{"Start":"16:49.840 ","End":"16:54.115","Text":"and this is g. Derived is 2x,"},{"Start":"16:54.115 ","End":"16:57.940","Text":"underived y squared, and vice versa."},{"Start":"16:57.940 ","End":"17:01.870","Text":"x squared as it is, y squared derived."},{"Start":"17:01.870 ","End":"17:04.270","Text":"y squared derived is simply 2y, isn\u0027t it?"},{"Start":"17:04.270 ","End":"17:07.850","Text":"Oh, no, we need to add the y prime."},{"Start":"17:09.660 ","End":"17:15.250","Text":"All this is over this thing which we\u0027re dragging"},{"Start":"17:15.250 ","End":"17:20.900","Text":"along with us 3x squared, y squared minus 1 all squared."},{"Start":"17:21.510 ","End":"17:24.175","Text":"Now other than simplification,"},{"Start":"17:24.175 ","End":"17:30.040","Text":"why don\u0027t I accept this really as an answer because when I write y prime,"},{"Start":"17:30.040 ","End":"17:33.700","Text":"I really expect to have it just in terms of x and y."},{"Start":"17:33.700 ","End":"17:36.565","Text":"It\u0027s bad enough that it\u0027s not just x, it\u0027s x and y."},{"Start":"17:36.565 ","End":"17:38.530","Text":"Why do I need the y prime in here?"},{"Start":"17:38.530 ","End":"17:41.470","Text":"I\u0027ll just highlight it."},{"Start":"17:41.470 ","End":"17:44.665","Text":"To make it stand out there is a y prime here."},{"Start":"17:44.665 ","End":"17:46.195","Text":"Are there anymore?"},{"Start":"17:46.195 ","End":"17:48.205","Text":"There is another 1 here."},{"Start":"17:48.205 ","End":"17:50.620","Text":"I don\u0027t want the y prime in there,"},{"Start":"17:50.620 ","End":"17:54.865","Text":"but I do have what y prime is in terms of this."},{"Start":"17:54.865 ","End":"17:58.405","Text":"I\u0027m going to substitute y prime is this."},{"Start":"17:58.405 ","End":"18:00.610","Text":"But after that, I\u0027m going to leave the thing alone,"},{"Start":"18:00.610 ","End":"18:04.465","Text":"and not simply going to simplify because it\u0027s a whole mess in algebra."},{"Start":"18:04.465 ","End":"18:06.100","Text":"Let\u0027s just write it first,"},{"Start":"18:06.100 ","End":"18:08.420","Text":"and you\u0027ll see what a mess it is."},{"Start":"18:08.700 ","End":"18:19.270","Text":"What I have here is that y double prime equals minus"},{"Start":"18:19.270 ","End":"18:24.115","Text":"2 times y cubed"},{"Start":"18:24.115 ","End":"18:34.239","Text":"plus 3xy squared,"},{"Start":"18:34.239 ","End":"18:37.660","Text":"then y prime, which is"},{"Start":"18:37.660 ","End":"18:46.250","Text":"1 minus 2xy cubed over 3x squared y squared minus 1."},{"Start":"18:51.990 ","End":"18:55.570","Text":"That was just putting the y prime in,"},{"Start":"18:55.570 ","End":"19:05.815","Text":"and still times 3x squared y squared minus 1 minus,"},{"Start":"19:05.815 ","End":"19:08.350","Text":"I\u0027ll just put the 3 in front there."},{"Start":"19:08.350 ","End":"19:11.020","Text":"Actually, while I\u0027m at it,"},{"Start":"19:11.020 ","End":"19:13.795","Text":"I might as well take the 2 out of here also."},{"Start":"19:13.795 ","End":"19:17.155","Text":"The 2 and the 3 make a 6 if you multiply them."},{"Start":"19:17.155 ","End":"19:20.485","Text":"So minus 6 of"},{"Start":"19:20.485 ","End":"19:29.120","Text":"1 minus 2xy cubed times"},{"Start":"19:35.550 ","End":"19:44.150","Text":"xy squared plus 2x squared y,"},{"Start":"19:47.700 ","End":"19:51.130","Text":"and again times y prime,"},{"Start":"19:51.130 ","End":"19:57.730","Text":"which is 1 minus 2xy cubed"},{"Start":"20:01.770 ","End":"20:06.680","Text":"over 3x squared y squared minus 1."},{"Start":"20:14.430 ","End":"20:18.445","Text":"That\u0027s it, y prime in terms of x, and y."},{"Start":"20:18.445 ","End":"20:22.195","Text":"Now, it is possible to simplify this algebraically."},{"Start":"20:22.195 ","End":"20:24.100","Text":"I don\u0027t feel like doing it."},{"Start":"20:24.100 ","End":"20:26.200","Text":"I mean, and what will you get out of it?"},{"Start":"20:26.200 ","End":"20:30.205","Text":"Another algebra exercise, we\u0027re doing calculus."},{"Start":"20:30.205 ","End":"20:35.095","Text":"I\u0027m sneaking out of doing this,"},{"Start":"20:35.095 ","End":"20:38.455","Text":"but I really don\u0027t think it has any value,"},{"Start":"20:38.455 ","End":"20:41.140","Text":"this is the double prime,"},{"Start":"20:41.140 ","End":"20:43.000","Text":"it looks confusing there,"},{"Start":"20:43.000 ","End":"20:45.505","Text":"which equals blah, blah, blah, blah, blah."},{"Start":"20:45.505 ","End":"20:46.885","Text":"But that\u0027s the answer,"},{"Start":"20:46.885 ","End":"20:50.990","Text":"and we\u0027re done with Number 2."}],"ID":28441},{"Watched":false,"Name":"Exercise 2 - Parts 3-4","Duration":"6m 57s","ChapterTopicVideoID":10170,"CourseChapterTopicPlaylistID":8714,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.160","Text":"Number 3, natural log of"},{"Start":"00:05.160 ","End":"00:11.430","Text":"x plus natural log of y is equal to 1."},{"Start":"00:11.430 ","End":"00:14.055","Text":"Differentiating natural log of x,"},{"Start":"00:14.055 ","End":"00:16.920","Text":"1 over x, natural log of y,"},{"Start":"00:16.920 ","End":"00:18.929","Text":"1 over y, almost."},{"Start":"00:18.929 ","End":"00:22.890","Text":"Just don\u0027t forget the y prime 1 gives us 0."},{"Start":"00:22.890 ","End":"00:28.335","Text":"All we have to do is isolate the y prime,"},{"Start":"00:28.335 ","End":"00:33.704","Text":"which means that we can put the 1 over x to the other side."},{"Start":"00:33.704 ","End":"00:35.220","Text":"That\u0027s minus 1 over x,"},{"Start":"00:35.220 ","End":"00:38.675","Text":"then multiply by y. That\u0027s all."},{"Start":"00:38.675 ","End":"00:45.990","Text":"Y prime is equal to minus y over x,"},{"Start":"00:45.990 ","End":"00:47.550","Text":"if you actually do it."},{"Start":"00:47.550 ","End":"00:51.165","Text":"Of course, there\u0027s a domain and the restrictions."},{"Start":"00:51.165 ","End":"00:56.625","Text":"Here we have to have that x not be equal to 0."},{"Start":"00:56.625 ","End":"00:58.260","Text":"That\u0027s part of the exercise,"},{"Start":"00:58.260 ","End":"00:59.460","Text":"that\u0027s the first derivative."},{"Start":"00:59.460 ","End":"01:01.910","Text":"Now we need the second derivative,"},{"Start":"01:01.910 ","End":"01:07.355","Text":"but I have this habit of marking the y primes and let me just do that."},{"Start":"01:07.355 ","End":"01:12.165","Text":"Now I can write y double-prime."},{"Start":"01:12.165 ","End":"01:18.185","Text":"Over here I shall write that f over g prime is"},{"Start":"01:18.185 ","End":"01:23.975","Text":"f prime g minus f g prime over g squared,"},{"Start":"01:23.975 ","End":"01:27.920","Text":"and apply it here where this is going to be my f, this is my g,"},{"Start":"01:27.920 ","End":"01:33.480","Text":"the minus I\u0027ll just keep as a minus in front of the fraction,"},{"Start":"01:33.480 ","End":"01:35.100","Text":"I\u0027ll put the fraction here."},{"Start":"01:35.100 ","End":"01:36.930","Text":"Sometimes I like to start with the bottom,"},{"Start":"01:36.930 ","End":"01:39.360","Text":"that\u0027s easiest, it\u0027s just squared."},{"Start":"01:39.360 ","End":"01:41.000","Text":"This is f of course,"},{"Start":"01:41.000 ","End":"01:44.180","Text":"and this is g. So f prime,"},{"Start":"01:44.180 ","End":"01:46.520","Text":"y prime is not just 1,"},{"Start":"01:46.520 ","End":"01:52.745","Text":"it\u0027s 1 times y prime times g which is x minus f,"},{"Start":"01:52.745 ","End":"01:54.535","Text":"which is just y,"},{"Start":"01:54.535 ","End":"01:57.225","Text":"and g prime is 1."},{"Start":"01:57.225 ","End":"01:59.070","Text":"The f and g were without the minus,"},{"Start":"01:59.070 ","End":"02:01.720","Text":"the minus I took separately,"},{"Start":"02:03.500 ","End":"02:07.850","Text":"there\u0027s not much simplification here to do."},{"Start":"02:07.850 ","End":"02:11.960","Text":"But I could notice that there is a y prime here,"},{"Start":"02:11.960 ","End":"02:15.080","Text":"and I would rather not have y-prime in my answer."},{"Start":"02:15.080 ","End":"02:16.610","Text":"I want just x and y."},{"Start":"02:16.610 ","End":"02:20.750","Text":"So this is equal to minus, thing I like to do a lot."},{"Start":"02:20.750 ","End":"02:22.570","Text":"If the minus in front of a difference,"},{"Start":"02:22.570 ","End":"02:26.240","Text":"I just change the order of these and then it comes out okay."},{"Start":"02:26.240 ","End":"02:31.880","Text":"So we have y minus x, y prime."},{"Start":"02:31.880 ","End":"02:37.585","Text":"Let me leave the y-prime there a second, over x squared."},{"Start":"02:37.585 ","End":"02:39.275","Text":"Instead of this y prime,"},{"Start":"02:39.275 ","End":"02:43.070","Text":"I\u0027ll just put minus y over x."},{"Start":"02:43.070 ","End":"02:46.400","Text":"What does this give me after simplification?"},{"Start":"02:46.400 ","End":"02:48.590","Text":"The x cancels,"},{"Start":"02:48.590 ","End":"02:53.044","Text":"and so what I get is y minus minus y,"},{"Start":"02:53.044 ","End":"02:58.660","Text":"which is 2y over x squared."},{"Start":"02:59.470 ","End":"03:01.820","Text":"Things are a little bit close there,"},{"Start":"03:01.820 ","End":"03:09.645","Text":"I\u0027ll just emphasize that this part is the solution for y double-prime."},{"Start":"03:09.645 ","End":"03:14.400","Text":"Of course, there are restrictions on the domain,"},{"Start":"03:14.400 ","End":"03:16.835","Text":"this won\u0027t make sense if x squared is 0,"},{"Start":"03:16.835 ","End":"03:21.315","Text":"which means that x cannot be equal to 0."},{"Start":"03:21.315 ","End":"03:22.760","Text":"That\u0027s what we had before anyway,"},{"Start":"03:22.760 ","End":"03:24.290","Text":"and it carries on."},{"Start":"03:24.290 ","End":"03:25.955","Text":"That was 3,"},{"Start":"03:25.955 ","End":"03:29.105","Text":"and we\u0027re left with 1 more,"},{"Start":"03:29.105 ","End":"03:31.535","Text":"which is number 4."},{"Start":"03:31.535 ","End":"03:38.635","Text":"Sine x plus sine y is equal to x."},{"Start":"03:38.635 ","End":"03:42.105","Text":"Again, implicit differentiation."},{"Start":"03:42.105 ","End":"03:45.450","Text":"Sine x gives us cosine x,"},{"Start":"03:45.450 ","End":"03:52.035","Text":"sine y will give us cosine y but times y prime,"},{"Start":"03:52.035 ","End":"03:54.705","Text":"and x will give us 1."},{"Start":"03:54.705 ","End":"03:58.610","Text":"We want to isolate y prime,"},{"Start":"03:58.610 ","End":"04:04.955","Text":"and so all we do is bring this cosine x to the other side and divide by cosine y."},{"Start":"04:04.955 ","End":"04:10.610","Text":"We get y prime equals 1 minus"},{"Start":"04:10.610 ","End":"04:17.120","Text":"cosine x all over cosine of y."},{"Start":"04:17.120 ","End":"04:20.420","Text":"That\u0027s the answer for the first derivative part."},{"Start":"04:20.420 ","End":"04:26.105","Text":"Started to make it a habit for my y primes to be highlight."},{"Start":"04:26.105 ","End":"04:29.750","Text":"Proceeding, y double-prime,"},{"Start":"04:29.750 ","End":"04:32.455","Text":"this looks like a quotient to me."},{"Start":"04:32.455 ","End":"04:36.035","Text":"When I see a quotient and I\u0027m differentiating,"},{"Start":"04:36.035 ","End":"04:37.975","Text":"I write the quotient rule,"},{"Start":"04:37.975 ","End":"04:43.070","Text":"f over g prime is f prime g"},{"Start":"04:43.070 ","End":"04:48.650","Text":"minus f g prime all over g squared."},{"Start":"04:48.650 ","End":"04:51.320","Text":"F is a numerator, g is a denominator."},{"Start":"04:51.320 ","End":"04:57.030","Text":"Following this, f prime is 1 minus,"},{"Start":"04:57.030 ","End":"05:00.255","Text":"derivative of cosine x is minus sine x,"},{"Start":"05:00.255 ","End":"05:06.575","Text":"so instead minus minus we\u0027ll just make it a plus sine x over,"},{"Start":"05:06.575 ","End":"05:11.375","Text":"cosine of y gives us minus sine y."},{"Start":"05:11.375 ","End":"05:13.460","Text":"But that\u0027s not all, remember,"},{"Start":"05:13.460 ","End":"05:15.995","Text":"times y prime,"},{"Start":"05:15.995 ","End":"05:20.690","Text":"and y prime will immediately be colored."},{"Start":"05:20.690 ","End":"05:25.030","Text":"Like I said, this is the answer for the second derivative,"},{"Start":"05:25.030 ","End":"05:30.590","Text":"but it is customary not to have y-prime just in terms of x and y is what we wanted."},{"Start":"05:30.590 ","End":"05:40.845","Text":"This is equal to 1 plus sine x over minus sine y."},{"Start":"05:40.845 ","End":"05:48.080","Text":"My y-prime, I\u0027ll take it from here as 1 minus cosine x over cosine y."},{"Start":"05:48.080 ","End":"05:51.175","Text":"Just a bit more simplification."},{"Start":"05:51.175 ","End":"05:53.294","Text":"What would I do to simplify?"},{"Start":"05:53.294 ","End":"05:56.555","Text":"I don\u0027t like the fraction inside a fraction."},{"Start":"05:56.555 ","End":"05:58.585","Text":"We\u0027ll just multiply top and bottom."},{"Start":"05:58.585 ","End":"06:01.505","Text":"Otherwise, I\u0027ll multiply the top by cosine y,"},{"Start":"06:01.505 ","End":"06:04.565","Text":"and I\u0027ll multiply the bottom by cosine y,"},{"Start":"06:04.565 ","End":"06:06.880","Text":"and I haven\u0027t changed anything."},{"Start":"06:06.880 ","End":"06:15.910","Text":"On the numerator, I get 1 plus sine x cosine y."},{"Start":"06:15.910 ","End":"06:18.420","Text":"On the denominator,"},{"Start":"06:18.420 ","End":"06:22.760","Text":"the cosine y cancels with the cosine y,"},{"Start":"06:22.760 ","End":"06:24.410","Text":"and I get minus,"},{"Start":"06:24.410 ","End":"06:28.450","Text":"which I would like to put in front of the fraction."},{"Start":"06:28.450 ","End":"06:30.885","Text":"You know what? Change my mind,"},{"Start":"06:30.885 ","End":"06:32.974","Text":"there\u0027s another thing I can do with the minus."},{"Start":"06:32.974 ","End":"06:37.580","Text":"I can also change the order of a difference that will swallow a minus."},{"Start":"06:37.580 ","End":"06:39.665","Text":"So over."},{"Start":"06:39.665 ","End":"06:45.760","Text":"It\u0027s sine y times cosine x minus 1."},{"Start":"06:45.760 ","End":"06:48.590","Text":"I just changed the order to get rid of the minus."},{"Start":"06:48.590 ","End":"06:49.910","Text":"I think we\u0027ll just leave it like that,"},{"Start":"06:49.910 ","End":"06:52.685","Text":"I can\u0027t see any great simplification."},{"Start":"06:52.685 ","End":"06:55.610","Text":"That\u0027s the end of number 4,"},{"Start":"06:55.610 ","End":"06:58.200","Text":"and that\u0027s the end of the set."}],"ID":10475}],"Thumbnail":null,"ID":8714},{"Name":"Calculations Using The Definition of Derivative","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 - Parts 1-2","Duration":"9m 29s","ChapterTopicVideoID":10159,"CourseChapterTopicPlaylistID":8715,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10159.jpeg","UploadDate":"2017-08-21T20:29:46.3170000","DurationForVideoObject":"PT9M29S","Description":null,"MetaTitle":"Exercise 1 - Parts 1-2 - Calculations Using The Definition of Derivative: Practice Makes Perfect | Proprep","MetaDescription":"Studied the topic name and want to practice? Here are some exercises on Calculations Using The Definition of Derivative practice questions for you to maximize your understanding.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/the-derivative-of-a-function/calculations-using-the-definition-of-derivative/vid10465","VideoComments":[],"Subtitles":[{"Start":"00:00.650 ","End":"00:04.650","Text":"In all the exercises on differentiation up to now,"},{"Start":"00:04.650 ","End":"00:07.785","Text":"we\u0027ve been using various formulae for e^x,"},{"Start":"00:07.785 ","End":"00:12.885","Text":"for natural log of x, for polynomials."},{"Start":"00:12.885 ","End":"00:14.850","Text":"But in this set of exercises,"},{"Start":"00:14.850 ","End":"00:16.695","Text":"and there are 12 of them in all,"},{"Start":"00:16.695 ","End":"00:22.935","Text":"we\u0027re going to use the basic definition of the derivative to solve the exercises."},{"Start":"00:22.935 ","End":"00:25.770","Text":"Of course, we can always check our answers"},{"Start":"00:25.770 ","End":"00:29.460","Text":"using the other means that we know of differentiation."},{"Start":"00:29.460 ","End":"00:35.010","Text":"I\u0027ll just remind you what it means to differentiate from the definition,"},{"Start":"00:35.010 ","End":"00:40.300","Text":"or sometimes we say from first principles or whatever."},{"Start":"00:40.300 ","End":"00:42.305","Text":"If we have a function f of x,"},{"Start":"00:42.305 ","End":"00:46.790","Text":"we define the derivative function f prime of x according to this formula."},{"Start":"00:46.790 ","End":"00:49.010","Text":"The limit as h goes to 0,"},{"Start":"00:49.010 ","End":"00:50.450","Text":"which means that h is something very"},{"Start":"00:50.450 ","End":"00:55.085","Text":"small, of the function of x plus h,"},{"Start":"00:55.085 ","End":"00:58.040","Text":"meaning we substitute x plus h instead of x,"},{"Start":"00:58.040 ","End":"01:02.570","Text":"less f of x over h. I would"},{"Start":"01:02.570 ","End":"01:07.325","Text":"just also like to mention that there are some books that don\u0027t use h,"},{"Start":"01:07.325 ","End":"01:10.530","Text":"instead they use Delta x."},{"Start":"01:10.530 ","End":"01:16.280","Text":"If you encounter a lecturer or a book which may have the same thing,"},{"Start":"01:16.280 ","End":"01:17.480","Text":"but with Delta x,"},{"Start":"01:17.480 ","End":"01:19.700","Text":"it\u0027s just 1 concept also for something very,"},{"Start":"01:19.700 ","End":"01:22.410","Text":"very small, which we add to x."},{"Start":"01:22.410 ","End":"01:25.260","Text":"Anyone who knows physics and in the engineering,"},{"Start":"01:25.260 ","End":"01:27.465","Text":"Delta usually means addition."},{"Start":"01:27.465 ","End":"01:29.910","Text":"A very small addition to something."},{"Start":"01:29.910 ","End":"01:33.930","Text":"I\u0027m going to erase that though so as not to confuse."},{"Start":"01:36.620 ","End":"01:40.130","Text":"In case you see it in another book and you say, \"What\u0027s Delta x?"},{"Start":"01:40.130 ","End":"01:46.290","Text":"I know h.\" There we go and we\u0027ll start with the whole set of 8."},{"Start":"01:46.290 ","End":"01:49.665","Text":"We\u0027ll begin with the number 1,"},{"Start":"01:49.665 ","End":"01:58.690","Text":"f of x equals x squared."},{"Start":"02:03.460 ","End":"02:05.990","Text":"I know the answer supposed to be 2x,"},{"Start":"02:05.990 ","End":"02:07.340","Text":"and if we do get 2x,"},{"Start":"02:07.340 ","End":"02:10.620","Text":"we\u0027ll have a confirmation that we\u0027ve done it right."},{"Start":"02:11.500 ","End":"02:19.155","Text":"So f prime of x is according to this definition."},{"Start":"02:19.155 ","End":"02:23.510","Text":"I\u0027m wondering if I shouldn\u0027t put it in a box maybe."},{"Start":"02:23.510 ","End":"02:25.460","Text":"You know what? I\u0027ll even go for red."},{"Start":"02:25.460 ","End":"02:31.180","Text":"It\u0027s so important that I shall put a box around this."},{"Start":"02:31.760 ","End":"02:36.915","Text":"This is our definition of the derivative."},{"Start":"02:36.915 ","End":"02:47.430","Text":"So f prime of x is equal to the limit as h goes to 0. Let\u0027s look at this carefully."},{"Start":"02:47.430 ","End":"02:49.505","Text":"You know how to do substitutions."},{"Start":"02:49.505 ","End":"02:52.265","Text":"If f of x is x squared,"},{"Start":"02:52.265 ","End":"02:59.660","Text":"then f of x plus h is just x plus h squared less the original f of x,"},{"Start":"02:59.660 ","End":"03:01.570","Text":"which is x squared."},{"Start":"03:01.570 ","End":"03:06.195","Text":"All this over h and then we have to take the limit."},{"Start":"03:06.195 ","End":"03:11.290","Text":"Let\u0027s just keep doing some same algebraic simplifications."},{"Start":"03:14.170 ","End":"03:16.425","Text":"Delta [inaudible] limit,"},{"Start":"03:16.425 ","End":"03:19.050","Text":"[inaudible] I\u0027m going to write it so many times."},{"Start":"03:19.050 ","End":"03:24.075","Text":"It will always be of x squared plus"},{"Start":"03:24.075 ","End":"03:29.770","Text":"2xh plus h squared."},{"Start":"03:29.770 ","End":"03:31.750","Text":"That\u0027s the a plus b squared formula,"},{"Start":"03:31.750 ","End":"03:34.975","Text":"which I hope you memorize and you know it from your algebra."},{"Start":"03:34.975 ","End":"03:37.150","Text":"I mean, I don\u0027t have to tell you,"},{"Start":"03:37.150 ","End":"03:39.100","Text":"or maybe I do, who knows?"},{"Start":"03:39.100 ","End":"03:46.680","Text":"That a plus b squared is a squared plus 2ab plus b squared."},{"Start":"03:46.680 ","End":"03:49.275","Text":"Don\u0027t tell me you haven\u0027t seen that."},{"Start":"03:49.275 ","End":"03:59.655","Text":"The way up to here, less x squared all over h. Scroll up a bit there."},{"Start":"03:59.655 ","End":"04:07.515","Text":"This equals the limit of x squared and x squared cancel."},{"Start":"04:07.515 ","End":"04:11.660","Text":"We can even take h outside the brackets and what\u0027s left."},{"Start":"04:11.660 ","End":"04:13.430","Text":"We have h,"},{"Start":"04:13.430 ","End":"04:22.755","Text":"2x plus h all over h. Now, oh,"},{"Start":"04:22.755 ","End":"04:30.500","Text":"even more luck, this h cancels with this h. Notice that h is not 0 because h tends to 0,"},{"Start":"04:30.500 ","End":"04:33.860","Text":"it isn\u0027t 0 and everything makes sense here."},{"Start":"04:33.860 ","End":"04:39.005","Text":"At this point, we now are able to use the substitution method."},{"Start":"04:39.005 ","End":"04:48.075","Text":"We just substitute h equals 0, so we get 2x plus 0."},{"Start":"04:48.075 ","End":"04:53.045","Text":"In other words, 2x, which is what we expected to get any way."},{"Start":"04:53.045 ","End":"04:55.235","Text":"That ends number 1."},{"Start":"04:55.235 ","End":"04:57.760","Text":"Now, we\u0027ll go to number 2."},{"Start":"04:57.760 ","End":"04:59.460","Text":"But I don\u0027t remember what it is,"},{"Start":"04:59.460 ","End":"05:05.640","Text":"so I\u0027ll go up and see that number 2 is x squared plus 4x plus 1."},{"Start":"05:05.640 ","End":"05:15.720","Text":"Repeat that x squared plus 4x plus 1."},{"Start":"05:15.720 ","End":"05:20.055","Text":"x squared plus 4x plus 1."},{"Start":"05:20.055 ","End":"05:21.530","Text":"What are we hoping to get?"},{"Start":"05:21.530 ","End":"05:24.005","Text":"We can do this in our heads with the standard methods,"},{"Start":"05:24.005 ","End":"05:26.445","Text":"2x plus 4,"},{"Start":"05:26.445 ","End":"05:28.665","Text":"then I\u0027ll know I\u0027ve got it right."},{"Start":"05:28.665 ","End":"05:32.105","Text":"Looking at that red formula,"},{"Start":"05:32.105 ","End":"05:34.835","Text":"the top, and I suggest you memorize it,"},{"Start":"05:34.835 ","End":"05:42.140","Text":"we get that f prime of x is f of x plus h."},{"Start":"05:42.140 ","End":"05:43.970","Text":"Now what does that mean?"},{"Start":"05:43.970 ","End":"05:47.750","Text":"It means I put x plus h instead of x everywhere."},{"Start":"05:47.750 ","End":"05:52.760","Text":"We have x plus h squared"},{"Start":"05:52.760 ","End":"05:59.030","Text":"plus 4 times x plus h plus 1."},{"Start":"05:59.030 ","End":"06:03.660","Text":"All that is f of x plus h. In fact,"},{"Start":"06:03.660 ","End":"06:05.355","Text":"I\u0027ll put it in square brackets."},{"Start":"06:05.355 ","End":"06:06.675","Text":"That\u0027s the first piece."},{"Start":"06:06.675 ","End":"06:08.760","Text":"Less f of x,"},{"Start":"06:08.760 ","End":"06:15.480","Text":"which is just x squared plus 4x plus 1."},{"Start":"06:15.480 ","End":"06:23.620","Text":"All this is over h. I forgot to say limit as h goes to 0."},{"Start":"06:23.620 ","End":"06:26.240","Text":"Hang on a second, I\u0027ll alter that."},{"Start":"06:26.390 ","End":"06:31.120","Text":"I just shifted it over to the right and got room."},{"Start":"06:31.120 ","End":"06:35.840","Text":"It\u0027s very important this limit as h goes to 0, silly me."},{"Start":"06:36.830 ","End":"06:40.300","Text":"Let\u0027s continue."},{"Start":"06:40.300 ","End":"06:43.780","Text":"This is the limit of"},{"Start":"06:43.780 ","End":"06:50.390","Text":"fraction over h and c."},{"Start":"06:50.390 ","End":"06:52.295","Text":"x plus h squared,"},{"Start":"06:52.295 ","End":"06:55.300","Text":"again, using the a plus b squared formula,"},{"Start":"06:55.300 ","End":"07:00.510","Text":"is x squared plus 2xh"},{"Start":"07:00.520 ","End":"07:08.565","Text":"plus h squared plus 4x plus 4h."},{"Start":"07:08.565 ","End":"07:10.710","Text":"Let\u0027s just leave this in brackets."},{"Start":"07:10.710 ","End":"07:15.795","Text":"4x plus 4h, then this,"},{"Start":"07:15.795 ","End":"07:25.030","Text":"plus 1 minus,"},{"Start":"07:25.070 ","End":"07:27.540","Text":"just the same thing,"},{"Start":"07:27.540 ","End":"07:33.660","Text":"x squared plus 4x plus 1."},{"Start":"07:33.660 ","End":"07:36.760","Text":"Now let\u0027s see if anything cancels."},{"Start":"07:36.760 ","End":"07:39.610","Text":"Well, let me see."},{"Start":"07:39.610 ","End":"07:44.415","Text":"This x squared goes with this minus x squared,"},{"Start":"07:44.415 ","End":"07:51.800","Text":"4x minus 4x and plus 1 minus 1."},{"Start":"07:51.800 ","End":"07:58.750","Text":"All that we are left with is limit of, let\u0027s see,"},{"Start":"07:58.750 ","End":"08:04.240","Text":"2xh plus h"},{"Start":"08:04.240 ","End":"08:11.860","Text":"squared plus 4h,"},{"Start":"08:11.860 ","End":"08:20.710","Text":"and that\u0027s it, all over h. That equals the limit."},{"Start":"08:20.710 ","End":"08:24.010","Text":"We can take h outside the brackets,"},{"Start":"08:24.010 ","End":"08:33.895","Text":"h times 2x plus h, after we take h out, plus 4 all"},{"Start":"08:33.895 ","End":"08:38.740","Text":"over h. Then this h"},{"Start":"08:38.740 ","End":"08:44.620","Text":"cancels with this h. Let\u0027s put it in the h tends to 0,"},{"Start":"08:44.620 ","End":"08:48.410","Text":"otherwise someone will complain or get confused."},{"Start":"08:49.410 ","End":"08:51.970","Text":"Now, that\u0027s what it is."},{"Start":"08:51.970 ","End":"08:55.300","Text":"It\u0027s just repetitive to keep writing h goes to 0,"},{"Start":"08:55.300 ","End":"09:00.530","Text":"h goes to 0, h goes to 0, but that\u0027s what it is."},{"Start":"09:00.870 ","End":"09:07.340","Text":"Now, it\u0027s simple enough to just do use the substitution, and substitution because there\u0027s"},{"Start":"09:07.340 ","End":"09:17.140","Text":"no reason h couldn\u0027t be put 0 here and we just get 2x plus 0 plus 4."},{"Start":"09:17.140 ","End":"09:20.885","Text":"I\u0027m not going to make that into 2 separate lines, it\u0027s just 2x plus 4."},{"Start":"09:20.885 ","End":"09:24.100","Text":"That\u0027s what we expected anyway, 2x plus 4."},{"Start":"09:24.100 ","End":"09:27.800","Text":"That does number 2."}],"ID":10465},{"Watched":false,"Name":"Exercise 1 - Parts 3-4","Duration":"9m 5s","ChapterTopicVideoID":10160,"CourseChapterTopicPlaylistID":8715,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.640","Text":"Next, after number 2,"},{"Start":"00:06.410 ","End":"00:13.965","Text":"there will be number 3. You know what,"},{"Start":"00:13.965 ","End":"00:18.705","Text":"I\u0027ll just go and check, and stop a minute."},{"Start":"00:18.705 ","End":"00:22.815","Text":"I just took time out to check and it\u0027s x cubed."},{"Start":"00:22.815 ","End":"00:30.885","Text":"Exercise 3 is f of x is equal to x cubed."},{"Start":"00:30.885 ","End":"00:35.175","Text":"Again, the same formula with the x plus h and so forth."},{"Start":"00:35.175 ","End":"00:38.460","Text":"We\u0027ll try and see what happens here."},{"Start":"00:38.460 ","End":"00:41.365","Text":"F prime of x,"},{"Start":"00:41.365 ","End":"00:45.215","Text":"differentiating from scratch or from the formula,"},{"Start":"00:45.215 ","End":"00:50.895","Text":"is going to be f of x plus h,"},{"Start":"00:50.895 ","End":"00:58.410","Text":"which is x plus h cubed because our function x cubed if x is x plus h less f of x,"},{"Start":"00:58.410 ","End":"01:04.680","Text":"which is just x cubed all over h. Silly me,"},{"Start":"01:04.680 ","End":"01:07.990","Text":"I\u0027ve again forgotten to put the limit, so hang on."},{"Start":"01:08.030 ","End":"01:13.740","Text":"I just rewrote it with the limit as h goes to 0."},{"Start":"01:13.740 ","End":"01:16.455","Text":"We\u0027ll continue."},{"Start":"01:16.455 ","End":"01:21.165","Text":"This is equal to the limit."},{"Start":"01:21.165 ","End":"01:28.970","Text":"Now, there is a formula for a plus b cubed and I\u0027ll just show you what it is."},{"Start":"01:28.970 ","End":"01:36.660","Text":"A plus b cubed is a cubed plus 3a"},{"Start":"01:36.660 ","End":"01:45.600","Text":"squared b plus 3ab squared plus b cubed."},{"Start":"01:45.600 ","End":"01:47.610","Text":"If I apply that here,"},{"Start":"01:47.610 ","End":"01:51.165","Text":"I\u0027ll get x cubed,"},{"Start":"01:51.165 ","End":"01:52.560","Text":"I\u0027m just using here,"},{"Start":"01:52.560 ","End":"01:59.220","Text":"plus 3x squared h plus"},{"Start":"01:59.220 ","End":"02:06.375","Text":"3xh squared plus h cubed."},{"Start":"02:06.375 ","End":"02:08.490","Text":"That\u0027s for this. Then,"},{"Start":"02:08.490 ","End":"02:11.055","Text":"minus x cubed,"},{"Start":"02:11.055 ","End":"02:21.725","Text":"and all this over h. What we get is the limit."},{"Start":"02:21.725 ","End":"02:25.880","Text":"Now, luckily, you\u0027ll see why luckily,"},{"Start":"02:25.880 ","End":"02:28.415","Text":"the x cubed cancels."},{"Start":"02:28.415 ","End":"02:31.250","Text":"All the other terms have an h in them,"},{"Start":"02:31.250 ","End":"02:34.445","Text":"so we can take h outside the brackets."},{"Start":"02:34.445 ","End":"02:38.020","Text":"From here, we\u0027re left with 3x squared."},{"Start":"02:38.020 ","End":"02:44.690","Text":"Here, we\u0027re left with 3xh. Here we\u0027re"},{"Start":"02:44.690 ","End":"02:49.745","Text":"left with h squared."},{"Start":"02:49.745 ","End":"03:00.600","Text":"All this is over h. The h\u0027s cancel."},{"Start":"03:01.340 ","End":"03:06.120","Text":"Remember, here it\u0027s the h goes to 0."},{"Start":"03:06.120 ","End":"03:08.280","Text":"I\u0027ll put it here also,"},{"Start":"03:08.280 ","End":"03:11.115","Text":"h goes to 0,"},{"Start":"03:11.115 ","End":"03:14.690","Text":"which means that we can use substitution here"},{"Start":"03:14.690 ","End":"03:17.690","Text":"because there\u0027s no reason why we can\u0027t just put our h equals 0."},{"Start":"03:17.690 ","End":"03:20.960","Text":"That\u0027s 0 and that\u0027s 0 because it has an h and all we\u0027re left"},{"Start":"03:20.960 ","End":"03:24.765","Text":"with is the familiar 3x squared."},{"Start":"03:24.765 ","End":"03:30.590","Text":"That\u0027s number 3 and then onto number 4."},{"Start":"03:30.590 ","End":"03:35.300","Text":"But let me just quickly go and see what number 4 is."},{"Start":"03:35.300 ","End":"03:40.550","Text":"I went up to take a peek and it\u0027s f of x is 1 over x."},{"Start":"03:40.550 ","End":"03:48.255","Text":"F of x equals 1 over x."},{"Start":"03:48.255 ","End":"03:50.100","Text":"Just scroll up."},{"Start":"03:50.100 ","End":"03:53.830","Text":"Again, we\u0027re going to use the formula,"},{"Start":"03:53.900 ","End":"03:56.510","Text":"but I\u0027m not going to repeat the formula."},{"Start":"03:56.510 ","End":"04:01.680","Text":"Just check back the notes as to what the formula is."},{"Start":"04:03.200 ","End":"04:07.085","Text":"Make sure you have that formula in front of you."},{"Start":"04:07.085 ","End":"04:16.370","Text":"What we get is that f prime of x is f of x plus h"},{"Start":"04:16.370 ","End":"04:19.580","Text":"and f of x plus h will be"},{"Start":"04:19.580 ","End":"04:27.405","Text":"1 over x plus h less f of x,"},{"Start":"04:27.405 ","End":"04:30.570","Text":"which is just 1 over x,"},{"Start":"04:30.570 ","End":"04:33.390","Text":"and all this over h."},{"Start":"04:33.390 ","End":"04:40.670","Text":"Again, I\u0027ve forgotten to write limit as x goes to 0, so hang on."},{"Start":"04:41.360 ","End":"04:43.915","Text":"I just stuck in the limit,"},{"Start":"04:43.915 ","End":"04:46.000","Text":"h goes to 0 here."},{"Start":"04:46.000 ","End":"04:48.830","Text":"Now, let\u0027s continue,"},{"Start":"04:52.910 ","End":"04:56.450","Text":"f prime of x is going to be equal."},{"Start":"04:56.450 ","End":"04:59.310","Text":"Now, I\u0027ll just do a bit of algebra here."},{"Start":"05:01.460 ","End":"05:05.025","Text":"The limit of,"},{"Start":"05:05.025 ","End":"05:09.360","Text":"it\u0027s going to be over h, that\u0027s for sure."},{"Start":"05:09.360 ","End":"05:13.530","Text":"If we do our fractions,"},{"Start":"05:13.530 ","End":"05:19.440","Text":"this minus this, we multiply this times this."},{"Start":"05:19.440 ","End":"05:20.970","Text":"There are many ways of doing it."},{"Start":"05:20.970 ","End":"05:25.460","Text":"1 of them is saying common denominator is x times x plus h,"},{"Start":"05:25.460 ","End":"05:27.485","Text":"so we multiply this by,"},{"Start":"05:27.485 ","End":"05:29.345","Text":"whatever way you\u0027re used to doing it."},{"Start":"05:29.345 ","End":"05:35.315","Text":"I\u0027m used to doing it by taking this and putting it here, and that\u0027s x."},{"Start":"05:35.315 ","End":"05:38.450","Text":"Taking this and multiplying by that,"},{"Start":"05:38.450 ","End":"05:45.220","Text":"which is x plus h over the product,"},{"Start":"05:45.220 ","End":"05:47.810","Text":"the common denominator, which is,"},{"Start":"05:47.810 ","End":"05:49.175","Text":"I\u0027ll put the x first,"},{"Start":"05:49.175 ","End":"05:57.065","Text":"x times x plus h and just complete this a bit longer."},{"Start":"05:57.065 ","End":"06:03.060","Text":"That\u0027s it. What we"},{"Start":"06:03.060 ","End":"06:09.905","Text":"get is the top comes out to be the x minus x,"},{"Start":"06:09.905 ","End":"06:12.690","Text":"which is minus h,"},{"Start":"06:14.510 ","End":"06:18.900","Text":"and here over x, x plus h."},{"Start":"06:18.900 ","End":"06:32.230","Text":"Now, I want to put a big dividing line with an h on the bottom,"},{"Start":"06:32.230 ","End":"06:35.649","Text":"but there\u0027s no need to do that because I could just put the h"},{"Start":"06:35.649 ","End":"06:39.460","Text":"together with the denominator of the numerator, if that makes sense."},{"Start":"06:39.460 ","End":"06:42.100","Text":"Anyway, let me just say,"},{"Start":"06:42.100 ","End":"06:43.840","Text":"in algebra, if you\u0027re not following,"},{"Start":"06:43.840 ","End":"06:49.990","Text":"that if I have a over b and all this is over c,"},{"Start":"06:49.990 ","End":"06:53.320","Text":"then this is the same as a over bc."},{"Start":"06:53.320 ","End":"06:56.270","Text":"Just look at it a while and you\u0027ll see."},{"Start":"06:57.180 ","End":"07:02.120","Text":"H cancels with this."},{"Start":"07:03.510 ","End":"07:05.845","Text":"I\u0027ll write it again though."},{"Start":"07:05.845 ","End":"07:07.300","Text":"After the h cancels,"},{"Start":"07:07.300 ","End":"07:11.120","Text":"we\u0027re left with minus 1 and limit, of course."},{"Start":"07:11.120 ","End":"07:16.640","Text":"I have a tendency to forget to write the limit and also here, the limit."},{"Start":"07:16.640 ","End":"07:26.820","Text":"We get minus 1 over x times x plus h. This is a very simple one."},{"Start":"07:26.820 ","End":"07:28.490","Text":"This is just a substitution."},{"Start":"07:28.490 ","End":"07:32.075","Text":"There\u0027s no reason that h can\u0027t be 0."},{"Start":"07:32.075 ","End":"07:33.965","Text":"Right from the beginning,"},{"Start":"07:33.965 ","End":"07:40.580","Text":"we knew that x can\u0027t be 0 because in the definition of 1 over x,"},{"Start":"07:40.580 ","End":"07:45.665","Text":"I should have said it\u0027s only defined when x is not equal to 0"},{"Start":"07:45.665 ","End":"07:57.225","Text":"and so [inaudible] back here."},{"Start":"07:57.225 ","End":"07:59.505","Text":"X is not 0."},{"Start":"07:59.505 ","End":"08:01.170","Text":"If I put h equals 0,"},{"Start":"08:01.170 ","End":"08:02.250","Text":"this won\u0027t be 0."},{"Start":"08:02.250 ","End":"08:04.900","Text":"I\u0027m not dividing by 0."},{"Start":"08:05.930 ","End":"08:10.140","Text":"Well, x plus h you might say could be 0,"},{"Start":"08:10.140 ","End":"08:13.350","Text":"but if x is not 0,"},{"Start":"08:13.350 ","End":"08:18.665","Text":"h can be half of x or smaller than that,"},{"Start":"08:18.665 ","End":"08:21.780","Text":"or something much, much smaller than x."},{"Start":"08:21.890 ","End":"08:24.215","Text":"This won\u0027t be 0 either."},{"Start":"08:24.215 ","End":"08:28.185","Text":"This is getting technical. It works."},{"Start":"08:28.185 ","End":"08:31.320","Text":"We can put x equals h,"},{"Start":"08:31.320 ","End":"08:33.940","Text":"we can put equals 0."},{"Start":"08:37.790 ","End":"08:41.265","Text":"I might as well put it in here too."},{"Start":"08:41.265 ","End":"08:49.920","Text":"This leaves us with minus 1 over x times x is x squared."},{"Start":"08:49.920 ","End":"08:55.955","Text":"This is the familiar derivative of for sure the whole exercise here."},{"Start":"08:55.955 ","End":"09:00.050","Text":"1 over x, we know that its derivative is minus 1 over x squared,"},{"Start":"09:00.050 ","End":"09:03.100","Text":"so this has got to be right."}],"ID":10466},{"Watched":false,"Name":"Exercise 1 - Part 5","Duration":"6m 22s","ChapterTopicVideoID":10155,"CourseChapterTopicPlaylistID":8715,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.410 ","End":"00:05.760","Text":"Continuing, then after number 4,"},{"Start":"00:05.760 ","End":"00:08.874","Text":"we should get number 5."},{"Start":"00:08.874 ","End":"00:16.005","Text":"Number 5 will be,"},{"Start":"00:16.005 ","End":"00:20.730","Text":"and I already went and checked beforehand that it is that f"},{"Start":"00:20.730 ","End":"00:27.015","Text":"of x equals the square root of x."},{"Start":"00:27.015 ","End":"00:32.685","Text":"Now, I happen to remember the answer which is 1 over twice the square root of x,"},{"Start":"00:32.685 ","End":"00:34.125","Text":"but I didn\u0027t say that."},{"Start":"00:34.125 ","End":"00:38.505","Text":"That\u0027s just in my head for checking that l know when I got the right answer."},{"Start":"00:38.505 ","End":"00:46.545","Text":"Perhaps it is time to repeat the formula for the derivative."},{"Start":"00:46.545 ","End":"00:48.725","Text":"Once in a while, I\u0027ll remind you of it,"},{"Start":"00:48.725 ","End":"00:58.715","Text":"that f prime of x is the limit as h goes to 0."},{"Start":"00:58.715 ","End":"01:01.910","Text":"I\u0027m going to also make the same remark that in some books instead of h,"},{"Start":"01:01.910 ","End":"01:08.120","Text":"they use Delta x, so it\u0027s f of x plus h"},{"Start":"01:08.120 ","End":"01:16.115","Text":"minus f of x all over h. Over here,"},{"Start":"01:16.115 ","End":"01:21.244","Text":"we will have that f prime of x"},{"Start":"01:21.244 ","End":"01:29.030","Text":"equals the limit as h goes to 0 of f of x plus h,"},{"Start":"01:29.030 ","End":"01:35.030","Text":"which is the square root of x plus h minus f of"},{"Start":"01:35.030 ","End":"01:41.970","Text":"x all over h. Now,"},{"Start":"01:41.970 ","End":"01:43.745","Text":"what do we do with this?"},{"Start":"01:43.745 ","End":"01:47.990","Text":"Well, it looks to me like it\u0027s 1 of those cases where we need to use"},{"Start":"01:47.990 ","End":"01:53.090","Text":"conjugates because we have the square root on 1 or more of the terms,"},{"Start":"01:53.090 ","End":"01:56.345","Text":"and if we multiply by the conjugate,"},{"Start":"01:56.345 ","End":"01:57.940","Text":"then that should do the trick."},{"Start":"01:57.940 ","End":"02:00.950","Text":"I will remind you what a conjugate is like."},{"Start":"02:00.950 ","End":"02:04.440","Text":"For example, here we have something of the form A minus B,"},{"Start":"02:04.440 ","End":"02:10.040","Text":"and the conjugate of A minus B is A plus B, and vice versa."},{"Start":"02:10.040 ","End":"02:13.100","Text":"The thing is that if we multiply something by its conjugate,"},{"Start":"02:13.100 ","End":"02:17.430","Text":"we have this difference of squares formula where this product is equal to"},{"Start":"02:17.430 ","End":"02:21.680","Text":"A squared minus B squared, and that will be just fantastic for us"},{"Start":"02:21.680 ","End":"02:25.355","Text":"because if A is the square root of something and we square it,"},{"Start":"02:25.355 ","End":"02:27.035","Text":"we won\u0027t have a square root."},{"Start":"02:27.035 ","End":"02:29.570","Text":"Let\u0027s see what happens with this."},{"Start":"02:29.570 ","End":"02:35.750","Text":"This is equal to the limit of"},{"Start":"02:35.750 ","End":"02:43.180","Text":"the square root of x plus h minus the square root of x,"},{"Start":"02:43.180 ","End":"02:46.430","Text":"and I\u0027ll put that in brackets."},{"Start":"02:47.360 ","End":"02:51.270","Text":"It\u0027s going to be over h still,"},{"Start":"02:51.270 ","End":"02:52.620","Text":"but what I\u0027m going to do,"},{"Start":"02:52.620 ","End":"02:57.960","Text":"is I want to multiply this thing by its conjugate."},{"Start":"02:57.960 ","End":"03:06.990","Text":"I\u0027ll multiply it by the square root of x plus h plus the square root of x."},{"Start":"03:08.600 ","End":"03:14.000","Text":"Now, I can\u0027t just go ahead and multiply by something"},{"Start":"03:14.000 ","End":"03:15.769","Text":"because it changes the exercise,"},{"Start":"03:15.769 ","End":"03:18.175","Text":"but if I also divide by it,"},{"Start":"03:18.175 ","End":"03:20.495","Text":"if I divide by the same thing,"},{"Start":"03:20.495 ","End":"03:27.110","Text":"the square root of x plus h plus the square root of x,"},{"Start":"03:27.110 ","End":"03:30.415","Text":"then I haven\u0027t changed the exercise at all."},{"Start":"03:30.415 ","End":"03:34.215","Text":"I didn\u0027t write this very neatly, hang on."},{"Start":"03:34.215 ","End":"03:38.330","Text":"If I also divide by it and just multiply this fraction by 1,"},{"Start":"03:38.330 ","End":"03:43.204","Text":"by A over A or something because it\u0027s just 1, so that\u0027s fine."},{"Start":"03:43.204 ","End":"03:51.530","Text":"Now, let\u0027s see. If we multiply out, this times this, by this formula is"},{"Start":"03:51.530 ","End":"03:53.210","Text":"A squared minus B squared,"},{"Start":"03:53.210 ","End":"03:55.070","Text":"so it\u0027s this thing squared,"},{"Start":"03:55.070 ","End":"04:00.460","Text":"which is just x plus h without the square root,"},{"Start":"04:00.460 ","End":"04:08.570","Text":"minus B squared is minus square root of x squared is just x without the square root over"},{"Start":"04:08.570 ","End":"04:14.480","Text":"h times square root of x plus"},{"Start":"04:14.480 ","End":"04:22.760","Text":"h plus square root of x."},{"Start":"04:22.760 ","End":"04:24.215","Text":"Again, I\u0027ve forgotten the limit."},{"Start":"04:24.215 ","End":"04:27.800","Text":"Hang on. There,"},{"Start":"04:27.800 ","End":"04:32.130","Text":"I\u0027ve put it back in what I had before with limit."},{"Start":"04:34.150 ","End":"04:36.815","Text":"Yeah, there we go."},{"Start":"04:36.815 ","End":"04:41.670","Text":"Now we can continue doing some algebra."},{"Start":"04:41.670 ","End":"04:46.125","Text":"Limit, h goes to 0,"},{"Start":"04:46.125 ","End":"04:50.800","Text":"x plus h minus x is just h."},{"Start":"04:51.560 ","End":"04:56.510","Text":"Then in the denominator we have h times whatever we had before,"},{"Start":"04:56.510 ","End":"05:03.435","Text":"square root of x plus h. Continue this line a bit,"},{"Start":"05:03.435 ","End":"05:06.865","Text":"plus square root of x."},{"Start":"05:06.865 ","End":"05:11.000","Text":"But this thing, the h cancels."},{"Start":"05:11.000 ","End":"05:13.370","Text":"Remember h goes to 0, means h is not 0."},{"Start":"05:13.370 ","End":"05:14.870","Text":"After h cancels,"},{"Start":"05:14.870 ","End":"05:19.385","Text":"we just leave 1 here in its place."},{"Start":"05:19.385 ","End":"05:22.295","Text":"This is equal to."},{"Start":"05:22.295 ","End":"05:27.600","Text":"Now, there\u0027s no reason here that we can\u0027t just put h equals 0."},{"Start":"05:29.800 ","End":"05:33.335","Text":"After putting this as a substitution part,"},{"Start":"05:33.335 ","End":"05:39.139","Text":"we get 1 over, f, h is 0."},{"Start":"05:39.139 ","End":"05:41.855","Text":"It\u0027s the square root of x plus"},{"Start":"05:41.855 ","End":"05:51.990","Text":"0 plus the square root of x."},{"Start":"05:52.210 ","End":"05:55.580","Text":"Square root of x plus 0 is just the square root of x,"},{"Start":"05:55.580 ","End":"05:56.885","Text":"and we have 2 of them."},{"Start":"05:56.885 ","End":"06:04.600","Text":"I just write the answer as 1 over twice the square root of x."},{"Start":"06:04.600 ","End":"06:10.325","Text":"That\u0027s the derivative and that\u0027s what we\u0027ve always known."},{"Start":"06:10.325 ","End":"06:14.179","Text":"Some of us have remembered it that the derivative"},{"Start":"06:14.179 ","End":"06:20.480","Text":"of square root of x is 1 over twice the square root of x."},{"Start":"06:20.480 ","End":"06:22.770","Text":"That\u0027s number 5."}],"ID":10467},{"Watched":false,"Name":"Exercise 1 - Part 6","Duration":"11m 44s","ChapterTopicVideoID":10156,"CourseChapterTopicPlaylistID":8715,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"I did number 5 and I jumped straight to number"},{"Start":"00:03.180 ","End":"00:07.335","Text":"6 and copied it and even slightly started it."},{"Start":"00:07.335 ","End":"00:12.450","Text":"Just note that instead of no particular reason,"},{"Start":"00:12.450 ","End":"00:17.040","Text":"instead of the letter f for the function they using the letter z,"},{"Start":"00:17.040 ","End":"00:19.560","Text":"which if you\u0027re from England is called zed."},{"Start":"00:19.560 ","End":"00:22.875","Text":"Zed z will use the American form z."},{"Start":"00:22.875 ","End":"00:30.900","Text":"So z of x is natural log of x and using the same formula for the derivative,"},{"Start":"00:30.900 ","End":"00:40.005","Text":"the z prime of x is the limit as h goes to 0 of the function at the point x plus"},{"Start":"00:40.005 ","End":"00:43.130","Text":"h minus the function at the point x over"},{"Start":"00:43.130 ","End":"00:50.060","Text":"h. We seem to be a bit stuck here what tools are we going to use?"},{"Start":"00:50.060 ","End":"00:56.070","Text":"What I suggest is to put a dividing line"},{"Start":"00:56.070 ","End":"01:03.475","Text":"here and we\u0027ll write some formulae which could help."},{"Start":"01:03.475 ","End":"01:09.400","Text":"Now this one reminds me of something from algebra with"},{"Start":"01:09.400 ","End":"01:15.510","Text":"logarithms is that the natural log there\u0027s 1 for product and 1 for quotient,"},{"Start":"01:15.510 ","End":"01:20.485","Text":"so I\u0027ll use the 1 for quotient because that will suit us."},{"Start":"01:20.485 ","End":"01:25.340","Text":"Natural log of a over b is"},{"Start":"01:25.340 ","End":"01:30.995","Text":"natural log of a minus natural log of b."},{"Start":"01:30.995 ","End":"01:36.140","Text":"You can see by the numerator why I\u0027m thinking of this rule and there\u0027s another rule"},{"Start":"01:36.140 ","End":"01:42.440","Text":"that the natural log of a to the power of something,"},{"Start":"01:42.440 ","End":"01:51.360","Text":"a to the power of b is just b times natural log of a."},{"Start":"01:51.790 ","End":"01:54.245","Text":"With this in mind,"},{"Start":"01:54.245 ","End":"02:04.440","Text":"I\u0027m going to continue here and say that this is equal to the limit as h goes to 0."},{"Start":"02:05.000 ","End":"02:09.530","Text":"Now I can actually use these 2 formulas at once."},{"Start":"02:09.530 ","End":"02:10.850","Text":"No, I won\u0027t be lazy,"},{"Start":"02:10.850 ","End":"02:13.620","Text":"I\u0027ll do it in 2 goes."},{"Start":"02:14.020 ","End":"02:18.690","Text":"The numerator is the natural log"},{"Start":"02:20.900 ","End":"02:26.080","Text":"of x plus h over x,"},{"Start":"02:28.930 ","End":"02:32.840","Text":"and then divided by h I would rather instead of dividing by h,"},{"Start":"02:32.840 ","End":"02:37.400","Text":"multiply by 1 over h. Because my 1 over h is going to"},{"Start":"02:37.400 ","End":"02:43.195","Text":"be the b in the second formula and so what I get is the limit"},{"Start":"02:43.195 ","End":"02:53.110","Text":"as h goes to 0 of the natural log of this thing to"},{"Start":"02:53.110 ","End":"03:03.235","Text":"the power of 1 over h. If I also do a division here,"},{"Start":"03:03.235 ","End":"03:07.955","Text":"this thing is 1 plus h over x,"},{"Start":"03:07.955 ","End":"03:11.610","Text":"and this thing is the b that\u0027s, well,"},{"Start":"03:11.610 ","End":"03:15.580","Text":"it\u0027s same as an art form."},{"Start":"03:15.580 ","End":"03:18.000","Text":"It comes afterwards times b,"},{"Start":"03:18.000 ","End":"03:19.690","Text":"I could have written that as well,"},{"Start":"03:19.690 ","End":"03:22.420","Text":"but I could just put it in front anyway."},{"Start":"03:22.420 ","End":"03:32.870","Text":"To the power of b is to the power of 1 over h. What I\u0027d like to do is"},{"Start":"03:32.870 ","End":"03:41.460","Text":"a little side exercise and let\u0027s think of a color that\u0027s a green and"},{"Start":"03:41.460 ","End":"03:46.910","Text":"this thing I\u0027m going to do over here and"},{"Start":"03:46.910 ","End":"03:52.395","Text":"simplify it and so I\u0027m going to write it like this."},{"Start":"03:52.395 ","End":"03:55.455","Text":"1 plus h over"},{"Start":"03:55.455 ","End":"04:03.845","Text":"x to the power of 1 over h is equal to."},{"Start":"04:03.845 ","End":"04:07.160","Text":"Now, the reason I\u0027m doing all this at the side is because it"},{"Start":"04:07.160 ","End":"04:10.340","Text":"really reminds me of something and it reminds me of"},{"Start":"04:10.340 ","End":"04:14.400","Text":"another formula which says"},{"Start":"04:19.160 ","End":"04:28.830","Text":"that the limit as t goes to 0 of"},{"Start":"04:28.830 ","End":"04:34.630","Text":"1 plus t to the power of t is equal to"},{"Start":"04:34.630 ","End":"04:38.410","Text":"e. What I\u0027m trying to do is get this to"},{"Start":"04:38.410 ","End":"04:44.510","Text":"somehow look like 1 plus t to the power of t and use this formula."},{"Start":"04:44.550 ","End":"04:50.765","Text":"What I can do is if I make a certain substitution."},{"Start":"04:50.765 ","End":"05:00.470","Text":"I will just do"},{"Start":"05:00.470 ","End":"05:04.010","Text":"it over below here just a continuation here."},{"Start":"05:04.010 ","End":"05:09.380","Text":"If I let t is equal to h over x,"},{"Start":"05:09.380 ","End":"05:21.080","Text":"then what I get is that as t goes to 0,"},{"Start":"05:21.080 ","End":"05:24.680","Text":"also h goes to 0 or the other way around."},{"Start":"05:24.680 ","End":"05:26.150","Text":"If h goes to 0,"},{"Start":"05:26.150 ","End":"05:28.919","Text":"then t goes to 0."},{"Start":"05:29.630 ","End":"05:32.210","Text":"In some ways X is our constant,"},{"Start":"05:32.210 ","End":"05:34.890","Text":"x is not changing h is changing in numerator."},{"Start":"05:34.890 ","End":"05:38.675","Text":"H goes to 0, t goes to 0 and vice versa,"},{"Start":"05:38.675 ","End":"05:45.180","Text":"that\u0027s 1 thing I want to point out and also that I could write."},{"Start":"05:50.600 ","End":"05:53.510","Text":"I\u0027m sorry. I wrote something wrong."},{"Start":"05:53.510 ","End":"05:54.950","Text":"Just 1 second,"},{"Start":"05:54.950 ","End":"05:59.990","Text":"I\u0027ll correct that that\u0027s supposed to be 1 over t. Yes,"},{"Start":"05:59.990 ","End":"06:03.320","Text":"I thought something was wrong there."},{"Start":"06:03.320 ","End":"06:05.885","Text":"T is h over x,"},{"Start":"06:05.885 ","End":"06:10.880","Text":"so 1 over t is x over"},{"Start":"06:10.880 ","End":"06:18.185","Text":"h. What I could do,"},{"Start":"06:18.185 ","End":"06:23.850","Text":"since what I have in the numerator is 1 over h,"},{"Start":"06:23.850 ","End":"06:31.025","Text":"1 over h is like x over t. It might be a little bit of a shorter way to do this,"},{"Start":"06:31.025 ","End":"06:34.739","Text":"1 over h is just"},{"Start":"06:48.140 ","End":"06:51.585","Text":"1 over tx."},{"Start":"06:51.585 ","End":"06:54.075","Text":"Yeah, that\u0027s right, 1 over h,"},{"Start":"06:54.075 ","End":"06:57.360","Text":"of course would be 1 over"},{"Start":"06:57.360 ","End":"07:04.550","Text":"tx and I\u0027m all the time looking at this formula trying to get that."},{"Start":"07:04.550 ","End":"07:06.455","Text":"What I finally get,"},{"Start":"07:06.455 ","End":"07:10.175","Text":"if I do this is I have 1 plus,"},{"Start":"07:10.175 ","End":"07:11.960","Text":"now I\u0027m going to do some substitutions."},{"Start":"07:11.960 ","End":"07:17.990","Text":"1 plus t and we said that t goes to 0 when h goes to 0."},{"Start":"07:17.990 ","End":"07:21.425","Text":"So it\u0027s going to be a limit with t going to 0,"},{"Start":"07:21.425 ","End":"07:25.235","Text":"and then the 1 over h is the 1 over tx,"},{"Start":"07:25.235 ","End":"07:31.595","Text":"which is 1 over t. But instead of putting the x here,"},{"Start":"07:31.595 ","End":"07:38.870","Text":"I could put this whole thing to the power of 1 over"},{"Start":"07:38.870 ","End":"07:47.720","Text":"x. I think that does it and then finally,"},{"Start":"07:47.720 ","End":"07:51.770","Text":"so what I can do is I\u0027ve worked out the green bit and I can just"},{"Start":"07:51.770 ","End":"07:56.285","Text":"put it instead of what I\u0027ve marked in green here."},{"Start":"07:56.285 ","End":"08:00.960","Text":"This is going to equal limit."},{"Start":"08:01.870 ","End":"08:06.680","Text":"Now remember saying that h goes to 0 is like saying that t goes"},{"Start":"08:06.680 ","End":"08:09.995","Text":"to 0 of"},{"Start":"08:09.995 ","End":"08:17.780","Text":"the natural log of,"},{"Start":"08:17.780 ","End":"08:19.440","Text":"let me keep the square brackets of"},{"Start":"08:19.440 ","End":"08:29.470","Text":"1 plus t to the power of 1 over t and all this,"},{"Start":"08:30.040 ","End":"08:35.070","Text":"m to the power of 1 over x."},{"Start":"08:37.550 ","End":"08:43.250","Text":"Now it\u0027s one of those things that you can put the limit inside a function,"},{"Start":"08:43.250 ","End":"08:46.310","Text":"like if the limit of natural log of"},{"Start":"08:46.310 ","End":"08:49.505","Text":"something is the same as the natural log of the limit of something."},{"Start":"08:49.505 ","End":"08:53.945","Text":"The same thing when I say something to the power of something,"},{"Start":"08:53.945 ","End":"08:55.600","Text":"that will be to the power of."},{"Start":"08:55.600 ","End":"09:02.160","Text":"In short, what I\u0027m trying to say is that if I take just the limit,"},{"Start":"09:02.240 ","End":"09:05.000","Text":"I\u0027ll leave outside the log,"},{"Start":"09:05.000 ","End":"09:08.045","Text":"I\u0027ll leave outside the exponent of 1 over x,"},{"Start":"09:08.045 ","End":"09:13.475","Text":"and I\u0027ll get the limit of 1 plus t to the 1 over t,"},{"Start":"09:13.475 ","End":"09:20.250","Text":"which I\u0027ll compute and then I\u0027ll take"},{"Start":"09:20.250 ","End":"09:28.610","Text":"the natural log of"},{"Start":"09:28.610 ","End":"09:38.720","Text":"this and to the power of 1 over x."},{"Start":"09:38.720 ","End":"09:40.220","Text":"Just to make it really clear,"},{"Start":"09:40.220 ","End":"09:44.720","Text":"this thing is to the 1 over the x and then I take the log,"},{"Start":"09:44.720 ","End":"09:47.560","Text":"natural log of all this."},{"Start":"09:47.560 ","End":"09:53.895","Text":"Now here\u0027s where it starts to get simpler though it looks like a mess."},{"Start":"09:53.895 ","End":"09:57.230","Text":"Again, just to briefly summarize what I\u0027m doing here,"},{"Start":"09:57.230 ","End":"09:59.120","Text":"I can do things in any order."},{"Start":"09:59.120 ","End":"10:04.070","Text":"I can first of all, take the limit and then take the natural log of that,"},{"Start":"10:04.070 ","End":"10:07.640","Text":"or if I have something to the power of 1 over x,"},{"Start":"10:07.640 ","End":"10:11.300","Text":"I can also do that with the limit."},{"Start":"10:11.300 ","End":"10:14.690","Text":"I can take the limit of something to the power and then take it to the power of 1"},{"Start":"10:14.690 ","End":"10:18.620","Text":"over x or instead of just leaving the 1 over x inside."},{"Start":"10:18.620 ","End":"10:22.885","Text":"I just interchange the order of operations and it worked somehow with limits,"},{"Start":"10:22.885 ","End":"10:28.100","Text":"it alternates with natural log or exponent."},{"Start":"10:28.100 ","End":"10:29.705","Text":"This is equal to."},{"Start":"10:29.705 ","End":"10:33.635","Text":"Now, this one from this formula is e,"},{"Start":"10:33.635 ","End":"10:36.395","Text":"so we have the natural log."},{"Start":"10:36.395 ","End":"10:41.300","Text":"This whole limit, and I should have written t goes to 0, of course,"},{"Start":"10:41.300 ","End":"10:48.600","Text":"is equal to e"},{"Start":"10:48.650 ","End":"10:52.215","Text":"and it\u0027s e to the 1 over x."},{"Start":"10:52.215 ","End":"10:59.095","Text":"It\u0027s natural log of e to the 1 over x and this is equal to,"},{"Start":"10:59.095 ","End":"11:05.165","Text":"well, the natural log of anything to the natural log of e to anything."},{"Start":"11:05.165 ","End":"11:10.975","Text":"Say a it\u0027s just a because they\u0027re reverse operations,"},{"Start":"11:10.975 ","End":"11:13.880","Text":"e to the power of natural log or reverse of each other."},{"Start":"11:13.880 ","End":"11:15.515","Text":"So this is just a."},{"Start":"11:15.515 ","End":"11:18.680","Text":"This natural log and e as if they cancel out because they\u0027re"},{"Start":"11:18.680 ","End":"11:22.940","Text":"inverse functions and we\u0027re left with just 1 over x."},{"Start":"11:22.940 ","End":"11:24.830","Text":"We worked hard at that,"},{"Start":"11:24.830 ","End":"11:26.000","Text":"but we got a good 1,"},{"Start":"11:26.000 ","End":"11:31.950","Text":"because natural log is a very important function."},{"Start":"11:31.950 ","End":"11:36.785","Text":"We just shown by definition that the derivative of log x,"},{"Start":"11:36.785 ","End":"11:38.300","Text":"natural log x is 1 over x."},{"Start":"11:38.300 ","End":"11:40.760","Text":"So well done for us for that."},{"Start":"11:40.760 ","End":"11:43.680","Text":"That\u0027s done for number 6."}],"ID":10468},{"Watched":false,"Name":"Exercise 1 - Part 7","Duration":"8m 1s","ChapterTopicVideoID":10157,"CourseChapterTopicPlaylistID":8715,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.740","Text":"Moving on to number 7,"},{"Start":"00:01.740 ","End":"00:04.830","Text":"we just did number 6."},{"Start":"00:04.830 ","End":"00:07.725","Text":"I just scroll down and I even wrote the exercise,"},{"Start":"00:07.725 ","End":"00:15.610","Text":"and I also recorded again the formula for the definition of the derivative."},{"Start":"00:15.680 ","End":"00:18.570","Text":"I\u0027ll just mention again that some books,"},{"Start":"00:18.570 ","End":"00:21.855","Text":"some teachers use Delta x instead of"},{"Start":"00:21.855 ","End":"00:29.315","Text":"h. If we plug into this formula where our f of x this time is e^x,"},{"Start":"00:29.315 ","End":"00:39.020","Text":"then we get that y prime is f of x plus h,"},{"Start":"00:39.020 ","End":"00:41.000","Text":"which is e^x,"},{"Start":"00:41.000 ","End":"00:45.320","Text":"plus h minus e^x,"},{"Start":"00:45.320 ","End":"00:51.500","Text":"which is just f of x over h. Now,"},{"Start":"00:51.500 ","End":"00:57.570","Text":"let\u0027s just remember a little rule from algebra."},{"Start":"00:57.570 ","End":"00:59.060","Text":"I don\u0027t even know if you need it,"},{"Start":"00:59.060 ","End":"01:01.160","Text":"but I\u0027ll just write it at the side here,"},{"Start":"01:01.160 ","End":"01:03.365","Text":"perhaps in another color,"},{"Start":"01:03.365 ","End":"01:14.690","Text":"that a^b plus c"},{"Start":"01:14.690 ","End":"01:20.300","Text":"is equal to a^b times a^c."},{"Start":"01:20.300 ","End":"01:29.300","Text":"In our case, e^x plus h is e^x times e^h minus e^x,"},{"Start":"01:29.300 ","End":"01:32.370","Text":"all over h, which equals,"},{"Start":"01:32.370 ","End":"01:39.460","Text":"I\u0027ll take e^x outside the brackets and I get e^x times,"},{"Start":"01:39.460 ","End":"01:42.235","Text":"now I forgot to write the limits,"},{"Start":"01:42.235 ","End":"01:44.180","Text":"I have to do this again."},{"Start":"01:44.180 ","End":"01:52.180","Text":"Wait, here they are. Occasionally, I forget to write limit."},{"Start":"01:52.180 ","End":"01:55.840","Text":"It\u0027s taking e^x outside the brackets,"},{"Start":"01:55.840 ","End":"02:02.260","Text":"we get e^h minus 1 and all this over h,"},{"Start":"02:02.260 ","End":"02:05.270","Text":"I would like to put the h over here."},{"Start":"02:06.930 ","End":"02:12.295","Text":"Actually, since x doesn\u0027t involve h at all,"},{"Start":"02:12.295 ","End":"02:14.860","Text":"it\u0027s equal to e^x."},{"Start":"02:14.860 ","End":"02:21.740","Text":"I can pull it like a constant because it\u0027s a constant as far as h goes of the limit."},{"Start":"02:21.900 ","End":"02:24.970","Text":"As h goes to 0,"},{"Start":"02:24.970 ","End":"02:43.079","Text":"e^h minus 1 over h. I would like to work on this separately."},{"Start":"02:43.300 ","End":"02:47.240","Text":"This bit here that I put a green rectangle around,"},{"Start":"02:47.240 ","End":"02:50.660","Text":"if I can do this at the side and show that this is equal to 1,"},{"Start":"02:50.660 ","End":"02:54.510","Text":"then I\u0027ve got my answer that this will equal then e^x,"},{"Start":"02:54.510 ","End":"02:56.130","Text":"which we know it is."},{"Start":"02:56.130 ","End":"03:03.820","Text":"Let\u0027s see. Let\u0027s work on this at the side over here,"},{"Start":"03:04.520 ","End":"03:07.785","Text":"say, over here somewhere,"},{"Start":"03:07.785 ","End":"03:13.065","Text":"and we\u0027ll work on that and I\u0027ll use the same color, why not?"},{"Start":"03:13.065 ","End":"03:16.655","Text":"Before I continue,"},{"Start":"03:16.655 ","End":"03:26.715","Text":"I\u0027d like to remind you that we have a formula and the formula says that the limit as"},{"Start":"03:26.715 ","End":"03:35.970","Text":"t goes to 0 of 1 plus t to the power of 1/t"},{"Start":"03:35.970 ","End":"03:45.385","Text":"is equal to e. The same thing would be true if instead of t, I put h."},{"Start":"03:45.385 ","End":"03:49.020","Text":"So if I instead of t put h,"},{"Start":"03:49.020 ","End":"03:54.840","Text":"so let me just now do that, and then I\u0027ve just changed all the ts to hs."},{"Start":"03:54.840 ","End":"04:00.410","Text":"Now, if I notice it\u0027s still h goes to 0 just like we had here."},{"Start":"04:00.410 ","End":"04:06.800","Text":"What this means is that instead of the e, let\u0027s see,"},{"Start":"04:06.800 ","End":"04:13.940","Text":"we have an e here and we have an e here."},{"Start":"04:13.940 ","End":"04:17.780","Text":"If instead of e, I replace it with this limit,"},{"Start":"04:17.780 ","End":"04:20.015","Text":"I don\u0027t even have to put the limit."},{"Start":"04:20.015 ","End":"04:23.255","Text":"If I replace e with this expression,"},{"Start":"04:23.255 ","End":"04:26.644","Text":"because in any event it\u0027s under an h goes to 0,"},{"Start":"04:26.644 ","End":"04:29.050","Text":"then I should get the same answer."},{"Start":"04:29.050 ","End":"04:34.760","Text":"What I\u0027m saying is that I can"},{"Start":"04:34.760 ","End":"04:48.270","Text":"write this thing, which I was doing in green as,"},{"Start":"04:48.270 ","End":"04:54.690","Text":"okay, I really should have started here, but its limit."},{"Start":"04:55.660 ","End":"04:58.640","Text":"In other words, this thing, it\u0027s via this,"},{"Start":"04:58.640 ","End":"05:01.040","Text":"but really to here,"},{"Start":"05:01.040 ","End":"05:06.529","Text":"it\u0027s the limit as h goes to 0."},{"Start":"05:06.529 ","End":"05:11.195","Text":"Now instead of the e, I\u0027m going to put in 1 plus h to the 1/h,"},{"Start":"05:11.195 ","End":"05:19.980","Text":"so it\u0027s 1 plus h^1/h. That\u0027s this from here."},{"Start":"05:19.980 ","End":"05:22.980","Text":"But there\u0027s also another h there as you see,"},{"Start":"05:22.980 ","End":"05:25.160","Text":"so I\u0027ll put that in square brackets,"},{"Start":"05:25.160 ","End":"05:30.120","Text":"and put it also to the power of h minus 1"},{"Start":"05:30.120 ","End":"05:36.350","Text":"and all over h. Now,"},{"Start":"05:36.350 ","End":"05:38.165","Text":"so this thing is equal to this,"},{"Start":"05:38.165 ","End":"05:40.870","Text":"and then this is equal to,"},{"Start":"05:40.870 ","End":"05:46.180","Text":"look, if we have something to the 1/h times h, again,"},{"Start":"05:46.180 ","End":"05:48.425","Text":"because of the exponents,"},{"Start":"05:48.425 ","End":"06:02.510","Text":"what I have is something like a^b to the power of"},{"Start":"06:02.510 ","End":"06:09.425","Text":"c is equal to a^bc."},{"Start":"06:09.425 ","End":"06:13.430","Text":"If b is 1/h and c is h,"},{"Start":"06:13.430 ","End":"06:14.914","Text":"then these 2 together,"},{"Start":"06:14.914 ","End":"06:16.325","Text":"this is my b"},{"Start":"06:16.325 ","End":"06:17.660","Text":"and this is my c,"},{"Start":"06:17.660 ","End":"06:20.885","Text":"from that formula, then b times c is 1."},{"Start":"06:20.885 ","End":"06:31.000","Text":"So what I get is limit h goes to 0,"},{"Start":"06:31.000 ","End":"06:38.555","Text":"1 plus h is all I\u0027m left with to the power of 1 minus 1 over h,"},{"Start":"06:38.555 ","End":"06:41.765","Text":"which equals the limit."},{"Start":"06:41.765 ","End":"06:47.240","Text":"This is just 1 minus 1 cancels h/h, this is just the limit."},{"Start":"06:47.240 ","End":"06:49.050","Text":"We can just substitute,"},{"Start":"06:49.050 ","End":"06:50.655","Text":"in fact, we don\u0027t even need the limit,"},{"Start":"06:50.655 ","End":"06:55.520","Text":"so I\u0027m just going to cross that out because when we can substitute,"},{"Start":"06:55.520 ","End":"07:00.000","Text":"we don\u0027t need second back-to-back pen."},{"Start":"07:00.940 ","End":"07:04.440","Text":"Yeah, I can just substitute."},{"Start":"07:07.370 ","End":"07:10.186","Text":"Well, not exactly. No."},{"Start":"07:10.186 ","End":"07:12.410","Text":"Perhaps, after all,"},{"Start":"07:12.410 ","End":"07:17.325","Text":"I do need to show that this, 1 with 1,"},{"Start":"07:17.325 ","End":"07:22.650","Text":"and 1 with 1 cancels h/h is 1,"},{"Start":"07:22.650 ","End":"07:28.935","Text":"so it\u0027s the limit as h goes to 0 of 1,"},{"Start":"07:28.935 ","End":"07:33.200","Text":"and the limit of a constant is just the constant which is 1."},{"Start":"07:33.200 ","End":"07:41.060","Text":"This 1 now plugs back into the [inaudible] of this whole green rectangle,"},{"Start":"07:41.060 ","End":"07:46.760","Text":"and so what we get is that this is equal to e^x times 1,"},{"Start":"07:46.760 ","End":"07:49.165","Text":"which is just e^x,"},{"Start":"07:49.165 ","End":"07:51.950","Text":"and that is what we\u0027ve always known,"},{"Start":"07:51.950 ","End":"07:54.290","Text":"that derivative of e^x is e^x,"},{"Start":"07:54.290 ","End":"07:56.915","Text":"only this time we\u0027ve done it from basics."},{"Start":"07:56.915 ","End":"08:00.420","Text":"That ends number 7."}],"ID":10469},{"Watched":false,"Name":"Exercise 1 - Part 8","Duration":"5m 14s","ChapterTopicVideoID":10158,"CourseChapterTopicPlaylistID":8715,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"That takes care of number 7."},{"Start":"00:02.520 ","End":"00:09.390","Text":"So let\u0027s go on to number 8 which I\u0027ve already pre-written."},{"Start":"00:09.390 ","End":"00:12.540","Text":"That is that y equals sine 2x."},{"Start":"00:12.540 ","End":"00:16.050","Text":"I also want to remind you in general of the definition of"},{"Start":"00:16.050 ","End":"00:17.400","Text":"the derivative of a function"},{"Start":"00:17.400 ","End":"00:27.870","Text":"f is f of x plus h minus f of x all over h."},{"Start":"00:27.870 ","End":"00:34.440","Text":"The limit of that,"},{"Start":"00:34.440 ","End":"00:36.465","Text":"hang on a second,"},{"Start":"00:36.465 ","End":"00:38.880","Text":"something went wrong there."},{"Start":"00:38.880 ","End":"00:42.945","Text":"Third time, lucky. Here we are."},{"Start":"00:42.945 ","End":"00:52.845","Text":"Limit as h goes to 0 of this thing. Fair enough."},{"Start":"00:52.845 ","End":"00:57.050","Text":"In our case, what we have is that y prime"},{"Start":"00:57.050 ","End":"01:07.470","Text":"is going to equal the limit as h goes to 0."},{"Start":"01:07.470 ","End":"01:09.645","Text":"Am I missing a bracket here?"},{"Start":"01:09.645 ","End":"01:16.954","Text":"Yes. If I put x plus h, then I get sine of,"},{"Start":"01:16.954 ","End":"01:20.820","Text":"let\u0027s already multiply out by 2,"},{"Start":"01:20.820 ","End":"01:23.130","Text":"2x plus 2h,"},{"Start":"01:23.130 ","End":"01:31.185","Text":"that\u0027s twice x plus h minus f of x, which is sine of 2x,"},{"Start":"01:31.185 ","End":"01:39.980","Text":"and all this over h. We\u0027ll need a bit of trigonometry here."},{"Start":"01:39.980 ","End":"01:43.045","Text":"I\u0027m going to write down a trigonometric formula"},{"Start":"01:43.045 ","End":"01:46.820","Text":"because what we have is sine of something minus sine of something else."},{"Start":"01:46.820 ","End":"01:49.640","Text":"Luckily, there is a formula that will help us."},{"Start":"01:49.640 ","End":"01:58.290","Text":"That is that sine of Alpha minus sine of Beta is"},{"Start":"01:58.290 ","End":"02:08.600","Text":"twice the sine of Alpha minus Beta over 2"},{"Start":"02:08.600 ","End":"02:16.220","Text":"times the cosine of Alpha plus Beta over 2."},{"Start":"02:16.220 ","End":"02:21.170","Text":"What I\u0027m going to do is this 2x plus 2h will be the Alpha,"},{"Start":"02:21.170 ","End":"02:23.360","Text":"2x will be the Beta,"},{"Start":"02:23.360 ","End":"02:26.090","Text":"and I\u0027ll just use this formula here."},{"Start":"02:26.090 ","End":"02:33.395","Text":"What we will get will be that this thing is equal to limit,"},{"Start":"02:33.395 ","End":"02:37.115","Text":"again, as h is still going to 0."},{"Start":"02:37.115 ","End":"02:40.340","Text":"Let\u0027s see, Alpha minus Beta over 2,"},{"Start":"02:40.340 ","End":"02:42.880","Text":"this minus this over 2,"},{"Start":"02:42.880 ","End":"02:45.800","Text":"you can see that that comes out to be just h,"},{"Start":"02:45.800 ","End":"02:47.420","Text":"because 2x minus 2x cancels,"},{"Start":"02:47.420 ","End":"02:54.360","Text":"2h over 2 is h. It\u0027s sine of h."},{"Start":"02:54.360 ","End":"02:57.620","Text":"Here, Alpha plus Beta over 2,"},{"Start":"02:57.620 ","End":"03:01.650","Text":"so it\u0027s 4x plus 2h all over 2."},{"Start":"03:03.520 ","End":"03:10.430","Text":"4x plus 2h over 2 is 2x plus h."},{"Start":"03:10.430 ","End":"03:20.400","Text":"So it\u0027s cosine of 2x plus h."},{"Start":"03:20.400 ","End":"03:22.845","Text":"I have forgotten the 2 there."},{"Start":"03:22.845 ","End":"03:24.900","Text":"There, we have room for it."},{"Start":"03:24.900 ","End":"03:30.660","Text":"All this over h."},{"Start":"03:30.660 ","End":"03:38.570","Text":"Now, basically what we have here is if I just circle, let\u0027s say,"},{"Start":"03:38.570 ","End":"03:41.860","Text":"this sine h over h,"},{"Start":"03:41.860 ","End":"03:46.025","Text":"this is very familiar to us because the limit is,"},{"Start":"03:46.025 ","End":"03:48.335","Text":"in fact, I\u0027ll write it down,"},{"Start":"03:48.335 ","End":"04:00.615","Text":"the limit as h goes to 0 of sine h over h is equal to 1."},{"Start":"04:00.615 ","End":"04:04.865","Text":"This will come in very useful because all the rest of it,"},{"Start":"04:04.865 ","End":"04:08.375","Text":"there\u0027s no problem in just substituting h equals 0."},{"Start":"04:08.375 ","End":"04:15.840","Text":"When h is 0, we just get twice cosine 2x,"},{"Start":"04:15.840 ","End":"04:17.850","Text":"because this thing goes to 1,"},{"Start":"04:17.850 ","End":"04:20.215","Text":"we can just put that as 1."},{"Start":"04:20.215 ","End":"04:41.310","Text":"This thing here is the 1 from here times 2 cosine 2x."},{"Start":"04:41.310 ","End":"04:43.710","Text":"If we just leave the 1 off,"},{"Start":"04:43.710 ","End":"04:51.315","Text":"then we just get 2 cosine 2x which we know is,"},{"Start":"04:51.315 ","End":"04:53.460","Text":"in any event, by other means,"},{"Start":"04:53.460 ","End":"04:54.725","Text":"so this is the right answer."},{"Start":"04:54.725 ","End":"04:56.615","Text":"If we use the chain rule,"},{"Start":"04:56.615 ","End":"04:59.280","Text":"the derivative of sine is cosine,"},{"Start":"04:59.280 ","End":"05:01.370","Text":"but because it\u0027s 2x and not x,"},{"Start":"05:01.370 ","End":"05:04.895","Text":"we have to multiply by the internal derivative which is 2,"},{"Start":"05:04.895 ","End":"05:06.995","Text":"and that\u0027s the right answer."},{"Start":"05:06.995 ","End":"05:09.120","Text":"We\u0027re done with number 8,"},{"Start":"05:09.120 ","End":"05:11.300","Text":"but that\u0027s the last in the series, so altogether,"},{"Start":"05:11.300 ","End":"05:14.160","Text":"we\u0027re done with this whole exercise."}],"ID":10470}],"Thumbnail":null,"ID":8715},{"Name":"The Derivative of an Inverse of a Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Derivative of an Inverse of a Function","Duration":"3m 48s","ChapterTopicVideoID":10149,"CourseChapterTopicPlaylistID":8716,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.820","Text":"In this clip, we\u0027ll talk about differentiating the inverse of a function."},{"Start":"00:05.820 ","End":"00:08.670","Text":"We have a function, it might have an inverse function,"},{"Start":"00:08.670 ","End":"00:13.170","Text":"and that would have a derivative possibly and we\u0027re going to show how to find it."},{"Start":"00:13.170 ","End":"00:22.980","Text":"For example, x as the y cubed and the inverse would be y equals cube root of x."},{"Start":"00:22.980 ","End":"00:25.635","Text":"If I call this f of y,"},{"Start":"00:25.635 ","End":"00:31.815","Text":"then the cube root of x is the inverse function of f is f minus 1 of x."},{"Start":"00:31.815 ","End":"00:37.320","Text":"The rule is that the derivative of the inverse function,"},{"Start":"00:37.320 ","End":"00:44.945","Text":"if I take f minus 1 and then I take the derivative of that at some point x."},{"Start":"00:44.945 ","End":"00:49.490","Text":"This is going to be 1 over the derivative of the regular function,"},{"Start":"00:49.490 ","End":"00:50.690","Text":"but not at the point x,"},{"Start":"00:50.690 ","End":"00:55.085","Text":"at the point y where x and y are pairs."},{"Start":"00:55.085 ","End":"00:56.660","Text":"When I say that they are pairs,"},{"Start":"00:56.660 ","End":"01:00.170","Text":"I mean that we get from x to y by applying f,"},{"Start":"01:00.170 ","End":"01:04.733","Text":"and we get from y to x by applying the inverse of f."},{"Start":"01:04.733 ","End":"01:09.380","Text":"Another slightly shorter way of writing this is not using the letter f,"},{"Start":"01:09.380 ","End":"01:15.410","Text":"just saying that suppose x is some unnamed function of y,"},{"Start":"01:15.410 ","End":"01:22.580","Text":"and suppose the inverse function is that y is the inverse function depending on x,"},{"Start":"01:22.580 ","End":"01:25.415","Text":"then what we want is the derivative of this,"},{"Start":"01:25.415 ","End":"01:31.610","Text":"and it\u0027s given by the reciprocal 1 over the derivative of the regular function."},{"Start":"01:31.610 ","End":"01:34.940","Text":"This helps assuming we know the derivative of the regular function,"},{"Start":"01:34.940 ","End":"01:37.085","Text":"we want to know the derivative of the inverse."},{"Start":"01:37.085 ","End":"01:38.510","Text":"I\u0027ll give an example in a moment,"},{"Start":"01:38.510 ","End":"01:40.385","Text":"let me just give a third notation."},{"Start":"01:40.385 ","End":"01:41.630","Text":"I won\u0027t be using it much,"},{"Start":"01:41.630 ","End":"01:42.980","Text":"but you might see it."},{"Start":"01:42.980 ","End":"01:45.860","Text":"What I can say is the derivative of y with respect to x,"},{"Start":"01:45.860 ","End":"01:50.450","Text":"I can call that dy by dx using the other notation,"},{"Start":"01:50.450 ","End":"01:51.935","Text":"the Leibniz\u0027s notation,"},{"Start":"01:51.935 ","End":"01:55.780","Text":"and this is derivative of x with respect to y is dx/dy."},{"Start":"01:55.780 ","End":"02:00.370","Text":"This is equal to 1 over dx/dy,"},{"Start":"02:00.770 ","End":"02:03.890","Text":"and it behaves as if it was a fraction because"},{"Start":"02:03.890 ","End":"02:06.575","Text":"1 over a fraction is the reciprocal fraction."},{"Start":"02:06.575 ","End":"02:08.900","Text":"That\u0027s another form, but I won\u0027t be using it."},{"Start":"02:08.900 ","End":"02:11.240","Text":"What I\u0027ll do is give an example of this."},{"Start":"02:11.240 ","End":"02:13.840","Text":"An example question might look like,"},{"Start":"02:13.840 ","End":"02:19.940","Text":"I\u0027ll ask you to show that the derivative of the cube root of x equals,"},{"Start":"02:19.940 ","End":"02:21.515","Text":"this is what we have to show,"},{"Start":"02:21.515 ","End":"02:30.020","Text":"that it\u0027s going to equal 1/3 times the cube root of x squared."},{"Start":"02:30.020 ","End":"02:32.750","Text":"I\u0027m just going to say in which show using the rule for"},{"Start":"02:32.750 ","End":"02:37.250","Text":"the derivative of the inverse function,"},{"Start":"02:37.250 ","End":"02:39.780","Text":"not to differentiate it, just regular."},{"Start":"02:39.780 ","End":"02:40.880","Text":"We could write this as,"},{"Start":"02:40.880 ","End":"02:44.125","Text":"let\u0027s say y equals the cube root of x,"},{"Start":"02:44.125 ","End":"02:49.039","Text":"and the cube root is obviously the inverse of the cube function."},{"Start":"02:49.039 ","End":"02:54.530","Text":"This is true, if and only if x equals y cubed."},{"Start":"02:54.530 ","End":"02:57.935","Text":"Now, the cube function is easy to differentiate."},{"Start":"02:57.935 ","End":"03:03.115","Text":"What we\u0027re going to say is that if I want to find y prime,"},{"Start":"03:03.115 ","End":"03:05.675","Text":"where y is a function of x,"},{"Start":"03:05.675 ","End":"03:07.910","Text":"I\u0027ll say it\u0027s 1/x prime,"},{"Start":"03:07.910 ","End":"03:11.299","Text":"where x is a function of y, because that\u0027s easier."},{"Start":"03:11.299 ","End":"03:16.040","Text":"Now x prime of y is going to be just 3y squared."},{"Start":"03:16.040 ","End":"03:18.680","Text":"But that\u0027s still not what we were asked to prove,"},{"Start":"03:18.680 ","End":"03:23.810","Text":"but y is equal to the cube root of x."},{"Start":"03:23.810 ","End":"03:30.140","Text":"This is equal to 1/3 times the cube root of x squared,"},{"Start":"03:30.140 ","End":"03:34.050","Text":"you can just put the 2 inside the radical."},{"Start":"03:34.050 ","End":"03:36.020","Text":"It doesn\u0027t matter if you take cube root and then square"},{"Start":"03:36.020 ","End":"03:38.615","Text":"root or square it and then take the cube root."},{"Start":"03:38.615 ","End":"03:42.515","Text":"In any case, it\u0027s x^2/3 in the denominator."},{"Start":"03:42.515 ","End":"03:45.380","Text":"That\u0027s a typical exercise and that\u0027s all I want to say."},{"Start":"03:45.380 ","End":"03:49.470","Text":"There are more exercises at the end. We\u0027re done."}],"ID":10459},{"Watched":false,"Name":"Exercise 1","Duration":"1m 35s","ChapterTopicVideoID":10150,"CourseChapterTopicPlaylistID":8716,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.920","Text":"In this exercise, we have to show that"},{"Start":"00:01.920 ","End":"00:05.625","Text":"the derivative of the natural log of x is 1 over x."},{"Start":"00:05.625 ","End":"00:09.630","Text":"But using the rule for the derivative of the inverse function,"},{"Start":"00:09.630 ","End":"00:11.205","Text":"which I\u0027ll remind you,"},{"Start":"00:11.205 ","End":"00:13.795","Text":"is this, at least that\u0027s one of the forms of writing it."},{"Start":"00:13.795 ","End":"00:15.230","Text":"Now in our case,"},{"Start":"00:15.230 ","End":"00:19.370","Text":"our function is natural log of x and we need its inverse."},{"Start":"00:19.370 ","End":"00:22.610","Text":"Now I claim that the inverse of y equals"},{"Start":"00:22.610 ","End":"00:26.720","Text":"natural log of x is that x is e to the y. I mean,"},{"Start":"00:26.720 ","End":"00:30.430","Text":"in general, if a is the natural log of b,"},{"Start":"00:30.430 ","End":"00:35.630","Text":"then that\u0027s the same as saying that b is e to the power of a."},{"Start":"00:35.630 ","End":"00:38.315","Text":"Just from the definition of the natural logarithm,"},{"Start":"00:38.315 ","End":"00:40.385","Text":"the exponent is the inverse."},{"Start":"00:40.385 ","End":"00:42.545","Text":"Plugging this into this formula,"},{"Start":"00:42.545 ","End":"00:45.950","Text":"y prime is the derivative of natural log of x."},{"Start":"00:45.950 ","End":"00:50.060","Text":"I\u0027m putting the x here for emphasis that we\u0027re differentiating with respect to x."},{"Start":"00:50.060 ","End":"00:53.480","Text":"The right-hand side, I need to differentiate x with respect to"},{"Start":"00:53.480 ","End":"00:58.295","Text":"y. X is e to the y. I need the derivative of e to the y,"},{"Start":"00:58.295 ","End":"00:59.990","Text":"but as a function of y."},{"Start":"00:59.990 ","End":"01:02.425","Text":"Now what is the derivative of e to the y?"},{"Start":"01:02.425 ","End":"01:04.020","Text":"One of those basic derivatives."},{"Start":"01:04.020 ","End":"01:05.720","Text":"The derivative is the function itself."},{"Start":"01:05.720 ","End":"01:08.990","Text":"Instead of e to the y, derivative of e to the y,"},{"Start":"01:08.990 ","End":"01:11.990","Text":"but y is natural log of x."},{"Start":"01:11.990 ","End":"01:18.110","Text":"We can simplify the expression because e to the natural log of x is just x itself."},{"Start":"01:18.110 ","End":"01:21.560","Text":"It follows from here that e to the power"},{"Start":"01:21.560 ","End":"01:25.460","Text":"of a is e to the power of natural log b is just b."},{"Start":"01:25.460 ","End":"01:27.530","Text":"We just get 1 over x,"},{"Start":"01:27.530 ","End":"01:31.310","Text":"which is what we knew any way without using the inverse function rule."},{"Start":"01:31.310 ","End":"01:33.350","Text":"That\u0027s the derivative of natural log of x."},{"Start":"01:33.350 ","End":"01:36.130","Text":"We\u0027re done."}],"ID":10460},{"Watched":false,"Name":"Exercise 2","Duration":"1m 14s","ChapterTopicVideoID":10151,"CourseChapterTopicPlaylistID":8716,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"In this exercise we have to prove that the derivative of"},{"Start":"00:03.180 ","End":"00:07.110","Text":"the square root of x is 1 over twice the square root of x."},{"Start":"00:07.110 ","End":"00:11.160","Text":"We have to prove it using the rule for the derivative of an inverse function."},{"Start":"00:11.160 ","End":"00:13.200","Text":"I mean, this is an immediate derivative,"},{"Start":"00:13.200 ","End":"00:16.320","Text":"but we want to use the inverse function concept."},{"Start":"00:16.320 ","End":"00:18.810","Text":"Recall that at least in 1 form,"},{"Start":"00:18.810 ","End":"00:24.140","Text":"this is the derivative of an inverse function where y is the inverse function of x,"},{"Start":"00:24.140 ","End":"00:25.470","Text":"and y is a function of x,"},{"Start":"00:25.470 ","End":"00:28.020","Text":"x is a function of y, and they\u0027re inverses of each other."},{"Start":"00:28.020 ","End":"00:29.955","Text":"Let\u0027s apply that here,"},{"Start":"00:29.955 ","End":"00:35.730","Text":"where we have that y is the square root of x. Let\u0027s call this y."},{"Start":"00:35.730 ","End":"00:37.860","Text":"If y is the square root of x,"},{"Start":"00:37.860 ","End":"00:40.435","Text":"what can I say about x in terms of y?"},{"Start":"00:40.435 ","End":"00:43.405","Text":"Clearly, x is y squared."},{"Start":"00:43.405 ","End":"00:45.405","Text":"Now I apply this,"},{"Start":"00:45.405 ","End":"00:49.280","Text":"and I get that the derivative of the square root of x, that\u0027s what I\u0027m looking for,"},{"Start":"00:49.280 ","End":"00:50.810","Text":"derivative with respect to x,"},{"Start":"00:50.810 ","End":"00:53.815","Text":"is 1 over this which I can compute,"},{"Start":"00:53.815 ","End":"00:56.855","Text":"the derivative of y squared with respect to y."},{"Start":"00:56.855 ","End":"01:00.665","Text":"Now the derivative of y squared is a very basic thing."},{"Start":"01:00.665 ","End":"01:02.360","Text":"It\u0027s just 2y."},{"Start":"01:02.360 ","End":"01:05.275","Text":"The thing is we have to get back to x,"},{"Start":"01:05.275 ","End":"01:08.590","Text":"but remember that y is the square root of x."},{"Start":"01:08.590 ","End":"01:09.965","Text":"That\u0027s this function."},{"Start":"01:09.965 ","End":"01:15.420","Text":"This is what we get when we replace y by square root of x. We are done."}],"ID":10461},{"Watched":false,"Name":"Exercise 3","Duration":"1m 40s","ChapterTopicVideoID":10152,"CourseChapterTopicPlaylistID":8716,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"In this exercise, we want to prove that the derivative of"},{"Start":"00:02.760 ","End":"00:05.765","Text":"arctangent x is 1 over x squared plus 1,"},{"Start":"00:05.765 ","End":"00:09.615","Text":"but we want to do it using the rule for the derivative of an inverse function."},{"Start":"00:09.615 ","End":"00:12.585","Text":"Here\u0027s one form of it that we\u0027re going to use it or other forms."},{"Start":"00:12.585 ","End":"00:16.155","Text":"Next we have to find what is the inverse of arctangent x."},{"Start":"00:16.155 ","End":"00:18.360","Text":"In other words, if y is arctangent of x,"},{"Start":"00:18.360 ","End":"00:20.140","Text":"what is x in terms of y?"},{"Start":"00:20.140 ","End":"00:24.830","Text":"I think it\u0027s fairly clear even from the name that if y is the arctangent of x,"},{"Start":"00:24.830 ","End":"00:27.920","Text":"it just means that x is the tangent of y."},{"Start":"00:27.920 ","End":"00:30.230","Text":"We have y of x and x of y,"},{"Start":"00:30.230 ","End":"00:32.495","Text":"now let\u0027s plug into the formula."},{"Start":"00:32.495 ","End":"00:34.880","Text":"The left-hand side is what we\u0027re looking for."},{"Start":"00:34.880 ","End":"00:38.539","Text":"Right-hand side, we need the derivative of tangent of y."},{"Start":"00:38.539 ","End":"00:42.395","Text":"It\u0027s one of those basic derivatives that you just have to remember."},{"Start":"00:42.395 ","End":"00:46.565","Text":"The derivative of tangent is 1 over cosine squared."},{"Start":"00:46.565 ","End":"00:49.670","Text":"The thing is we want to get back to x somehow."},{"Start":"00:49.670 ","End":"00:54.200","Text":"We\u0027re going to use some trigonometry and I\u0027m going to show you that we end up with this."},{"Start":"00:54.200 ","End":"00:57.140","Text":"I put an asterisk here means I have to verify this,"},{"Start":"00:57.140 ","End":"01:00.050","Text":"so I\u0027m just giving you the answer that 1 over cosine squared"},{"Start":"01:00.050 ","End":"01:03.420","Text":"y is x squared plus 1. Let me show you how."},{"Start":"01:03.420 ","End":"01:07.350","Text":"We\u0027re just using some trigonometry here. Here it is."},{"Start":"01:07.350 ","End":"01:09.210","Text":"If x is tangent of y,"},{"Start":"01:09.210 ","End":"01:10.370","Text":"square both sides,"},{"Start":"01:10.370 ","End":"01:12.274","Text":"x squared is tangent squared."},{"Start":"01:12.274 ","End":"01:15.410","Text":"We want to build up to the x squared plus 1 to this expression,"},{"Start":"01:15.410 ","End":"01:18.245","Text":"so x squared is tangent squared y."},{"Start":"01:18.245 ","End":"01:21.680","Text":"Then we add 1, x squared plus 1 is this,"},{"Start":"01:21.680 ","End":"01:28.970","Text":"and then we can get that x squared plus 1 is 1 over the cosine squared of y and"},{"Start":"01:28.970 ","End":"01:36.755","Text":"so now that just shows you that we could replace this by x squared plus 1 as here."},{"Start":"01:36.755 ","End":"01:40.950","Text":"I\u0027ve showed you that this is true and we\u0027re done."}],"ID":10462},{"Watched":false,"Name":"Exercise 4","Duration":"1m 40s","ChapterTopicVideoID":10153,"CourseChapterTopicPlaylistID":8716,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.800","Text":"In this exercise, you want to prove that the derivative of arcsine of x,"},{"Start":"00:04.800 ","End":"00:11.400","Text":"let\u0027s say this is y equals arcsine of x is 1 over the square root of 1 minus x squared."},{"Start":"00:11.400 ","End":"00:15.840","Text":"But we have to do it using the rule for the derivative of an inverse function."},{"Start":"00:15.840 ","End":"00:19.290","Text":"The 1 form of this rule is as follows."},{"Start":"00:19.290 ","End":"00:23.040","Text":"But for this, we need to put x in terms of y also."},{"Start":"00:23.040 ","End":"00:28.570","Text":"We know y is in terms of x but I think it\u0027s fairly clear that if y is arcsine of x,"},{"Start":"00:28.570 ","End":"00:30.155","Text":"the next is sine y."},{"Start":"00:30.155 ","End":"00:33.740","Text":"Now we just plug into this formula and we get that the derivative,"},{"Start":"00:33.740 ","End":"00:34.925","Text":"this is what we\u0027re looking for,"},{"Start":"00:34.925 ","End":"00:38.015","Text":"is 1 over the derivative of sine y,"},{"Start":"00:38.015 ","End":"00:39.755","Text":"derivative with respect to y."},{"Start":"00:39.755 ","End":"00:44.880","Text":"Now, this is an easy 1 because we know that the derivative of sine is cosine."},{"Start":"00:44.880 ","End":"00:46.805","Text":"The derivative is 1 over cosine,"},{"Start":"00:46.805 ","End":"00:48.650","Text":"but we need to get back to x."},{"Start":"00:48.650 ","End":"00:50.870","Text":"How do we get from cosine y to x?"},{"Start":"00:50.870 ","End":"00:52.490","Text":"If we had sine y,"},{"Start":"00:52.490 ","End":"00:54.965","Text":"that would just be extra. What is cosine y?"},{"Start":"00:54.965 ","End":"00:57.530","Text":"Well, we\u0027ll do a bit of trigonometry and remember"},{"Start":"00:57.530 ","End":"01:00.170","Text":"1 of the fundamental trigonometric identities."},{"Start":"01:00.170 ","End":"01:02.240","Text":"You know what? Let me just give you the answer first."},{"Start":"01:02.240 ","End":"01:05.465","Text":"I claim that if x is sine y,"},{"Start":"01:05.465 ","End":"01:09.060","Text":"then square root of 1 minus x squared is cosine y."},{"Start":"01:09.060 ","End":"01:12.605","Text":"Let me demonstrate this. x is sine y."},{"Start":"01:12.605 ","End":"01:15.680","Text":"Raise both sides to the power of 2 and I get"},{"Start":"01:15.680 ","End":"01:18.875","Text":"x squared is sine squared y then subtract from 1."},{"Start":"01:18.875 ","End":"01:21.230","Text":"You see I want to get to 1 minus sine squared,"},{"Start":"01:21.230 ","End":"01:23.195","Text":"and use the famous identity."},{"Start":"01:23.195 ","End":"01:24.710","Text":"We get this,"},{"Start":"01:24.710 ","End":"01:27.665","Text":"1 minus sine squared is cosine squared."},{"Start":"01:27.665 ","End":"01:32.210","Text":"Now I can get to the cosine by just taking the square root of both sides."},{"Start":"01:32.210 ","End":"01:35.840","Text":"That shows the passage from the denominator here,"},{"Start":"01:35.840 ","End":"01:41.610","Text":"cosine y to square root of 1 minus x squared, and we\u0027re done."}],"ID":10463},{"Watched":false,"Name":"Exercise 5","Duration":"1m 38s","ChapterTopicVideoID":10154,"CourseChapterTopicPlaylistID":8716,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we\u0027re given a function f. I mean,"},{"Start":"00:03.630 ","End":"00:04.770","Text":"we\u0027re told that f is a function,"},{"Start":"00:04.770 ","End":"00:06.360","Text":"we\u0027re not given what it is,"},{"Start":"00:06.360 ","End":"00:08.010","Text":"and that this is its inverse."},{"Start":"00:08.010 ","End":"00:10.425","Text":"That\u0027s how we write the inverse to the minus 1."},{"Start":"00:10.425 ","End":"00:17.810","Text":"We\u0027re also told 2 other things that f of 2 is 7 and the derivative of f at 2 is root 7."},{"Start":"00:17.810 ","End":"00:23.285","Text":"Our task is to find the derivative of the inverse function at 7."},{"Start":"00:23.285 ","End":"00:27.814","Text":"The derivative of the inverse function is by our rule,"},{"Start":"00:27.814 ","End":"00:30.035","Text":"the reciprocal of the derivative of the function."},{"Start":"00:30.035 ","End":"00:31.880","Text":"Reciprocal means 1 over ,"},{"Start":"00:31.880 ","End":"00:33.950","Text":"and this is the derivative of the function,"},{"Start":"00:33.950 ","End":"00:36.635","Text":"but they have to be taken at the appropriate points."},{"Start":"00:36.635 ","End":"00:39.110","Text":"The inverse is applied at 7,"},{"Start":"00:39.110 ","End":"00:43.130","Text":"but the function itself is applied at 2 because 2 corresponds to 7."},{"Start":"00:43.130 ","End":"00:45.080","Text":"The answer is 1 over root 7."},{"Start":"00:45.080 ","End":"00:48.185","Text":"But let me go into a bit more detail because this was a bit quick."},{"Start":"00:48.185 ","End":"00:58.220","Text":"The usual version of the rule is that y prime of x is 1 over x prime of y."},{"Start":"00:58.220 ","End":"01:04.250","Text":"Let\u0027s suppose in our case that we\u0027re given that x is some function f of y,"},{"Start":"01:04.250 ","End":"01:08.750","Text":"which means that y is f inverse of x."},{"Start":"01:08.750 ","End":"01:17.660","Text":"What this becomes is the inverse of f prime at x is 1 over f prime of y."},{"Start":"01:17.660 ","End":"01:20.190","Text":"Now, this could be at any points,"},{"Start":"01:20.190 ","End":"01:22.965","Text":"let\u0027s say x naught, y naught."},{"Start":"01:22.965 ","End":"01:25.290","Text":"If y naught is 2,"},{"Start":"01:25.290 ","End":"01:29.340","Text":"then x naught is f of 2, which is 7."},{"Start":"01:29.340 ","End":"01:31.085","Text":"If we apply that here,"},{"Start":"01:31.085 ","End":"01:33.320","Text":"basically we just get what we have here."},{"Start":"01:33.320 ","End":"01:35.915","Text":"This just explains it a bit further."},{"Start":"01:35.915 ","End":"01:38.490","Text":"We\u0027re done. This is the answer."}],"ID":10464}],"Thumbnail":null,"ID":8716},{"Name":"Logarithmic Differentiation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Logarithmic Differentiation","Duration":"7m 13s","ChapterTopicVideoID":8875,"CourseChapterTopicPlaylistID":5119,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"In this clip, we\u0027ll learn yet another technique for"},{"Start":"00:03.090 ","End":"00:07.170","Text":"differentiation called logarithmic differentiation."},{"Start":"00:07.170 ","End":"00:11.370","Text":"I\u0027ll show you an example of when you might want to use this."},{"Start":"00:11.370 ","End":"00:16.455","Text":"Suppose y is this expression and we want to find y prime."},{"Start":"00:16.455 ","End":"00:24.000","Text":"Now, what characterizes this function of x is that it has a lot of products and quotients"},{"Start":"00:24.000 ","End":"00:27.810","Text":"and exponents and this is exactly the kind of condition"},{"Start":"00:27.810 ","End":"00:31.500","Text":"where we would use logarithmic differentiation."},{"Start":"00:31.500 ","End":"00:36.960","Text":"Sure, you could try and just do it regularly with the rules that you have,"},{"Start":"00:36.960 ","End":"00:39.225","Text":"chain rule, product rule,"},{"Start":"00:39.225 ","End":"00:42.145","Text":"quotient rule, but it would be quite a mess."},{"Start":"00:42.145 ","End":"00:45.130","Text":"This is where logarithmic differentiation comes in handy."},{"Start":"00:45.130 ","End":"00:46.985","Text":"But before I show you what it is,"},{"Start":"00:46.985 ","End":"00:49.880","Text":"I want to just review some logarithm rules."},{"Start":"00:49.880 ","End":"00:52.805","Text":"We\u0027re going to use base e, natural logarithms."},{"Start":"00:52.805 ","End":"00:57.650","Text":"The first rule says that natural log of a"},{"Start":"00:57.650 ","End":"01:07.805","Text":"plus natural log of b is equal to the natural log of a times b."},{"Start":"01:07.805 ","End":"01:11.240","Text":"The second rule is when we have a minus here,"},{"Start":"01:11.240 ","End":"01:17.960","Text":"natural log of a minus natural log of b equals natural log."},{"Start":"01:17.960 ","End":"01:19.910","Text":"The minus becomes a quotient,"},{"Start":"01:19.910 ","End":"01:22.730","Text":"so we have a/b."},{"Start":"01:22.730 ","End":"01:31.760","Text":"The third rule is if I have a number times the natural log of a,"},{"Start":"01:31.760 ","End":"01:41.630","Text":"then this is equal to the natural log of a to the power of n. For consistency,"},{"Start":"01:41.630 ","End":"01:44.050","Text":"I\u0027ll change the n to a b."},{"Start":"01:44.050 ","End":"01:49.860","Text":"These are important. I think I\u0027ll put them in a box, 3 rules, well,"},{"Start":"01:49.860 ","End":"01:52.010","Text":"actually it\u0027s 6 rules because each of them,"},{"Start":"01:52.010 ","End":"01:57.350","Text":"we sometimes use the equality from left to right and sometimes from right to left."},{"Start":"01:57.350 ","End":"02:01.190","Text":"This last 1, you usually remember that when you have a log of an exponent,"},{"Start":"02:01.190 ","End":"02:07.355","Text":"that the exponent comes out in front and the other way it goes into the exponent."},{"Start":"02:07.355 ","End":"02:10.820","Text":"Anyway, now we come to the logarithmic differentiation,"},{"Start":"02:10.820 ","End":"02:13.660","Text":"which means that we first of all take the logarithm,"},{"Start":"02:13.660 ","End":"02:17.185","Text":"then we tidy up a bit and then differentiate."},{"Start":"02:17.185 ","End":"02:20.780","Text":"Here it goes, natural log of"},{"Start":"02:20.780 ","End":"02:27.240","Text":"the left-hand side equals natural log of this whole thing here."},{"Start":"02:27.240 ","End":"02:29.830","Text":"I just did a copy-paste of course."},{"Start":"02:30.050 ","End":"02:38.314","Text":"This, I can breakup with the help of these rules into lots of little logarithm pieces."},{"Start":"02:38.314 ","End":"02:39.850","Text":"Well, I\u0027ll show you."},{"Start":"02:39.850 ","End":"02:42.530","Text":"I\u0027m going to be using a combination of this and"},{"Start":"02:42.530 ","End":"02:46.615","Text":"this and I\u0027m going to be going from right to left."},{"Start":"02:46.615 ","End":"02:54.170","Text":"What we get is here we have a product and these 2 on the bottom are quotients,"},{"Start":"02:54.170 ","End":"02:58.520","Text":"so we take these in plus and these in minus, I\u0027ll show you."},{"Start":"02:58.520 ","End":"03:00.890","Text":"We get natural log of the first 1,"},{"Start":"03:00.890 ","End":"03:06.560","Text":"which is x plus 1^10 and then plus,"},{"Start":"03:06.560 ","End":"03:11.375","Text":"because of this rule with a plus natural log of x"},{"Start":"03:11.375 ","End":"03:16.685","Text":"minus 4^11 and then the ones on the denominator,"},{"Start":"03:16.685 ","End":"03:17.710","Text":"we\u0027re using this rule,"},{"Start":"03:17.710 ","End":"03:19.120","Text":"they both get a minus."},{"Start":"03:19.120 ","End":"03:28.790","Text":"We\u0027ll have minus natural log of x minus 1^4 and minus natural log of the last 1,"},{"Start":"03:28.790 ","End":"03:34.450","Text":"x minus 10 to the power of 20."},{"Start":"03:34.450 ","End":"03:37.125","Text":"I just use the top 2 rules."},{"Start":"03:37.125 ","End":"03:38.420","Text":"Here I had a product,"},{"Start":"03:38.420 ","End":"03:40.415","Text":"perhaps I\u0027ll emphasize it with a dot."},{"Start":"03:40.415 ","End":"03:47.810","Text":"Here I had a product and this come out to be plus but here I have a big dividing sign,"},{"Start":"03:47.810 ","End":"03:50.405","Text":"so both of these will come out minus."},{"Start":"03:50.405 ","End":"03:56.270","Text":"Now we\u0027re going to use this last rule again in the direction from right to left."},{"Start":"03:56.270 ","End":"03:59.605","Text":"We take the exponent and pull it out front."},{"Start":"03:59.605 ","End":"04:03.180","Text":"We get the 10 comes to the front,"},{"Start":"04:03.180 ","End":"04:08.110","Text":"10 natural log of x plus 1 and then"},{"Start":"04:08.110 ","End":"04:14.345","Text":"11 natural log of x minus 4 minus 4,"},{"Start":"04:14.345 ","End":"04:17.975","Text":"natural log of x minus 1,"},{"Start":"04:17.975 ","End":"04:25.715","Text":"minus 20, natural log of x minus 10."},{"Start":"04:25.715 ","End":"04:28.505","Text":"Note that I haven\u0027t done any differentiation yet."},{"Start":"04:28.505 ","End":"04:33.575","Text":"I just took the logarithm and I\u0027ve been simplifying and got natural log of y,"},{"Start":"04:33.575 ","End":"04:35.495","Text":"natural log of y."},{"Start":"04:35.495 ","End":"04:38.210","Text":"Now we\u0027ll do the differentiation."},{"Start":"04:38.210 ","End":"04:40.685","Text":"Start with the right-hand side."},{"Start":"04:40.685 ","End":"04:46.145","Text":"The derivative of natural log of x plus 1 is just 1/x plus 1,"},{"Start":"04:46.145 ","End":"04:51.770","Text":"so here we\u0027ll get 10/x plus 1."},{"Start":"04:51.770 ","End":"04:57.245","Text":"Next 1, 11/x minus 4."},{"Start":"04:57.245 ","End":"05:02.565","Text":"The next 1, 4/x minus 1,"},{"Start":"05:02.565 ","End":"05:07.535","Text":"and the last d minus 20/x minus 10."},{"Start":"05:07.535 ","End":"05:09.665","Text":"Now what about the left-hand side?"},{"Start":"05:09.665 ","End":"05:12.440","Text":"Now we\u0027re differentiating with respect to x."},{"Start":"05:12.440 ","End":"05:17.735","Text":"Remember, y is a function of x. I\u0027ll just even emphasize it here,"},{"Start":"05:17.735 ","End":"05:20.135","Text":"y is a function of x."},{"Start":"05:20.135 ","End":"05:23.150","Text":"When we differentiate natural log of a function,"},{"Start":"05:23.150 ","End":"05:24.320","Text":"we need the chain rule,"},{"Start":"05:24.320 ","End":"05:31.550","Text":"it\u0027s 1/y, but that\u0027s not all because we have to multiply by the inner derivative,"},{"Start":"05:31.550 ","End":"05:33.785","Text":"which is y prime."},{"Start":"05:33.785 ","End":"05:36.050","Text":"Instead of writing 1/y times y prime,"},{"Start":"05:36.050 ","End":"05:39.510","Text":"I\u0027ll just write it as y prime over y."},{"Start":"05:39.560 ","End":"05:45.515","Text":"We\u0027re close to the end now because all we have to do is multiply both sides by y"},{"Start":"05:45.515 ","End":"05:54.000","Text":"and so y prime is equal to y times,"},{"Start":"05:55.370 ","End":"05:59.000","Text":"and I just copy-pasted this."},{"Start":"05:59.000 ","End":"06:03.365","Text":"Now at the end, we would normally replace y,"},{"Start":"06:03.365 ","End":"06:09.040","Text":"this y here by what it\u0027s equal to, which is here."},{"Start":"06:09.040 ","End":"06:14.185","Text":"This is usually acceptable as an answer but if you really want me to do it,"},{"Start":"06:14.185 ","End":"06:17.815","Text":"then I\u0027ll show you that it doesn\u0027t look any tidier."},{"Start":"06:17.815 ","End":"06:26.320","Text":"Here\u0027s what it looks like but it\u0027s just messier than this."},{"Start":"06:26.320 ","End":"06:28.180","Text":"We usually leave it with the y,"},{"Start":"06:28.180 ","End":"06:33.505","Text":"so let me go back to that and I\u0027ll just summarize the main steps again."},{"Start":"06:33.505 ","End":"06:37.105","Text":"We take the natural log of both sides,"},{"Start":"06:37.105 ","End":"06:40.070","Text":"simplify using the rules of logarithms"},{"Start":"06:40.070 ","End":"06:47.300","Text":"and we get some simple pieces then we take the derivative of both sides,"},{"Start":"06:47.300 ","End":"06:53.255","Text":"remembering that the derivative of natural log of y is y prime over y,"},{"Start":"06:53.255 ","End":"06:58.305","Text":"not just 1/y and here we get usually simple pieces."},{"Start":"06:58.305 ","End":"07:01.905","Text":"Then finally, we multiply by y."},{"Start":"07:01.905 ","End":"07:04.715","Text":"If you really want to,"},{"Start":"07:04.715 ","End":"07:10.550","Text":"then you can replace y by what it was originally if you like this better than this."},{"Start":"07:10.550 ","End":"07:14.400","Text":"That\u0027s logarithmic differentiation."}],"ID":9133},{"Watched":false,"Name":"Exercise 1","Duration":"5m 34s","ChapterTopicVideoID":8874,"CourseChapterTopicPlaylistID":5119,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"Here we have an exercise in logarithmic differentiation."},{"Start":"00:04.440 ","End":"00:06.575","Text":"We\u0027re given that y equals,"},{"Start":"00:06.575 ","End":"00:08.555","Text":"I\u0027m not going to read it all out."},{"Start":"00:08.555 ","End":"00:16.240","Text":"We have to find out what the derivative is presumably using logarithmic differentiation."},{"Start":"00:16.490 ","End":"00:22.815","Text":"The idea is to take the logarithm of both sides and then do the differentiation."},{"Start":"00:22.815 ","End":"00:25.080","Text":"But before I even take the logs,"},{"Start":"00:25.080 ","End":"00:27.120","Text":"I want to do a bit of simplification."},{"Start":"00:27.120 ","End":"00:28.770","Text":"I don\u0027t like these roots,"},{"Start":"00:28.770 ","End":"00:31.165","Text":"I prefer fractional exponents."},{"Start":"00:31.165 ","End":"00:38.194","Text":"I want to remind you that when we have the mth root of a to the n,"},{"Start":"00:38.194 ","End":"00:40.160","Text":"where a is some positive number,"},{"Start":"00:40.160 ","End":"00:49.980","Text":"then this is a to the power of n over m. Just the mth root of"},{"Start":"00:49.980 ","End":"00:52.320","Text":"m is a to the power"},{"Start":"00:52.320 ","End":"01:02.250","Text":"of 1 over m. if I apply those here,"},{"Start":"01:02.250 ","End":"01:06.105","Text":"then I get that y equals,"},{"Start":"01:06.105 ","End":"01:10.385","Text":"now this thing to the power of 1/4,"},{"Start":"01:10.385 ","End":"01:16.930","Text":"it\u0027s 10x minus 1 over x plus 1."},{"Start":"01:16.930 ","End":"01:21.004","Text":"Here because of the 10th root and the power of 7,"},{"Start":"01:21.004 ","End":"01:27.305","Text":"I have 2x plus 1 to the power of 7 over 10."},{"Start":"01:27.305 ","End":"01:30.679","Text":"Now we take the natural log of both sides."},{"Start":"01:30.679 ","End":"01:35.765","Text":"I have the natural log of y and I have a product."},{"Start":"01:35.765 ","End":"01:39.800","Text":"I hope you remember your rules of logarithms. Well, I\u0027ll remind you."},{"Start":"01:39.800 ","End":"01:49.230","Text":"In this case, we have natural log of a times b is natural log of a plus natural log of b."},{"Start":"01:49.540 ","End":"01:55.240","Text":"We have natural log of all this,"},{"Start":"01:55.540 ","End":"02:04.340","Text":"10x minus 1 over x plus 1 to the 1/4 perhaps put an extra brackets"},{"Start":"02:04.340 ","End":"02:11.480","Text":"plus natural log of"},{"Start":"02:11.480 ","End":"02:18.050","Text":"2x plus 1 to the power of 7/10."},{"Start":"02:18.050 ","End":"02:19.969","Text":"Now I\u0027m going to use another rule of logarithms."},{"Start":"02:19.969 ","End":"02:23.645","Text":"I want to pull the exponent in front here and here."},{"Start":"02:23.645 ","End":"02:27.520","Text":"I remember it pictorially by just putting some arrows like that,"},{"Start":"02:27.520 ","End":"02:29.555","Text":"but if you want the rule,"},{"Start":"02:29.555 ","End":"02:34.250","Text":"it\u0027s the natural log of a to the power of b,"},{"Start":"02:34.250 ","End":"02:35.555","Text":"bring the b in front."},{"Start":"02:35.555 ","End":"02:39.245","Text":"It\u0027s b natural log of a."},{"Start":"02:39.245 ","End":"02:41.435","Text":"If I do that,"},{"Start":"02:41.435 ","End":"02:52.454","Text":"we\u0027ll get 1/4 natural log of 10x minus 1"},{"Start":"02:52.454 ","End":"02:59.990","Text":"over x plus 1"},{"Start":"02:59.990 ","End":"03:03.995","Text":"plus 7/10 natural log"},{"Start":"03:03.995 ","End":"03:07.630","Text":"of 2x plus 1."},{"Start":"03:07.900 ","End":"03:10.910","Text":"For this quotient, we\u0027re going to use"},{"Start":"03:10.910 ","End":"03:19.690","Text":"the rule natural log of a over b is natural log of a minus natural log of b."},{"Start":"03:19.690 ","End":"03:26.115","Text":"We get 1/4 of log this minus log this will put the quarters in front 2."},{"Start":"03:26.115 ","End":"03:31.920","Text":"The 2 steps in 1/4 natural log of 10x minus 1,"},{"Start":"03:31.920 ","End":"03:37.905","Text":"minus 1/4 natural log of x plus 1."},{"Start":"03:37.905 ","End":"03:42.600","Text":"See how the 1/4 this minus this and I just multiply by the quarter"},{"Start":"03:42.600 ","End":"03:49.555","Text":"plus 7/10 natural log of 2x plus 1."},{"Start":"03:49.555 ","End":"03:52.640","Text":"Now it\u0027s time to do the differentiation."},{"Start":"03:52.640 ","End":"03:57.890","Text":"Let me just copy again natural log of y, differentiate both sides."},{"Start":"03:57.890 ","End":"04:00.235","Text":"The derivative of natural log of y."},{"Start":"04:00.235 ","End":"04:05.734","Text":"The tutorial I showed you y it\u0027s y prime over y."},{"Start":"04:05.734 ","End":"04:07.430","Text":"If you want a reminder,"},{"Start":"04:07.430 ","End":"04:09.140","Text":"y is a function of x."},{"Start":"04:09.140 ","End":"04:12.090","Text":"Derivative of natural log would normally be 1 over y,"},{"Start":"04:12.090 ","End":"04:13.670","Text":"but because of the chain rule,"},{"Start":"04:13.670 ","End":"04:15.980","Text":"you have to multiply by the inner derivative,"},{"Start":"04:15.980 ","End":"04:17.810","Text":"so y prime over y,"},{"Start":"04:17.810 ","End":"04:21.590","Text":"some people write it 1 over y times y prime, doesn\u0027t matter."},{"Start":"04:21.590 ","End":"04:23.510","Text":"Anyway, that\u0027s the left-hand side."},{"Start":"04:23.510 ","End":"04:24.710","Text":"As for the right-hand side,"},{"Start":"04:24.710 ","End":"04:31.760","Text":"we have 1/4 derivative of natural log normally would just be 1 over 10x minus 1."},{"Start":"04:31.760 ","End":"04:37.220","Text":"But there\u0027s an inner derivative, 10 minus 1/4."},{"Start":"04:37.220 ","End":"04:42.125","Text":"Here we just have 1 over x plus 1,"},{"Start":"04:42.125 ","End":"04:48.530","Text":"and here we have 7/10,"},{"Start":"04:48.530 ","End":"04:50.600","Text":"1 over 2x plus 1,"},{"Start":"04:50.600 ","End":"04:53.240","Text":"but times inner derivative 2."},{"Start":"04:53.240 ","End":"04:59.390","Text":"Another last step is to multiply both sides by y."},{"Start":"04:59.390 ","End":"05:02.240","Text":"You could do some simplification of the fractions."},{"Start":"05:02.240 ","End":"05:03.980","Text":"I\u0027m not going to bother with that."},{"Start":"05:03.980 ","End":"05:10.400","Text":"We\u0027ll get that y prime is equal to y times all of this."},{"Start":"05:10.400 ","End":"05:14.450","Text":"I just did a copy paste from here."},{"Start":"05:14.450 ","End":"05:17.675","Text":"Then finally, and this is optional,"},{"Start":"05:17.675 ","End":"05:25.070","Text":"you could replace the y here by the original y from here."},{"Start":"05:25.070 ","End":"05:28.910","Text":"But it would just look more messy and this is usually"},{"Start":"05:28.910 ","End":"05:34.620","Text":"acceptable just to leave it like this. We\u0027re done."}],"ID":9134},{"Watched":false,"Name":"Exercise 2","Duration":"6m 5s","ChapterTopicVideoID":8876,"CourseChapterTopicPlaylistID":5119,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.455","Text":"Now, another exercise in logarithmic differentiation."},{"Start":"00:04.455 ","End":"00:08.170","Text":"This time, we\u0027ll take y to be the"},{"Start":"00:08.170 ","End":"00:18.960","Text":"fourth root of 10x plus 1, all this to the power of 2^x."},{"Start":"00:18.960 ","End":"00:23.590","Text":"The question is, what is y prime equal to?"},{"Start":"00:24.110 ","End":"00:27.390","Text":"We don\u0027t have to use logarithmic differentiation,"},{"Start":"00:27.390 ","End":"00:33.525","Text":"but that\u0027s a good way to go because we have a lot of exponents and roots."},{"Start":"00:33.525 ","End":"00:37.110","Text":"The first step is to take the natural log of both sides,"},{"Start":"00:37.110 ","End":"00:40.440","Text":"but I suggest tidying up a bit or"},{"Start":"00:40.440 ","End":"00:46.505","Text":"rearranging this route in terms of fractional exponents."},{"Start":"00:46.505 ","End":"00:48.950","Text":"In case you\u0027ve forgotten your rules,"},{"Start":"00:48.950 ","End":"00:50.780","Text":"the fourth root or in general,"},{"Start":"00:50.780 ","End":"00:55.295","Text":"the nth root of a is a to the power of 1 over n,"},{"Start":"00:55.295 ","End":"00:56.900","Text":"and in our case,"},{"Start":"00:56.900 ","End":"01:03.020","Text":"we have that y equals 10x plus 1,"},{"Start":"01:03.020 ","End":"01:09.210","Text":"to the power of 1/4, and all this to the power of 2^x."},{"Start":"01:09.210 ","End":"01:14.665","Text":"We\u0027re going to need another rule of logarithms."},{"Start":"01:14.665 ","End":"01:16.580","Text":"You should know this,"},{"Start":"01:16.580 ","End":"01:18.100","Text":"but I\u0027ll write it anyway."},{"Start":"01:18.100 ","End":"01:20.330","Text":"If you have a power of a power,"},{"Start":"01:20.330 ","End":"01:23.810","Text":"let\u0027s say, a to the power of n to the power of m,"},{"Start":"01:23.810 ","End":"01:26.120","Text":"you just multiply the exponents."},{"Start":"01:26.120 ","End":"01:33.425","Text":"In our case, we have 10x plus 1 to the power of 1/4 times 2^x."},{"Start":"01:33.425 ","End":"01:36.560","Text":"I could write it as 2^x over 4,"},{"Start":"01:36.560 ","End":"01:40.520","Text":"could simplify it further because this is 2^2, no need to."},{"Start":"01:40.520 ","End":"01:44.160","Text":"This is now y."},{"Start":"01:44.160 ","End":"01:48.105","Text":"Now, let\u0027s take the natural logarithm."},{"Start":"01:48.105 ","End":"01:55.250","Text":"We get natural logarithm of y equals natural logarithm of the right-hand side,"},{"Start":"01:55.250 ","End":"02:04.710","Text":"which is 10x plus 1 to the power of 2^x over 4."},{"Start":"02:04.860 ","End":"02:07.570","Text":"Today, I\u0027m giving you all the rules you need,"},{"Start":"02:07.570 ","End":"02:08.770","Text":"though you should know these."},{"Start":"02:08.770 ","End":"02:13.195","Text":"That the natural log of a^b,"},{"Start":"02:13.195 ","End":"02:15.130","Text":"just bring the b out front."},{"Start":"02:15.130 ","End":"02:19.960","Text":"It\u0027s b natural log of a. I don\u0027t remember this rule."},{"Start":"02:19.960 ","End":"02:22.540","Text":"I remember it pictorially that when you have an exponent,"},{"Start":"02:22.540 ","End":"02:24.665","Text":"you bring it out front."},{"Start":"02:24.665 ","End":"02:36.549","Text":"What we get is 1/4 2^x natural log of 10x plus 1,"},{"Start":"02:36.549 ","End":"02:39.370","Text":"and this is still natural log of y."},{"Start":"02:39.370 ","End":"02:42.800","Text":"Still haven\u0027t done any differentiation yet."},{"Start":"02:42.980 ","End":"02:48.265","Text":"Now, it\u0027s time to take the derivative of each side."},{"Start":"02:48.265 ","End":"02:50.765","Text":"The derivative of natural log of y,"},{"Start":"02:50.765 ","End":"02:53.340","Text":"we\u0027ve talked about it before,"},{"Start":"02:53.340 ","End":"02:56.180","Text":"it\u0027s not just 1 over y,"},{"Start":"02:56.180 ","End":"02:58.535","Text":"because y is a function of x,"},{"Start":"02:58.535 ","End":"03:00.230","Text":"you need also the inner derivative,"},{"Start":"03:00.230 ","End":"03:02.135","Text":"which is y prime."},{"Start":"03:02.135 ","End":"03:06.785","Text":"The derivative of this is y prime over y,"},{"Start":"03:06.785 ","End":"03:10.264","Text":"and for the right-hand side,"},{"Start":"03:10.264 ","End":"03:15.590","Text":"I\u0027ve got 1/4 and I need the derivative."},{"Start":"03:15.590 ","End":"03:23.070","Text":"You know what? I\u0027ll just copy this and write a derivative sign outside the brackets."},{"Start":"03:23.070 ","End":"03:24.750","Text":"Now, this is a product,"},{"Start":"03:24.750 ","End":"03:26.550","Text":"there\u0027s a product here."},{"Start":"03:26.550 ","End":"03:28.790","Text":"We\u0027re going to use the product rule."},{"Start":"03:28.790 ","End":"03:32.840","Text":"The difficulty is that I don\u0027t remember,"},{"Start":"03:32.840 ","End":"03:34.100","Text":"well, I do, but let\u0027s say,"},{"Start":"03:34.100 ","End":"03:38.215","Text":"we don\u0027t remember what is the derivative of 2^x."},{"Start":"03:38.215 ","End":"03:41.630","Text":"I\u0027m being generous today with all the formulas that you need."},{"Start":"03:41.630 ","End":"03:45.634","Text":"Not 2. I\u0027m going to give, in general,"},{"Start":"03:45.634 ","End":"03:53.340","Text":"the derivative of a^x is a^x times natural log of a."},{"Start":"03:53.340 ","End":"03:55.830","Text":"a could be 2."},{"Start":"03:55.830 ","End":"03:59.265","Text":"What we get is 1/4,"},{"Start":"03:59.265 ","End":"04:02.180","Text":"now, I\u0027m not going to remind you of the product rule,"},{"Start":"04:02.180 ","End":"04:03.770","Text":"that I expect you to know."},{"Start":"04:03.770 ","End":"04:07.835","Text":"We take the derivative of the first times the second and then vice versa."},{"Start":"04:07.835 ","End":"04:13.990","Text":"The derivative of this according to this is 2^x natural log of 2."},{"Start":"04:13.990 ","End":"04:16.530","Text":"The other bit as is,"},{"Start":"04:16.530 ","End":"04:20.465","Text":"natural log of 10x plus 1."},{"Start":"04:20.465 ","End":"04:22.745","Text":"Now, we do the other way around."},{"Start":"04:22.745 ","End":"04:24.665","Text":"The other one differentiated,"},{"Start":"04:24.665 ","End":"04:29.300","Text":"so we\u0027ll take the 2^x as is and the derivative of"},{"Start":"04:29.300 ","End":"04:36.795","Text":"natural log of 10x plus 1 starts out by being 1 over 10x plus 1,"},{"Start":"04:36.795 ","End":"04:38.540","Text":"but the inner derivative is 10,"},{"Start":"04:38.540 ","End":"04:43.250","Text":"which I\u0027ll put instead of the 1 there, close brackets."},{"Start":"04:43.250 ","End":"04:45.560","Text":"We don\u0027t have to simplify,"},{"Start":"04:45.560 ","End":"04:47.890","Text":"I\u0027ll just do a little bit of simplification."},{"Start":"04:47.890 ","End":"04:52.330","Text":"The 2^x is common so why don\u0027t I take that out?"},{"Start":"04:52.330 ","End":"04:56.950","Text":"I have 1/4 times 2^x times"},{"Start":"04:56.950 ","End":"05:03.765","Text":"natural log of 2 natural log of 10x plus 1,"},{"Start":"05:03.765 ","End":"05:08.805","Text":"plus 10 over 10x plus 1."},{"Start":"05:08.805 ","End":"05:11.760","Text":"This is not y prime,"},{"Start":"05:11.760 ","End":"05:14.070","Text":"it\u0027s y prime over y."},{"Start":"05:14.070 ","End":"05:19.210","Text":"All I have to do is multiply both sides by y"},{"Start":"05:19.210 ","End":"05:24.869","Text":"and then we have that y prime is equal to"},{"Start":"05:24.869 ","End":"05:30.600","Text":"1/4 2^x times y,"},{"Start":"05:30.600 ","End":"05:34.140","Text":"doesn\u0027t really matter where you put the y, times."},{"Start":"05:34.150 ","End":"05:37.790","Text":"Now, generally it\u0027s okay to leave it like this,"},{"Start":"05:37.790 ","End":"05:40.460","Text":"but if you really want it only in terms of x,"},{"Start":"05:40.460 ","End":"05:42.960","Text":"then you can take this y."},{"Start":"05:42.960 ","End":"05:51.090","Text":"Instead of y, write what\u0027s here."},{"Start":"05:51.090 ","End":"05:56.180","Text":"Just plug this for y and get the answer."},{"Start":"05:56.180 ","End":"05:57.950","Text":"I won\u0027t bother doing it, it\u0027s messy,"},{"Start":"05:57.950 ","End":"06:03.410","Text":"and usually, most professors will allow you to leave it like this."},{"Start":"06:03.410 ","End":"06:05.970","Text":"So we\u0027re done."}],"ID":9135}],"Thumbnail":null,"ID":5119}]

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