[{"Name":"Introduction to Implicit Differentiation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Differentiation of Implicit Function Theory 1","Duration":"16m 6s","ChapterTopicVideoID":8669,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8669.jpeg","UploadDate":"2020-02-26T11:54:57.6830000","DurationForVideoObject":"PT16M6S","Description":null,"MetaTitle":"Differentiation of Implicit Function Theory 1: Video + Workbook | Proprep","MetaDescription":"Implicit Differentiation - Introduction to Implicit Differentiation. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/implicit-differentiation/introduction-to-implicit-differentiation/vid8943","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.525","Text":"In this clip, I\u0027ll be talking about partial derivatives of implicit functions."},{"Start":"00:06.525 ","End":"00:08.415","Text":"I\u0027m going to build up to this."},{"Start":"00:08.415 ","End":"00:12.750","Text":"Let\u0027s, first of all, take an explicit function,"},{"Start":"00:12.750 ","End":"00:19.170","Text":"z is equal to x squared plus y squared."},{"Start":"00:19.170 ","End":"00:22.680","Text":"What I want is the 2 partial derivatives."},{"Start":"00:22.680 ","End":"00:24.630","Text":"That\u0027s, first of all, what I\u0027m looking for."},{"Start":"00:24.630 ","End":"00:26.820","Text":"Let us say there\u0027s no problem here."},{"Start":"00:26.820 ","End":"00:33.780","Text":"The derivative with respect to x means that we keep y as a constant and then we"},{"Start":"00:33.780 ","End":"00:41.745","Text":"get 2x and z with respect to y is similar,"},{"Start":"00:41.745 ","End":"00:47.170","Text":"x is treated as a constant and we differentiate y so we get 2y."},{"Start":"00:47.170 ","End":"00:49.040","Text":"That\u0027s straight enough."},{"Start":"00:49.040 ","End":"00:51.860","Text":"Now, let\u0027s make it a little bit more difficult."},{"Start":"00:51.860 ","End":"00:57.719","Text":"Suppose I gave you 4z minus x"},{"Start":"00:57.719 ","End":"01:04.560","Text":"plus y equals 0 and now I ask the same question,"},{"Start":"01:04.560 ","End":"01:08.195","Text":"what is the derivative of z with respect to x?"},{"Start":"01:08.195 ","End":"01:11.735","Text":"What is the derivative of z with respect to y?"},{"Start":"01:11.735 ","End":"01:16.030","Text":"You\u0027d say, we can isolate z. What\u0027s the problem?"},{"Start":"01:16.030 ","End":"01:21.440","Text":"We can say that z equals I bring the x and y to the other side."},{"Start":"01:21.440 ","End":"01:28.400","Text":"It\u0027s x minus y divide by 4, 1/4x minus 1/4y."},{"Start":"01:28.400 ","End":"01:32.010","Text":"Now I say z with respect to x"},{"Start":"01:32.010 ","End":"01:39.195","Text":"is 1/4 and z with respect to y is minus 1/4."},{"Start":"01:39.195 ","End":"01:43.130","Text":"No problem. I\u0027ll take it a step further."},{"Start":"01:43.130 ","End":"01:50.845","Text":"Natural log of z plus y squared equals"},{"Start":"01:50.845 ","End":"01:56.240","Text":"2x minus z^4."},{"Start":"01:56.240 ","End":"02:02.705","Text":"Now, how would you like to give me the partial derivative of z with respect to x?"},{"Start":"02:02.705 ","End":"02:04.880","Text":"This time, not so easy."},{"Start":"02:04.880 ","End":"02:10.010","Text":"We can\u0027t easily isolate z in terms of y and x."},{"Start":"02:10.010 ","End":"02:11.510","Text":"It\u0027s just too difficult."},{"Start":"02:11.510 ","End":"02:15.155","Text":"What do we do if we can\u0027t isolate z?"},{"Start":"02:15.155 ","End":"02:19.070","Text":"That\u0027s where the implicit differentiation comes in."},{"Start":"02:19.070 ","End":"02:22.490","Text":"Later we\u0027ll also find z with respect to y."},{"Start":"02:22.490 ","End":"02:24.735","Text":"But let\u0027s even try one of them."},{"Start":"02:24.735 ","End":"02:31.430","Text":"We need a whole new technique for this and let me start this over again."},{"Start":"02:31.430 ","End":"02:33.750","Text":"The first thing you do,"},{"Start":"02:33.880 ","End":"02:38.780","Text":"we bring everything to the left-hand side and leave 0 on the right,"},{"Start":"02:38.780 ","End":"02:43.850","Text":"so natural log of z plus y"},{"Start":"02:43.850 ","End":"02:51.960","Text":"squared minus 2x plus z^4 equals 0."},{"Start":"02:51.960 ","End":"02:55.220","Text":"That\u0027s the first step and this is what we do in general,"},{"Start":"02:55.220 ","End":"02:58.860","Text":"not just in this example, Step 1."},{"Start":"02:58.880 ","End":"03:03.740","Text":"Then Step 2 is to set"},{"Start":"03:03.740 ","End":"03:10.580","Text":"the left-hand side equal to a function and this function is going to be a function of x,"},{"Start":"03:10.580 ","End":"03:19.775","Text":"y, and z, a function of 3 variables and we set this equal to whatever it says here."},{"Start":"03:19.775 ","End":"03:26.660","Text":"Natural log of z plus y squared minus 2x plus z^4,"},{"Start":"03:26.660 ","End":"03:28.910","Text":"the next step, well,"},{"Start":"03:28.910 ","End":"03:31.205","Text":"it\u0027s really part of the same step."},{"Start":"03:31.205 ","End":"03:35.690","Text":"What it is is to use the formulas for the derivatives."},{"Start":"03:35.690 ","End":"03:37.295","Text":"When I write them over here,"},{"Start":"03:37.295 ","End":"03:45.635","Text":"the derivative of z with respect to x equals minus the derivative"},{"Start":"03:45.635 ","End":"03:54.695","Text":"of f with respect to x divided by the derivative of f with respect to z."},{"Start":"03:54.695 ","End":"04:00.935","Text":"The derivative of z with respect to y is equal to minus"},{"Start":"04:00.935 ","End":"04:08.610","Text":"f with respect to y over f with respect to z."},{"Start":"04:08.610 ","End":"04:14.500","Text":"Perhaps a bit of highlighting won\u0027t hurt either, z in green,"},{"Start":"04:14.840 ","End":"04:23.210","Text":"and let\u0027s do x in yellow and let\u0027s do y in this color,"},{"Start":"04:23.210 ","End":"04:26.690","Text":"this turquoise, I don\u0027t know."},{"Start":"04:26.690 ","End":"04:31.790","Text":"These are the 3 partial derivatives of f here."},{"Start":"04:31.790 ","End":"04:37.385","Text":"I don\u0027t know if we\u0027ve actually practiced differentiating a function of 3 variables,"},{"Start":"04:37.385 ","End":"04:40.805","Text":"but it\u0027s exactly the same thing as in 2 variables."},{"Start":"04:40.805 ","End":"04:43.445","Text":"Whenever you do it with respect to x,"},{"Start":"04:43.445 ","End":"04:44.960","Text":"y and z are constants."},{"Start":"04:44.960 ","End":"04:46.370","Text":"When you do it with respect to y,"},{"Start":"04:46.370 ","End":"04:48.665","Text":"x and z are constants and with respect to z,"},{"Start":"04:48.665 ","End":"04:50.975","Text":"you treat x and y as constants."},{"Start":"04:50.975 ","End":"04:53.435","Text":"That\u0027s the general formula."},{"Start":"04:53.435 ","End":"04:56.300","Text":"Let\u0027s apply it to our case."},{"Start":"04:56.300 ","End":"05:04.970","Text":"So z with respect to x is equal to minus."},{"Start":"05:04.970 ","End":"05:07.925","Text":"Then on the numerator,"},{"Start":"05:07.925 ","End":"05:11.315","Text":"I put the derivative of f with respect to x."},{"Start":"05:11.315 ","End":"05:14.390","Text":"This is our f. With respect to x,"},{"Start":"05:14.390 ","End":"05:18.155","Text":"we get, this is nothing because it\u0027s constant."},{"Start":"05:18.155 ","End":"05:20.090","Text":"This is a constant."},{"Start":"05:20.090 ","End":"05:23.940","Text":"With respect to x, this becomes minus 2,"},{"Start":"05:23.940 ","End":"05:26.220","Text":"and this is also 0."},{"Start":"05:26.220 ","End":"05:30.350","Text":"On the denominator, we need f with respect to z."},{"Start":"05:30.350 ","End":"05:33.904","Text":"From here we get 1 over"},{"Start":"05:33.904 ","End":"05:42.000","Text":"z and nothing from here and nothing from here and plus 4z cubed."},{"Start":"05:42.560 ","End":"05:46.145","Text":"That\u0027s the derivative of z with respect to x."},{"Start":"05:46.145 ","End":"05:49.265","Text":"Although it does involve z itself,"},{"Start":"05:49.265 ","End":"05:51.515","Text":"it\u0027s not in terms of x and y."},{"Start":"05:51.515 ","End":"05:56.825","Text":"Continuing derivative of z with respect to y,"},{"Start":"05:56.825 ","End":"06:01.490","Text":"very similar, minus something over something."},{"Start":"06:01.490 ","End":"06:04.270","Text":"The denominator\u0027s the same."},{"Start":"06:04.270 ","End":"06:07.910","Text":"The numerator this time I only have to look for the terms with y."},{"Start":"06:07.910 ","End":"06:09.935","Text":"It\u0027s here, it\u0027s 2y."},{"Start":"06:09.935 ","End":"06:12.080","Text":"That solves the question."},{"Start":"06:12.080 ","End":"06:17.510","Text":"I\u0027d like to point out that we don\u0027t always have the variables x, y,"},{"Start":"06:17.510 ","End":"06:18.980","Text":"and z. I mean,"},{"Start":"06:18.980 ","End":"06:21.300","Text":"they could be a, b,"},{"Start":"06:21.300 ","End":"06:25.995","Text":"and c, u, v and w are the variables of possible."},{"Start":"06:25.995 ","End":"06:30.845","Text":"It\u0027s not necessarily a good idea to remember it just this way."},{"Start":"06:30.845 ","End":"06:35.525","Text":"What you might notice is that when I take the derivative of z with respect to x,"},{"Start":"06:35.525 ","End":"06:37.490","Text":"the x goes on the top,"},{"Start":"06:37.490 ","End":"06:39.090","Text":"the z goes on the bottom."},{"Start":"06:39.090 ","End":"06:40.160","Text":"That\u0027s a crossing."},{"Start":"06:40.160 ","End":"06:42.245","Text":"This goes here, this goes there."},{"Start":"06:42.245 ","End":"06:45.230","Text":"Similarly here, the thing that\u0027s the subscript"},{"Start":"06:45.230 ","End":"06:48.620","Text":"that goes on the top and the big letter goes on the bottom."},{"Start":"06:48.620 ","End":"06:49.925","Text":"But in all cases,"},{"Start":"06:49.925 ","End":"06:55.085","Text":"we have the function here and it\u0027s always minus something over something,"},{"Start":"06:55.085 ","End":"06:56.945","Text":"minus something over something."},{"Start":"06:56.945 ","End":"07:02.015","Text":"Perhaps better to remember it in schematic terms and you know what,"},{"Start":"07:02.015 ","End":"07:05.045","Text":"I\u0027ll do an example where we change the letters."},{"Start":"07:05.045 ","End":"07:08.120","Text":"This time, I\u0027ll use a, b, and c,"},{"Start":"07:08.120 ","End":"07:11.075","Text":"even though they\u0027re normally reserved for constants in mathematics,"},{"Start":"07:11.075 ","End":"07:17.030","Text":"a squared plus b squared equals natural log of"},{"Start":"07:17.030 ","End":"07:23.405","Text":"c plus natural log of b minus c squared."},{"Start":"07:23.405 ","End":"07:27.425","Text":"I want to know what c with respect to"},{"Start":"07:27.425 ","End":"07:33.415","Text":"a equals and the partial derivative of c with respect to b."},{"Start":"07:33.415 ","End":"07:34.775","Text":"What does that equal?"},{"Start":"07:34.775 ","End":"07:37.610","Text":"What I\u0027m saying is that c is considered to"},{"Start":"07:37.610 ","End":"07:41.720","Text":"be the function of a and b. I imagine that I\u0027ve somehow"},{"Start":"07:41.720 ","End":"07:45.590","Text":"extracted c from this equation and it\u0027s"},{"Start":"07:45.590 ","End":"07:49.790","Text":"a function of a and b. I\u0027ll follow the steps above."},{"Start":"07:49.790 ","End":"07:55.425","Text":"Step 1 is to write everything on the left-hand side."},{"Start":"07:55.425 ","End":"08:02.765","Text":"I get a squared plus b squared minus log"},{"Start":"08:02.765 ","End":"08:13.245","Text":"c minus log b plus c squared equals 0."},{"Start":"08:13.245 ","End":"08:16.150","Text":"I think I\u0027ll lower this a bit."},{"Start":"08:16.150 ","End":"08:20.440","Text":"Now, we can go on to step number 2,"},{"Start":"08:20.440 ","End":"08:23.035","Text":"and that\u0027s where we let a function of,"},{"Start":"08:23.035 ","End":"08:24.910","Text":"what letters do we have here, a, b,"},{"Start":"08:24.910 ","End":"08:30.070","Text":"and c. This will equal what\u0027s written above,"},{"Start":"08:30.070 ","End":"08:33.220","Text":"a squared plus b squared,"},{"Start":"08:33.220 ","End":"08:36.655","Text":"I think I\u0027ll change the order a bit, it looks nicer,"},{"Start":"08:36.655 ","End":"08:45.955","Text":"minus natural log of c minus natural log of b."},{"Start":"08:45.955 ","End":"08:51.010","Text":"Using this formula but remembering that we now have a, b, and c,"},{"Start":"08:51.010 ","End":"08:57.820","Text":"we get that the derivative of c with respect to a equals it\u0027s minus."},{"Start":"08:57.820 ","End":"09:00.220","Text":"We still have the letter f although even that could be different."},{"Start":"09:00.220 ","End":"09:02.380","Text":"It could be g, I don\u0027t know."},{"Start":"09:02.380 ","End":"09:12.655","Text":"With respect to the bottom letter is a and divide it by f with respect to this letter c,"},{"Start":"09:12.655 ","End":"09:14.230","Text":"just the same as here,"},{"Start":"09:14.230 ","End":"09:23.260","Text":"and c with respect to b will equal minus the derivative of f with respect"},{"Start":"09:23.260 ","End":"09:32.785","Text":"to b over f with respect to c. Let\u0027s see what this equals."},{"Start":"09:32.785 ","End":"09:35.965","Text":"This equals minus."},{"Start":"09:35.965 ","End":"09:38.830","Text":"Now derivative of f with respect to a,"},{"Start":"09:38.830 ","End":"09:41.290","Text":"I just have to look for the terms with a because"},{"Start":"09:41.290 ","End":"09:43.990","Text":"all the rest are constants as far as that\u0027s concerned,"},{"Start":"09:43.990 ","End":"09:49.570","Text":"so it\u0027s 2a over the derivative with respect to c. From"},{"Start":"09:49.570 ","End":"09:56.545","Text":"here I get 2c and from here I get minus 1/c."},{"Start":"09:56.545 ","End":"10:00.685","Text":"This one equals, is also minus,"},{"Start":"10:00.685 ","End":"10:02.770","Text":"the denominator\u0027s always the same,"},{"Start":"10:02.770 ","End":"10:07.340","Text":"I\u0027m just copying it, 2c minus 1/c."},{"Start":"10:09.480 ","End":"10:17.300","Text":"The numerator this time will be 2b minus 1/b."},{"Start":"10:18.690 ","End":"10:21.520","Text":"We have what C_a equals,"},{"Start":"10:21.520 ","End":"10:24.115","Text":"and we have C_b equals."},{"Start":"10:24.115 ","End":"10:29.650","Text":"I\u0027d like to remark that here I considered c to be an implicit function of a and b,"},{"Start":"10:29.650 ","End":"10:32.500","Text":"but I could just as equally considered b to be"},{"Start":"10:32.500 ","End":"10:35.740","Text":"an implicit function of a and c. I could have also asked,"},{"Start":"10:35.740 ","End":"10:39.940","Text":"what is the derivative of b with respect to c?"},{"Start":"10:39.940 ","End":"10:41.950","Text":"Let\u0027s do that one also."},{"Start":"10:41.950 ","End":"10:47.635","Text":"The derivative of b with respect to c equals,"},{"Start":"10:47.635 ","End":"10:49.915","Text":"you can automatically put the minus,"},{"Start":"10:49.915 ","End":"10:52.494","Text":"put a dividing line,"},{"Start":"10:52.494 ","End":"10:55.225","Text":"we take the function f,"},{"Start":"10:55.225 ","End":"10:57.460","Text":"first the one on the bottom,"},{"Start":"10:57.460 ","End":"10:59.605","Text":"and then the big letter."},{"Start":"10:59.605 ","End":"11:03.865","Text":"Actually, we have fb and fc over here,"},{"Start":"11:03.865 ","End":"11:06.670","Text":"which leads me to an observation that b with"},{"Start":"11:06.670 ","End":"11:10.315","Text":"respect to c is just the reciprocal of c with respect to b."},{"Start":"11:10.315 ","End":"11:14.500","Text":"This would be minus,"},{"Start":"11:14.500 ","End":"11:16.630","Text":"I can just look at it right here,"},{"Start":"11:16.630 ","End":"11:25.030","Text":"2c minus 1/c, it\u0027s just the reverse over 2b minus 1/b."},{"Start":"11:25.030 ","End":"11:26.050","Text":"That\u0027s this example."},{"Start":"11:26.050 ","End":"11:28.630","Text":"Let\u0027s do one more."},{"Start":"11:28.630 ","End":"11:35.140","Text":"Here\u0027s the example, xy plus yz plus"},{"Start":"11:35.140 ","End":"11:41.080","Text":"zt equals log z plus e^y plus e^x."},{"Start":"11:41.080 ","End":"11:44.335","Text":"Notice that this time we have more variables than usual."},{"Start":"11:44.335 ","End":"11:46.420","Text":"We don\u0027t have just 3, xy, and z,"},{"Start":"11:46.420 ","End":"11:51.250","Text":"we also have t. This system works with any number of variables."},{"Start":"11:51.250 ","End":"11:53.020","Text":"This time I\u0027m asking,"},{"Start":"11:53.020 ","End":"11:56.800","Text":"what is z with respect to t, the derivative?"},{"Start":"11:56.800 ","End":"11:59.740","Text":"What\u0027s the partial of y with respect to z,"},{"Start":"11:59.740 ","End":"12:02.365","Text":"and what\u0027s the partial of x with respect to t."},{"Start":"12:02.365 ","End":"12:07.150","Text":"Each one of these variables could be considered a function of the remaining 3,"},{"Start":"12:07.150 ","End":"12:12.445","Text":"even though I can\u0027t isolate z or y or x,"},{"Start":"12:12.445 ","End":"12:15.670","Text":"the same steps apply."},{"Start":"12:15.670 ","End":"12:22.810","Text":"Step number 1 is to bring everything to one side and let it equals 0."},{"Start":"12:22.810 ","End":"12:26.980","Text":"I have x, y plus yz plus"},{"Start":"12:26.980 ","End":"12:34.150","Text":"zt minus log z minus e^y"},{"Start":"12:34.150 ","End":"12:38.275","Text":"minus e^x equals 0."},{"Start":"12:38.275 ","End":"12:45.355","Text":"Then in Step 2, I define a function this time of 4 variables,"},{"Start":"12:45.355 ","End":"12:47.695","Text":"x, y, z, and t,"},{"Start":"12:47.695 ","End":"12:50.335","Text":"and this is equal to the same thing as above."},{"Start":"12:50.335 ","End":"12:54.280","Text":"Let me just copy paste it. Here we are."},{"Start":"12:54.280 ","End":"13:00.460","Text":"I didn\u0027t copy the formulas from the previous page because it\u0027s a bit different now,"},{"Start":"13:00.460 ","End":"13:04.270","Text":"has a different number of variables and I want you to remember the pattern."},{"Start":"13:04.270 ","End":"13:12.010","Text":"What it is, is if I want z with respect to t,"},{"Start":"13:12.010 ","End":"13:17.869","Text":"it\u0027s always minus a dividing line,"},{"Start":"13:17.910 ","End":"13:24.115","Text":"f with respect to the little letter,"},{"Start":"13:24.115 ","End":"13:28.280","Text":"and on the bottom, the big letter."},{"Start":"13:29.190 ","End":"13:31.810","Text":"We\u0027ll do this in a moment."},{"Start":"13:31.810 ","End":"13:34.445","Text":"I just want to write all these."},{"Start":"13:34.445 ","End":"13:39.299","Text":"Next one we had is y with respect to z,"},{"Start":"13:39.299 ","End":"13:49.255","Text":"and this is going to be minus dividing line f with respect to this one first,"},{"Start":"13:49.255 ","End":"13:53.305","Text":"and then f with respect to this one."},{"Start":"13:53.305 ","End":"13:55.720","Text":"We\u0027ll figure it out in the moment."},{"Start":"13:55.720 ","End":"14:02.005","Text":"Lastly, x with respect to t equals,"},{"Start":"14:02.005 ","End":"14:07.300","Text":"we start with a minus dividing line f with respect to something,"},{"Start":"14:07.300 ","End":"14:08.590","Text":"f with respect to something,"},{"Start":"14:08.590 ","End":"14:12.175","Text":"this letter here, this letter here."},{"Start":"14:12.175 ","End":"14:14.620","Text":"Just purely technique."},{"Start":"14:14.620 ","End":"14:17.380","Text":"We\u0027ll figure out what this is in a minute."},{"Start":"14:17.380 ","End":"14:21.700","Text":"Now, it\u0027s time to do some partial differentiation with respect to"},{"Start":"14:21.700 ","End":"14:27.565","Text":"t. The only place I see t is over here."},{"Start":"14:27.565 ","End":"14:31.600","Text":"The derivative with respect to t is"},{"Start":"14:31.600 ","End":"14:38.605","Text":"just z. I\u0027m going to put the minus and the dividing line,"},{"Start":"14:38.605 ","End":"14:43.120","Text":"and the derivative with respect to z."},{"Start":"14:43.120 ","End":"14:45.835","Text":"Let\u0027s see where do I have z?"},{"Start":"14:45.835 ","End":"14:47.515","Text":"I have it here."},{"Start":"14:47.515 ","End":"14:56.290","Text":"This becomes y, I have it here so this becomes t when I differentiate it."},{"Start":"14:56.290 ","End":"15:04.550","Text":"I have it here where it becomes minus 1/z."},{"Start":"15:05.640 ","End":"15:10.270","Text":"Next one, minus with respect to z,"},{"Start":"15:10.270 ","End":"15:12.535","Text":"I have that already, so I\u0027ll just copy it,"},{"Start":"15:12.535 ","End":"15:16.015","Text":"y plus t minus 1/z."},{"Start":"15:16.015 ","End":"15:18.745","Text":"Now, with respect to y,"},{"Start":"15:18.745 ","End":"15:28.670","Text":"so I have from here x plus z minus e^y."},{"Start":"15:30.570 ","End":"15:33.084","Text":"What do I have here?"},{"Start":"15:33.084 ","End":"15:37.690","Text":"I have minus f with respect to t,"},{"Start":"15:37.690 ","End":"15:41.515","Text":"I have that already is z,"},{"Start":"15:41.515 ","End":"15:44.230","Text":"and with respect to x,"},{"Start":"15:44.230 ","End":"15:46.240","Text":"I don\u0027t have that yet,"},{"Start":"15:46.240 ","End":"15:50.800","Text":"so I get y."},{"Start":"15:50.800 ","End":"15:55.190","Text":"What else? Minus e^x."},{"Start":"15:56.190 ","End":"16:00.190","Text":"This solves this exercise as example,"},{"Start":"16:00.190 ","End":"16:07.340","Text":"and we\u0027re done with this introduction to implicit differentiation."}],"ID":8943},{"Watched":false,"Name":"Implicit Function Second Order","Duration":"24m 28s","ChapterTopicVideoID":8670,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"This clip is entitled implicit functions,"},{"Start":"00:04.335 ","End":"00:06.840","Text":"second-order partial derivatives."},{"Start":"00:06.840 ","End":"00:09.165","Text":"Wow, scary title."},{"Start":"00:09.165 ","End":"00:12.960","Text":"Not to worry all will be clear in good time."},{"Start":"00:12.960 ","End":"00:17.140","Text":"I want to begin with recalling that we used a nice trick,"},{"Start":"00:17.140 ","End":"00:23.690","Text":"when we had to find the derivatives of implicit functions of the first order."},{"Start":"00:23.690 ","End":"00:26.135","Text":"We had a nice trick."},{"Start":"00:26.135 ","End":"00:36.500","Text":"Just for example, we had stuff like z by x was equal to minus fx over fz,"},{"Start":"00:36.500 ","End":"00:39.935","Text":"where we put this variable here and this variable here and so on."},{"Start":"00:39.935 ","End":"00:45.610","Text":"Now this trick works all very well for our first-order partial derivatives,"},{"Start":"00:45.610 ","End":"00:51.635","Text":"but it somehow gets a bit sticky when we use it on second-order partial derivatives."},{"Start":"00:51.635 ","End":"00:57.635","Text":"In this clip, I want to show you how to do second-order partial derivatives."},{"Start":"00:57.635 ","End":"00:59.765","Text":"Derivatives of the second order."},{"Start":"00:59.765 ","End":"01:04.205","Text":"Like in 1 variable we had y double prime with second-order."},{"Start":"01:04.205 ","End":"01:06.020","Text":"Now you know what I mean."},{"Start":"01:06.020 ","End":"01:11.660","Text":"I\u0027d like to take you back to implicit differentiation in calculus 1,"},{"Start":"01:11.660 ","End":"01:13.715","Text":"where we just had y and x,"},{"Start":"01:13.715 ","End":"01:17.705","Text":"and I\u0027d like to show you that it\u0027s even possible to"},{"Start":"01:17.705 ","End":"01:25.145","Text":"differentiate something like y equals x squared using partial derivatives."},{"Start":"01:25.145 ","End":"01:30.050","Text":"Using the system that we learned in calculus 2."},{"Start":"01:30.050 ","End":"01:32.090","Text":"If this was an exercise in calculus 2,"},{"Start":"01:32.090 ","End":"01:34.805","Text":"and I asked you to find y prime,"},{"Start":"01:34.805 ","End":"01:37.174","Text":"or in this case not y prime,"},{"Start":"01:37.174 ","End":"01:41.765","Text":"but the partial derivative of y with respect to x."},{"Start":"01:41.765 ","End":"01:52.545","Text":"Then we could use that trick and say that it\u0027s equal to minus fx over fy,"},{"Start":"01:52.545 ","End":"01:54.660","Text":"except that we don\u0027t have our f yet."},{"Start":"01:54.660 ","End":"01:59.565","Text":"To get our f we put everything on 1 side and we get f"},{"Start":"01:59.565 ","End":"02:05.599","Text":"of xy equals y minus x squared."},{"Start":"02:05.599 ","End":"02:09.950","Text":"Because what we did is we put everything on the left side and make it equal to 0,"},{"Start":"02:09.950 ","End":"02:11.150","Text":"and what\u0027s on the left side,"},{"Start":"02:11.150 ","End":"02:12.920","Text":"we call f of x, y."},{"Start":"02:12.920 ","End":"02:18.515","Text":"Now we can compute this and say that this equals minus"},{"Start":"02:18.515 ","End":"02:25.080","Text":"partial of f with respect to x is minus 2x."},{"Start":"02:26.610 ","End":"02:37.335","Text":"The partial derivative of f with respect to y is just 1 because x is a constant,"},{"Start":"02:37.335 ","End":"02:41.470","Text":"and this minus and this minus cancel and dividing by 1."},{"Start":"02:41.470 ","End":"02:47.155","Text":"We get 2x. That\u0027s what we would have got if we just differentiated it regular."},{"Start":"02:47.155 ","End":"02:52.660","Text":"We can do regular differentiation using implicit differentiation."},{"Start":"02:52.660 ","End":"02:54.965","Text":"Back in the days of calculus 1,"},{"Start":"02:54.965 ","End":"02:56.655","Text":"we didn\u0027t have this trick."},{"Start":"02:56.655 ","End":"03:01.225","Text":"In fact we didn\u0027t have partial derivative functions of 2 variables and all that."},{"Start":"03:01.225 ","End":"03:06.680","Text":"I\u0027d like to remind you of how we used to do the implicit differentiation in calculus 1."},{"Start":"03:06.680 ","End":"03:08.279","Text":"I\u0027ll take an example."},{"Start":"03:08.279 ","End":"03:12.350","Text":"We would have something like x squared plus y squared"},{"Start":"03:12.350 ","End":"03:18.530","Text":"equal natural log of x plus natural log of y."},{"Start":"03:18.530 ","End":"03:20.405","Text":"The rule was this,"},{"Start":"03:20.405 ","End":"03:26.090","Text":"you differentiate each bit in terms of it\u0027s own variable."},{"Start":"03:26.090 ","End":"03:29.690","Text":"But if we differentiate something with y in it,"},{"Start":"03:29.690 ","End":"03:34.440","Text":"we throw in y prime alongside."},{"Start":"03:34.440 ","End":"03:38.880","Text":"I\u0027ll show you, x squared becomes 2x,"},{"Start":"03:38.880 ","End":"03:41.865","Text":"y squared is 2y."},{"Start":"03:41.865 ","End":"03:46.275","Text":"But because it\u0027s y we throw in a y prime."},{"Start":"03:46.275 ","End":"03:48.720","Text":"Here, natural log of x,"},{"Start":"03:48.720 ","End":"03:52.565","Text":"1 over x, natural log of y, 1 over y."},{"Start":"03:52.565 ","End":"03:55.810","Text":"But because it\u0027s y we throw in a y prime."},{"Start":"03:55.810 ","End":"03:58.805","Text":"Take a quickly review why this is so,"},{"Start":"03:58.805 ","End":"04:04.175","Text":"it\u0027s because we imagine that we\u0027ve isolated y as a function of x,"},{"Start":"04:04.175 ","End":"04:05.930","Text":"and if y is a function of x,"},{"Start":"04:05.930 ","End":"04:08.750","Text":"what we\u0027re really doing is using the chain rule."},{"Start":"04:08.750 ","End":"04:11.250","Text":"I\u0027m differentiating it with respect to x,"},{"Start":"04:11.250 ","End":"04:14.990","Text":"x squared of course is just 2x and the expression in x is normal."},{"Start":"04:14.990 ","End":"04:16.460","Text":"But when we have y-squared,"},{"Start":"04:16.460 ","End":"04:18.230","Text":"we\u0027re really using the chain rule."},{"Start":"04:18.230 ","End":"04:19.760","Text":"We have something squared,"},{"Start":"04:19.760 ","End":"04:25.070","Text":"so it\u0027s twice that something times something prime because y is like the inner function."},{"Start":"04:25.070 ","End":"04:26.840","Text":"The same here, natural log of x,"},{"Start":"04:26.840 ","End":"04:29.030","Text":"1 over x, natural log of y."},{"Start":"04:29.030 ","End":"04:30.530","Text":"If y is a function of x,"},{"Start":"04:30.530 ","End":"04:33.900","Text":"so it\u0027s 1 over that function of x times the anti-derivative."},{"Start":"04:33.900 ","End":"04:36.770","Text":"But it\u0027s best to remember just mechanically."},{"Start":"04:36.770 ","End":"04:41.090","Text":"Just proceed, and whenever we have a bit with y in it,"},{"Start":"04:41.090 ","End":"04:44.105","Text":"then we put y prime alongside."},{"Start":"04:44.105 ","End":"04:46.415","Text":"Once we get to this point,"},{"Start":"04:46.415 ","End":"04:50.300","Text":"then what we do is we take all the expressions with"},{"Start":"04:50.300 ","End":"04:54.260","Text":"y prime and throw them to the left-hand side,"},{"Start":"04:54.260 ","End":"04:55.320","Text":"all the others on the right,"},{"Start":"04:55.320 ","End":"05:00.410","Text":"and then we take y prime outside and divide by it."},{"Start":"05:00.410 ","End":"05:03.980","Text":"I\u0027m not going to continue this exercise."},{"Start":"05:03.980 ","End":"05:10.460","Text":"I just want to emphasize that y-prime is the same as what we now call"},{"Start":"05:10.460 ","End":"05:14.000","Text":"the partial of y with respect to x. I\u0027m writing it"},{"Start":"05:14.000 ","End":"05:17.570","Text":"this way because I want to extend this idea to more variables."},{"Start":"05:17.570 ","End":"05:21.225","Text":"I want to take now, this idea,"},{"Start":"05:21.225 ","End":"05:22.970","Text":"an example with x, y,"},{"Start":"05:22.970 ","End":"05:26.705","Text":"and z. I\u0027m going to take the example,"},{"Start":"05:26.705 ","End":"05:31.160","Text":"x squared plus y squared plus"},{"Start":"05:31.160 ","End":"05:40.820","Text":"z squared equals xyz."},{"Start":"05:40.820 ","End":"05:45.140","Text":"I\u0027m going to use the same trick that worked here,"},{"Start":"05:45.140 ","End":"05:51.365","Text":"and that is every time I see an expression with z,"},{"Start":"05:51.365 ","End":"05:55.055","Text":"after I\u0027ve differentiated it, regular,"},{"Start":"05:55.055 ","End":"05:59.325","Text":"I add not z prime,"},{"Start":"05:59.325 ","End":"06:02.350","Text":"but z with respect to x."},{"Start":"06:02.660 ","End":"06:06.760","Text":"The other thing I should note really is that y is"},{"Start":"06:06.760 ","End":"06:10.880","Text":"treated like a constant if I\u0027m differentiating with respect to x."},{"Start":"06:10.880 ","End":"06:14.250","Text":"Expressions in x are just done normally."},{"Start":"06:14.250 ","End":"06:16.645","Text":"Let\u0027s continue with this 1."},{"Start":"06:16.645 ","End":"06:19.720","Text":"Differentiating with respect to x the variables in x are"},{"Start":"06:19.720 ","End":"06:23.785","Text":"just regular derivatives so x squared becomes 2x."},{"Start":"06:23.785 ","End":"06:30.944","Text":"Expression with y is neither x nor z, so it becomes 0,"},{"Start":"06:30.944 ","End":"06:38.145","Text":"and the expression in z is differentiated as if it was just regular like 2z,"},{"Start":"06:38.145 ","End":"06:49.280","Text":"but because it\u0027s z, I throw in the extra z with respect to x and this equals."},{"Start":"06:50.240 ","End":"06:56.035","Text":"By the way, what I\u0027m doing can be considered as taking the"},{"Start":"06:56.035 ","End":"07:02.495","Text":"derivative with respect to x of both sides of the equation,"},{"Start":"07:02.495 ","End":"07:09.535","Text":"and remembering that z is considered to be a function of x and y,"},{"Start":"07:09.535 ","End":"07:13.225","Text":"like an implicit function that I haven\u0027t managed to quite isolate."},{"Start":"07:13.225 ","End":"07:15.640","Text":"Let\u0027s go on with the other side."},{"Start":"07:15.640 ","End":"07:19.475","Text":"The other side is a product."},{"Start":"07:19.475 ","End":"07:21.130","Text":"y is a constant,"},{"Start":"07:21.130 ","End":"07:24.910","Text":"so I can bring it outside the parentheses,"},{"Start":"07:24.910 ","End":"07:27.640","Text":"but I still have x and z as a product,"},{"Start":"07:27.640 ","End":"07:36.529","Text":"so I\u0027ll just write x times z and its derivative with respect to x,"},{"Start":"07:36.529 ","End":"07:41.540","Text":"and this equals y times product rule."},{"Start":"07:41.540 ","End":"07:50.630","Text":"Derivative of x is 1 times z plus x times derivative of z is 1."},{"Start":"07:50.630 ","End":"07:56.225","Text":"But because it was z, I have to multiply by zx."},{"Start":"07:56.225 ","End":"07:59.890","Text":"Finally, what we can do is,"},{"Start":"07:59.890 ","End":"08:01.880","Text":"we can open up the brackets,"},{"Start":"08:01.880 ","End":"08:04.200","Text":"which I\u0027m not going to do."},{"Start":"08:04.220 ","End":"08:07.700","Text":"All the terms with zx go to the left,"},{"Start":"08:07.700 ","End":"08:10.925","Text":"everything else to the right we take zx outside the brackets."},{"Start":"08:10.925 ","End":"08:15.845","Text":"Shouldn\u0027t be difficult for you to isolate what zx equals."},{"Start":"08:15.845 ","End":"08:17.810","Text":"But I\u0027m not going to do it."},{"Start":"08:17.810 ","End":"08:19.940","Text":"Just wanted to get to the first stage,"},{"Start":"08:19.940 ","End":"08:22.595","Text":"so you can then just do normal algebra."},{"Start":"08:22.595 ","End":"08:25.840","Text":"I\u0027d like to do another example,"},{"Start":"08:25.840 ","End":"08:30.795","Text":"and then I\u0027ll express some rule of how to do this."},{"Start":"08:30.795 ","End":"08:35.310","Text":"The following example will be"},{"Start":"08:35.310 ","End":"08:42.050","Text":"x cubed plus y cubed"},{"Start":"08:42.050 ","End":"08:47.490","Text":"plus z cubed equals x^5,"},{"Start":"08:47.490 ","End":"08:52.170","Text":"plus y^5, plus z^5."},{"Start":"08:52.940 ","End":"08:57.680","Text":"What I want to inquire about is,"},{"Start":"08:57.680 ","End":"09:00.420","Text":"and I\u0027ll make it confusing,"},{"Start":"09:00.420 ","End":"09:05.915","Text":"this time I want the partial derivative of x with respect to z,"},{"Start":"09:05.915 ","End":"09:08.110","Text":"not z with respect to x."},{"Start":"09:08.110 ","End":"09:11.685","Text":"Using the same system,"},{"Start":"09:11.685 ","End":"09:16.740","Text":"what I do is z is the variable."},{"Start":"09:16.740 ","End":"09:19.930","Text":"When I write x with respect to z,"},{"Start":"09:19.930 ","End":"09:25.090","Text":"implies that I\u0027ve got x as a function of y and z."},{"Start":"09:25.090 ","End":"09:27.805","Text":"The 1 I\u0027m differentiating is a function of the others."},{"Start":"09:27.805 ","End":"09:30.985","Text":"I want its partial according to z."},{"Start":"09:30.985 ","End":"09:36.085","Text":"What this means is that y is treated as a constant."},{"Start":"09:36.085 ","End":"09:39.250","Text":"Each of the 3 variables is treated differently."},{"Start":"09:39.250 ","End":"09:45.535","Text":"Just as before when we differentiated if an expression involving z,"},{"Start":"09:45.535 ","End":"09:48.235","Text":"we threw in zx."},{"Start":"09:48.235 ","End":"09:51.805","Text":"In this case, when we have an expression involving x,"},{"Start":"09:51.805 ","End":"09:55.630","Text":"we\u0027ll multiply it by xz."},{"Start":"09:55.630 ","End":"10:02.260","Text":"Z will be differentiated just regular as if it was the independent variable."},{"Start":"10:02.260 ","End":"10:06.939","Text":"Expressions in y will become 0 because y is treated like a constant."},{"Start":"10:06.939 ","End":"10:08.560","Text":"We have an example of each,"},{"Start":"10:08.560 ","End":"10:11.215","Text":"so here\u0027s an expression with x."},{"Start":"10:11.215 ","End":"10:15.790","Text":"We derive it, we get 3x squared."},{"Start":"10:15.790 ","End":"10:23.005","Text":"But because it\u0027s x, we multiply by x by z or something."},{"Start":"10:23.005 ","End":"10:26.260","Text":"Now, y is just a constant,"},{"Start":"10:26.260 ","End":"10:33.295","Text":"so it\u0027s 0 and z just regular 3z squared."},{"Start":"10:33.295 ","End":"10:35.935","Text":"On the other side, very similar."},{"Start":"10:35.935 ","End":"10:38.530","Text":"5x to the 4,"},{"Start":"10:38.530 ","End":"10:45.430","Text":"but because it\u0027s x by z, we multiply by."},{"Start":"10:45.430 ","End":"10:50.274","Text":"The y again is a constant, so we get a 0."},{"Start":"10:50.274 ","End":"10:56.680","Text":"The expression in z is just regular 5z to the 4."},{"Start":"10:56.680 ","End":"10:58.930","Text":"I say sometimes zed sometimes zee,"},{"Start":"10:58.930 ","End":"11:01.730","Text":"forgive me English or American."},{"Start":"11:02.250 ","End":"11:10.015","Text":"At this point, we can easily isolate what xz,"},{"Start":"11:10.015 ","End":"11:11.980","Text":"and I\u0027m not going to do that,"},{"Start":"11:11.980 ","End":"11:14.560","Text":"but we can certainly get what it is."},{"Start":"11:14.560 ","End":"11:16.270","Text":"Perhaps I\u0027m repeating myself,"},{"Start":"11:16.270 ","End":"11:18.430","Text":"but I\u0027d like to just go over it again."},{"Start":"11:18.430 ","End":"11:19.990","Text":"When you have something like this,"},{"Start":"11:19.990 ","End":"11:24.250","Text":"and we do the implicit differentiation like in calculus 1,"},{"Start":"11:24.250 ","End":"11:28.690","Text":"it\u0027s as if we\u0027re differentiating both sides."},{"Start":"11:28.690 ","End":"11:35.989","Text":"Each of these I\u0027m differentiating partially with respect to z."},{"Start":"11:36.480 ","End":"11:41.995","Text":"But imagine that x is a function of y and z."},{"Start":"11:41.995 ","End":"11:46.135","Text":"Every time I see an expression with x and I differentiated dy and"},{"Start":"11:46.135 ","End":"11:51.085","Text":"xz alongside as I did here and as I did here,"},{"Start":"11:51.085 ","End":"11:55.525","Text":"every time I see an expression with y I treat it like a constant, so this comes out 0."},{"Start":"11:55.525 ","End":"12:01.855","Text":"When I see z, I just treat it as z. I mean, just differentiate regularly."},{"Start":"12:01.855 ","End":"12:06.970","Text":"Perhaps I could generalize this somehow schematically,"},{"Start":"12:06.970 ","End":"12:08.755","Text":"I don\u0027t know which variable is what."},{"Start":"12:08.755 ","End":"12:17.890","Text":"Let\u0027s say I want to find the partial of square with respect to triangle derivative,"},{"Start":"12:17.890 ","End":"12:24.475","Text":"just like previously I did z with respect to x and here I had x with respect to z."},{"Start":"12:24.475 ","End":"12:26.050","Text":"This variable is the square,"},{"Start":"12:26.050 ","End":"12:27.700","Text":"this is the triangle."},{"Start":"12:27.700 ","End":"12:29.560","Text":"Here this is the square,"},{"Start":"12:29.560 ","End":"12:31.090","Text":"this is the triangle."},{"Start":"12:31.090 ","End":"12:36.775","Text":"What I do is if I see a square, I differentiate,"},{"Start":"12:36.775 ","End":"12:44.935","Text":"and then I multiply by this thing,"},{"Start":"12:44.935 ","End":"12:48.970","Text":"like z with respect to x or vice versa."},{"Start":"12:48.970 ","End":"12:53.515","Text":"If I see this,"},{"Start":"12:53.515 ","End":"12:57.790","Text":"it\u0027s just regular differentiation."},{"Start":"12:57.790 ","End":"13:01.465","Text":"If I see another variable like the y,"},{"Start":"13:01.465 ","End":"13:03.340","Text":"then it\u0027s a constant,"},{"Start":"13:03.340 ","End":"13:07.885","Text":"so it\u0027s just like 0 because it was like a constant."},{"Start":"13:07.885 ","End":"13:11.110","Text":"That\u0027s sort of schematically how I would"},{"Start":"13:11.110 ","End":"13:15.070","Text":"explain how to differentiate some function of square,"},{"Start":"13:15.070 ","End":"13:19.180","Text":"circle, triangle, and maybe some other shapes."},{"Start":"13:19.180 ","End":"13:21.745","Text":"When I say circle, I mean, and other shapes too."},{"Start":"13:21.745 ","End":"13:23.350","Text":"I\u0027ll do an example."},{"Start":"13:23.350 ","End":"13:26.425","Text":"Tell you I\u0027ll use the same equation,"},{"Start":"13:26.425 ","End":"13:29.290","Text":"but instead of x by z,"},{"Start":"13:29.290 ","End":"13:35.979","Text":"this time I want y by x with the same example."},{"Start":"13:35.979 ","End":"13:43.975","Text":"This time I\u0027m going to consider y as a function of x and z."},{"Start":"13:43.975 ","End":"13:46.780","Text":"This is my y and this is my x."},{"Start":"13:46.780 ","End":"13:54.430","Text":"That means that whenever I see y I differentiate and multiply by y with respect to x."},{"Start":"13:54.430 ","End":"13:57.655","Text":"When I see x just regular."},{"Start":"13:57.655 ","End":"14:03.370","Text":"When I see z becomes 0 because it\u0027s a constant."},{"Start":"14:03.370 ","End":"14:07.120","Text":"Doing this again, except instead of a z here,"},{"Start":"14:07.120 ","End":"14:10.630","Text":"I would put an x here this time, I\u0027ll just copy it."},{"Start":"14:10.630 ","End":"14:16.495","Text":"X cubed plus y cubed plus z cubed equals x to the 5,"},{"Start":"14:16.495 ","End":"14:18.485","Text":"plus y to the 5,"},{"Start":"14:18.485 ","End":"14:22.095","Text":"plus z to the 5."},{"Start":"14:22.095 ","End":"14:28.500","Text":"Both sides, I\u0027ll take the derivative with respect to x,"},{"Start":"14:28.500 ","End":"14:31.140","Text":"where y is the main variable,"},{"Start":"14:31.140 ","End":"14:33.365","Text":"that\u0027s a function of the other 2."},{"Start":"14:33.365 ","End":"14:36.295","Text":"Again, with respect to x,"},{"Start":"14:36.295 ","End":"14:41.215","Text":"but y is the special 1 that when I differentiate it,"},{"Start":"14:41.215 ","End":"14:46.900","Text":"every time something in y, I have to multiply by y with respect to x."},{"Start":"14:46.900 ","End":"14:51.085","Text":"In this example, y is my square,"},{"Start":"14:51.085 ","End":"14:53.770","Text":"x is my triangle,"},{"Start":"14:53.770 ","End":"14:59.395","Text":"and z is this other variable or the other circle."},{"Start":"14:59.395 ","End":"15:05.140","Text":"Same here, triangle, square, circle."},{"Start":"15:05.140 ","End":"15:06.565","Text":"I hope this helps, If not,"},{"Start":"15:06.565 ","End":"15:10.150","Text":"then don\u0027t use it or make up your own scheme for remembering."},{"Start":"15:10.150 ","End":"15:17.950","Text":"Let\u0027s continue, x is this triangle is just regular, 3x squared."},{"Start":"15:17.950 ","End":"15:21.355","Text":"Y is my main variable,"},{"Start":"15:21.355 ","End":"15:25.390","Text":"when I differentiate it I\u0027ve to multiply it by y by x."},{"Start":"15:25.390 ","End":"15:31.690","Text":"So 3y squared times y by x,"},{"Start":"15:31.690 ","End":"15:33.715","Text":"I want to emphasize that."},{"Start":"15:33.715 ","End":"15:39.759","Text":"Plus z is 0, it\u0027s a constant."},{"Start":"15:39.759 ","End":"15:44.215","Text":"Then similar here, 5x to the 4,"},{"Start":"15:44.215 ","End":"15:47.170","Text":"just regular, y is the main variable,"},{"Start":"15:47.170 ","End":"15:48.550","Text":"it\u0027s a function of the other 2,"},{"Start":"15:48.550 ","End":"15:54.175","Text":"so 5y to the 4 times y with respect to x."},{"Start":"15:54.175 ","End":"15:56.110","Text":"Once again, z is a constant,"},{"Start":"15:56.110 ","End":"15:59.695","Text":"so its derivative becomes 0."},{"Start":"15:59.695 ","End":"16:02.920","Text":"Then we isolate y with respect to x,"},{"Start":"16:02.920 ","End":"16:07.400","Text":"we could bring things to the other side and you can do that."},{"Start":"16:08.100 ","End":"16:11.200","Text":"I want to get to some more example,"},{"Start":"16:11.200 ","End":"16:13.580","Text":"let\u0027s get a bit crazier."},{"Start":"16:13.890 ","End":"16:18.235","Text":"Let\u0027s change the letters altogether,"},{"Start":"16:18.235 ","End":"16:19.720","Text":"let\u0027s use a, b, and c,"},{"Start":"16:19.720 ","End":"16:22.165","Text":"which are normally used for constants."},{"Start":"16:22.165 ","End":"16:30.715","Text":"I\u0027m going to take, a squared, plus b squared,"},{"Start":"16:30.715 ","End":"16:36.745","Text":"plus c squared equals natural log of a plus"},{"Start":"16:36.745 ","End":"16:43.840","Text":"the log of b plus the log of c. What do I want to find?"},{"Start":"16:43.840 ","End":"16:49.150","Text":"I want the partial derivative of b with respect to c. Which"},{"Start":"16:49.150 ","End":"16:55.675","Text":"means that I\u0027m thinking of b as an implicit function of a and c,"},{"Start":"16:55.675 ","End":"16:58.810","Text":"but b is my main variable."},{"Start":"16:58.810 ","End":"17:01.870","Text":"In this case, b is,"},{"Start":"17:01.870 ","End":"17:07.510","Text":"what I would call, write a legend here."},{"Start":"17:07.510 ","End":"17:11.035","Text":"B is what I think of as the main variable."},{"Start":"17:11.035 ","End":"17:14.830","Text":"That\u0027s the square."},{"Start":"17:14.830 ","End":"17:21.730","Text":"The 1 I\u0027m with respect to the triangle is this c here and the other variable,"},{"Start":"17:21.730 ","End":"17:25.780","Text":"which is treated like a constant, is the a."},{"Start":"17:25.780 ","End":"17:28.600","Text":"That\u0027s the legend in this thing."},{"Start":"17:28.600 ","End":"17:31.660","Text":"I\u0027ll now proceed to use these rules."},{"Start":"17:31.660 ","End":"17:33.550","Text":"Okay. Things with a circle,"},{"Start":"17:33.550 ","End":"17:35.875","Text":"a constant, so I differentiate this."},{"Start":"17:35.875 ","End":"17:39.970","Text":"It becomes 0."},{"Start":"17:40.140 ","End":"17:45.580","Text":"I\u0027d like to usually stress that what I\u0027m doing is with each side of the equation,"},{"Start":"17:45.580 ","End":"17:52.480","Text":"I\u0027m differentiating with respect to c. Okay."},{"Start":"17:52.480 ","End":"17:55.435","Text":"Now, b is my square."},{"Start":"17:55.435 ","End":"17:59.230","Text":"I\u0027d first differentiate as normal to b,"},{"Start":"17:59.230 ","End":"18:04.495","Text":"but then I must multiply by b with respect to"},{"Start":"18:04.495 ","End":"18:11.140","Text":"c. C is just the independent variable."},{"Start":"18:11.140 ","End":"18:13.420","Text":"It\u0027s 2c, just regular."},{"Start":"18:13.420 ","End":"18:15.340","Text":"Now this equals."},{"Start":"18:15.340 ","End":"18:18.040","Text":"A is the constant,"},{"Start":"18:18.040 ","End":"18:20.710","Text":"so it becomes also 0."},{"Start":"18:20.710 ","End":"18:24.205","Text":"B is my main variable,"},{"Start":"18:24.205 ","End":"18:26.095","Text":"so it\u0027s 1 over b,"},{"Start":"18:26.095 ","End":"18:31.765","Text":"but times the derivative of b with respect to c. Here,"},{"Start":"18:31.765 ","End":"18:36.970","Text":"just 1 over c. What I\u0027m going to do,"},{"Start":"18:36.970 ","End":"18:41.140","Text":"well, I\u0027m not going to do rather is to isolate b with respect to c,"},{"Start":"18:41.140 ","End":"18:43.780","Text":"put the terms with this on 1 side,"},{"Start":"18:43.780 ","End":"18:45.790","Text":"the rest of the terms on the other side,"},{"Start":"18:45.790 ","End":"18:51.220","Text":"and extract just like in algebra, etc."},{"Start":"18:51.220 ","End":"18:54.160","Text":"B with respect to c equals da da da."},{"Start":"18:54.160 ","End":"18:59.980","Text":"Now, it\u0027s finally time to talk about second-order derivatives."},{"Start":"18:59.980 ","End":"19:03.280","Text":"Let\u0027s take yet another example."},{"Start":"19:03.280 ","End":"19:11.650","Text":"The example will be x squared plus y squared plus z"},{"Start":"19:11.650 ","End":"19:20.770","Text":"squared is equal to e^x plus e^y plus e^z."},{"Start":"19:20.770 ","End":"19:29.140","Text":"What I\u0027d like to find is the second-order derivative of z with respect to x,"},{"Start":"19:29.140 ","End":"19:32.815","Text":"with respect to x, and what is this equal."},{"Start":"19:32.815 ","End":"19:38.020","Text":"Of course, I\u0027ll first of all have to find z with respect to x,"},{"Start":"19:38.020 ","End":"19:40.600","Text":"the first-order derivative before I can get to this."},{"Start":"19:40.600 ","End":"19:46.600","Text":"I need to differentiate by x partially and I do this twice."},{"Start":"19:46.600 ","End":"19:49.960","Text":"We have to use the system that I\u0027ve taught you today."},{"Start":"19:49.960 ","End":"19:52.915","Text":"System from the previous clip will not work."},{"Start":"19:52.915 ","End":"19:55.060","Text":"Now because it\u0027s z with respect to x,"},{"Start":"19:55.060 ","End":"19:59.485","Text":"you have to remember that z is a function of x, y."},{"Start":"19:59.485 ","End":"20:01.750","Text":"Every time we see z,"},{"Start":"20:01.750 ","End":"20:06.050","Text":"an expression in z, we have to multiply it by z_x."},{"Start":"20:06.690 ","End":"20:12.100","Text":"X squared is like my triangle in the previous,"},{"Start":"20:12.100 ","End":"20:14.780","Text":"I just differentiated regular."},{"Start":"20:16.230 ","End":"20:20.470","Text":"Y is the constant, so it\u0027s 0."},{"Start":"20:20.470 ","End":"20:23.965","Text":"Z squared becomes 2z."},{"Start":"20:23.965 ","End":"20:29.530","Text":"But because it\u0027s with z, I have to multiply by Z_X."},{"Start":"20:29.530 ","End":"20:31.509","Text":"Z is the square,"},{"Start":"20:31.509 ","End":"20:35.800","Text":"and y is the circle,"},{"Start":"20:35.800 ","End":"20:38.185","Text":"and x is the triangle if you like."},{"Start":"20:38.185 ","End":"20:44.980","Text":"I\u0027m differentiating this with respect to x and this with respect to x to do this."},{"Start":"20:44.980 ","End":"20:46.540","Text":"Now, the other side."},{"Start":"20:46.540 ","End":"20:50.290","Text":"With respect to x, just regular, e^x."},{"Start":"20:50.290 ","End":"20:52.855","Text":"Expression in y, 0."},{"Start":"20:52.855 ","End":"21:00.350","Text":"Expression in z, differentiate regular but then multiply by z with respect to x."},{"Start":"21:02.430 ","End":"21:05.140","Text":"Now we want to isolate z_x,"},{"Start":"21:05.140 ","End":"21:06.640","Text":"and this time I really will do it."},{"Start":"21:06.640 ","End":"21:11.545","Text":"We move the expressions in z_x to the left and everything else to the right."},{"Start":"21:11.545 ","End":"21:21.250","Text":"I get 2z times z with respect to x minus e^z"},{"Start":"21:21.250 ","End":"21:26.290","Text":"z by"},{"Start":"21:26.290 ","End":"21:34.220","Text":"x equals e^x minus 2x."},{"Start":"21:35.550 ","End":"21:37.750","Text":"I\u0027ll do 2 steps in 1."},{"Start":"21:37.750 ","End":"21:42.305","Text":"I\u0027ll take z with respect to x outside the brackets,"},{"Start":"21:42.305 ","End":"21:46.270","Text":"but then I\u0027ll straight away send them to the other side."},{"Start":"21:48.600 ","End":"21:52.540","Text":"If I took z to the x outside the brackets,"},{"Start":"21:52.540 ","End":"21:57.755","Text":"I would be left here with 2z minus e^z."},{"Start":"21:57.755 ","End":"22:02.245","Text":"Those go down at the bottom here, 2z minus e^z."},{"Start":"22:02.245 ","End":"22:05.840","Text":"I just skipped a step, but your algebra should be good enough."},{"Start":"22:06.120 ","End":"22:09.505","Text":"Okay. For the grand finale,"},{"Start":"22:09.505 ","End":"22:12.835","Text":"we\u0027ll now go on to the second derivative;"},{"Start":"22:12.835 ","End":"22:15.280","Text":"z with respect to x,"},{"Start":"22:15.280 ","End":"22:18.370","Text":"with respect to x, z_xx."},{"Start":"22:18.370 ","End":"22:22.120","Text":"What I have to do is just differentiate this,"},{"Start":"22:22.120 ","End":"22:24.760","Text":"but like we do with implicit functions,"},{"Start":"22:24.760 ","End":"22:27.975","Text":"is that x is as is."},{"Start":"22:27.975 ","End":"22:30.015","Text":"I mean, we differentiate it regular."},{"Start":"22:30.015 ","End":"22:33.390","Text":"Every time I see a z and I differentiate it,"},{"Start":"22:33.390 ","End":"22:37.950","Text":"I have to multiply by z_x just like we did here."},{"Start":"22:37.950 ","End":"22:40.935","Text":"Notice that y has disappeared completely,"},{"Start":"22:40.935 ","End":"22:42.690","Text":"and so we don\u0027t have to worry about that,"},{"Start":"22:42.690 ","End":"22:45.210","Text":"but it would be treated as a constant if it were here."},{"Start":"22:45.210 ","End":"22:47.940","Text":"Okay, quotient rule."},{"Start":"22:47.940 ","End":"22:50.040","Text":"I like to start with the denominator."},{"Start":"22:50.040 ","End":"22:51.735","Text":"It\u0027s always the easiest,"},{"Start":"22:51.735 ","End":"22:57.400","Text":"2z minus e^z squared. That\u0027s this bit."},{"Start":"22:57.400 ","End":"23:07.915","Text":"Derivative of the numerator is e^x minus 2 times the denominator,"},{"Start":"23:07.915 ","End":"23:13.540","Text":"2z minus e^z as is,"},{"Start":"23:13.540 ","End":"23:16.070","Text":"I need to extend this,"},{"Start":"23:19.710 ","End":"23:23.605","Text":"minus, and then the other way round,"},{"Start":"23:23.605 ","End":"23:29.185","Text":"this thing as is, e^x minus 2x."},{"Start":"23:29.185 ","End":"23:33.190","Text":"The derivative of the denominator,"},{"Start":"23:33.190 ","End":"23:35.350","Text":"this is an expression in z."},{"Start":"23:35.350 ","End":"23:39.880","Text":"We have to take 2 minus e^z,"},{"Start":"23:39.880 ","End":"23:41.680","Text":"which is regular what we would do,"},{"Start":"23:41.680 ","End":"23:45.680","Text":"but also to multiply it by z_x."},{"Start":"23:46.980 ","End":"23:49.300","Text":"Extend this still more."},{"Start":"23:49.300 ","End":"23:53.605","Text":"There\u0027s only 1 more step that I need to do,"},{"Start":"23:53.605 ","End":"23:56.530","Text":"and that is to replace,"},{"Start":"23:56.530 ","End":"23:59.335","Text":"I\u0027ll highlight it, this."},{"Start":"23:59.335 ","End":"24:03.550","Text":"But I have what it\u0027s equal to from here."},{"Start":"24:03.550 ","End":"24:08.800","Text":"If I replace this with this whole expression,"},{"Start":"24:08.800 ","End":"24:15.115","Text":"then I\u0027ll finally be left with just z and x and no partials."},{"Start":"24:15.115 ","End":"24:16.720","Text":"I\u0027m not going to do it,"},{"Start":"24:16.720 ","End":"24:19.165","Text":"and basically we\u0027re done."},{"Start":"24:19.165 ","End":"24:22.210","Text":"I admit it\u0027s not so simple."},{"Start":"24:22.210 ","End":"24:26.740","Text":"You should practice with the exercises following this tutorial."},{"Start":"24:26.740 ","End":"24:29.120","Text":"Okay, that\u0027s all."}],"ID":8944},{"Watched":false,"Name":"Differentiation of Implicit Function example-1","Duration":"4m 39s","ChapterTopicVideoID":8664,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.064","Text":"This is the first of 5 example clips,"},{"Start":"00:04.064 ","End":"00:08.040","Text":"all in the subject of differentiation of implicit function."},{"Start":"00:08.040 ","End":"00:13.815","Text":"Our first example says that x squared plus y to the fifth equals xy plus 1."},{"Start":"00:13.815 ","End":"00:19.695","Text":"We want to look upon this as an implicit function of y with respect to x."},{"Start":"00:19.695 ","End":"00:23.970","Text":"What we\u0027re asked is to find the derivative y prime and"},{"Start":"00:23.970 ","End":"00:28.590","Text":"also its value when x is 0. Let\u0027s start."},{"Start":"00:28.590 ","End":"00:34.155","Text":"The first thing to do is to write this all on the left-hand side,"},{"Start":"00:34.155 ","End":"00:43.674","Text":"x squared plus y to the fifth minus x y minus 1 equals 0."},{"Start":"00:43.674 ","End":"00:47.960","Text":"Then this expression, we call it f of xy."},{"Start":"00:47.960 ","End":"00:50.580","Text":"I\u0027ll do it all on the same line."},{"Start":"00:52.610 ","End":"01:02.195","Text":"Now we use something called the implicit function theorem. Here it is."},{"Start":"01:02.195 ","End":"01:05.345","Text":"What it says is if we have an implicit function"},{"Start":"01:05.345 ","End":"01:09.230","Text":"of y with respect to x written in this form,"},{"Start":"01:09.230 ","End":"01:13.579","Text":"then the derivative is given by this expression,"},{"Start":"01:13.579 ","End":"01:17.120","Text":"the partial of f with respect to x"},{"Start":"01:17.120 ","End":"01:23.760","Text":"over the partial of f with respect to y and a minus in front."},{"Start":"01:23.870 ","End":"01:28.000","Text":"In case you haven\u0027t learned this or don\u0027t remember it,"},{"Start":"01:28.000 ","End":"01:32.315","Text":"I can briefly tell you why this formula works."},{"Start":"01:32.315 ","End":"01:36.110","Text":"Because a regular derivative is much like a partial derivative,"},{"Start":"01:36.110 ","End":"01:37.940","Text":"only this 1 variable."},{"Start":"01:37.940 ","End":"01:44.740","Text":"Y prime is like the partial derivative of y with respect to x."},{"Start":"01:44.740 ","End":"01:50.150","Text":"We already learned that when we have the partial of 1 variable with respect to another,"},{"Start":"01:50.150 ","End":"01:52.340","Text":"we put a minus fraction sign,"},{"Start":"01:52.340 ","End":"01:57.350","Text":"we take the function and the thing that was down here, we put here,"},{"Start":"01:57.350 ","End":"01:59.225","Text":"and the letter that was up here,"},{"Start":"01:59.225 ","End":"02:02.630","Text":"we put down here, totally mechanically."},{"Start":"02:02.630 ","End":"02:05.810","Text":"This gives us the same expression as here."},{"Start":"02:05.810 ","End":"02:12.155","Text":"We\u0027ll just use this when we have f of xy. Let\u0027s do it."},{"Start":"02:12.155 ","End":"02:17.300","Text":"What we get is that y prime is equal to,"},{"Start":"02:17.300 ","End":"02:18.655","Text":"I\u0027ll just copy this,"},{"Start":"02:18.655 ","End":"02:24.810","Text":"minus fx over fy,"},{"Start":"02:24.940 ","End":"02:30.350","Text":"and this is equal to minus f with respect"},{"Start":"02:30.350 ","End":"02:38.935","Text":"to x is 2x minus y over,"},{"Start":"02:38.935 ","End":"02:41.070","Text":"let\u0027s see with respect to y,"},{"Start":"02:41.070 ","End":"02:48.545","Text":"we get 5y to the fourth minus x."},{"Start":"02:48.545 ","End":"02:51.515","Text":"That answers the first part."},{"Start":"02:51.515 ","End":"02:53.735","Text":"This we\u0027ve done."},{"Start":"02:53.735 ","End":"02:55.535","Text":"Now the second part,"},{"Start":"02:55.535 ","End":"02:59.400","Text":"we have to substitute x equals 0."},{"Start":"02:59.410 ","End":"03:04.760","Text":"We have to find out what y is when x is 0."},{"Start":"03:04.760 ","End":"03:07.670","Text":"Otherwise we can\u0027t compute y prime."},{"Start":"03:07.670 ","End":"03:12.290","Text":"What we do is we just substitute x equals 0 in the original equation."},{"Start":"03:12.290 ","End":"03:14.600","Text":"If we put it in the original equation,"},{"Start":"03:14.600 ","End":"03:18.470","Text":"we get x is 0,"},{"Start":"03:18.470 ","End":"03:23.390","Text":"so we get 0 squared plus y to the"},{"Start":"03:23.390 ","End":"03:30.780","Text":"fifth equals 0 times y plus 1."},{"Start":"03:30.780 ","End":"03:35.510","Text":"This just leaves us that y to the fifth is equal to 1,"},{"Start":"03:35.510 ","End":"03:39.850","Text":"and so y equals 1."},{"Start":"03:39.850 ","End":"03:44.415","Text":"Now we know that when x is 0, y is 1."},{"Start":"03:44.415 ","End":"03:47.440","Text":"I\u0027m going to erase the scratch."},{"Start":"03:47.960 ","End":"03:55.650","Text":"When x equals 0, y equals 1."},{"Start":"03:55.650 ","End":"04:03.060","Text":"Now I can compute what y prime of 0 is,"},{"Start":"04:03.060 ","End":"04:08.825","Text":"it\u0027s equal to what we have here when x is 0 and y is 1,"},{"Start":"04:08.825 ","End":"04:16.165","Text":"so it\u0027s minus 2 times 0"},{"Start":"04:16.165 ","End":"04:23.000","Text":"minus 1 over 5"},{"Start":"04:23.000 ","End":"04:29.875","Text":"times 1 to the fourth minus 0."},{"Start":"04:29.875 ","End":"04:33.524","Text":"What we get here is the fifth,"},{"Start":"04:33.524 ","End":"04:37.200","Text":"and that answers this part also."},{"Start":"04:37.200 ","End":"04:40.210","Text":"That\u0027s basically it."}],"ID":8945},{"Watched":false,"Name":"Differentiation of Implicit Function example-2","Duration":"4m 48s","ChapterTopicVideoID":8665,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.375","Text":"This is example 2 out of 5."},{"Start":"00:03.375 ","End":"00:08.775","Text":"Differentiation of implicit functions, is the exercise,"},{"Start":"00:08.775 ","End":"00:15.600","Text":"e^xy, yes, it should have been x squared times y squared,"},{"Start":"00:15.600 ","End":"00:18.435","Text":"equals 5x minus 4."},{"Start":"00:18.435 ","End":"00:27.720","Text":"The question is, what is the implicit derivative of y when x equals 1?"},{"Start":"00:27.720 ","End":"00:31.320","Text":"We first of all put everything on the left-hand side."},{"Start":"00:31.320 ","End":"00:38.699","Text":"We have e^xy plus x squared,"},{"Start":"00:38.699 ","End":"00:43.710","Text":"y squared, minus the 5x,"},{"Start":"00:43.710 ","End":"00:47.565","Text":"plus 4, equals 0."},{"Start":"00:47.565 ","End":"00:49.685","Text":"The expression on the left,"},{"Start":"00:49.685 ","End":"00:52.660","Text":"we call it f of xy."},{"Start":"00:52.660 ","End":"00:59.490","Text":"Now, I\u0027m going to use the implicit function theorem."},{"Start":"00:59.620 ","End":"01:05.030","Text":"This says that when we have an implicit function in this form,"},{"Start":"01:05.030 ","End":"01:08.030","Text":"that y-prime is given by this formula."},{"Start":"01:08.030 ","End":"01:10.415","Text":"Let\u0027s do that here."},{"Start":"01:10.415 ","End":"01:18.535","Text":"We get that y-prime is equal to, minus,"},{"Start":"01:18.535 ","End":"01:21.679","Text":"derivative of f with respect to x"},{"Start":"01:21.679 ","End":"01:29.525","Text":"is e^xy times the inner derivative,"},{"Start":"01:29.525 ","End":"01:33.899","Text":"which is y, plus,"},{"Start":"01:33.899 ","End":"01:36.270","Text":"y squared is just a constant,"},{"Start":"01:36.270 ","End":"01:39.645","Text":"so it\u0027s 2xy squared,"},{"Start":"01:39.645 ","End":"01:43.515","Text":"and then minus 5,"},{"Start":"01:43.515 ","End":"01:53.045","Text":"all that over the derivative of f with respect to y."},{"Start":"01:53.045 ","End":"01:56.730","Text":"That is xe^xy"},{"Start":"01:58.720 ","End":"02:07.255","Text":"plus 2yx squared."},{"Start":"02:07.255 ","End":"02:12.225","Text":"Nothing else. This is our expression for y-prime."},{"Start":"02:12.225 ","End":"02:17.200","Text":"Now I have to substitute, x equals 1."},{"Start":"02:17.320 ","End":"02:20.990","Text":"But it\u0027s not enough for me to put in x equals 1"},{"Start":"02:20.990 ","End":"02:24.265","Text":"because y-prime is really a function of x and y,"},{"Start":"02:24.265 ","End":"02:27.060","Text":"so I have to find out what y is."},{"Start":"02:27.060 ","End":"02:34.505","Text":"The best thing to do is to just substitute this in the original equation."},{"Start":"02:34.505 ","End":"02:42.275","Text":"We get that e^xy, so it\u0027s e^y,"},{"Start":"02:42.275 ","End":"02:44.550","Text":"x is 1 everywhere,"},{"Start":"02:45.580 ","End":"02:54.275","Text":"plus y squared is equal to 5 minus 4, which is 1."},{"Start":"02:54.275 ","End":"02:57.005","Text":"Not immediately clear how to solve this."},{"Start":"02:57.005 ","End":"03:00.325","Text":"Let\u0027s add the condition that this function,"},{"Start":"03:00.325 ","End":"03:05.930","Text":"only applies for y bigger or equal to 0."},{"Start":"03:05.930 ","End":"03:12.140","Text":"That helps us, because the first thing you\u0027d want to try is y equals 0."},{"Start":"03:12.140 ","End":"03:14.810","Text":"If you check y equals 0,"},{"Start":"03:14.810 ","End":"03:19.140","Text":"e^0 is 1, 0 squared is 0, so it works."},{"Start":"03:19.140 ","End":"03:25.115","Text":"This gives us that y equals 0."},{"Start":"03:25.115 ","End":"03:27.020","Text":"There\u0027s no other solution,"},{"Start":"03:27.020 ","End":"03:29.380","Text":"because if there was another solution,"},{"Start":"03:29.380 ","End":"03:33.530","Text":"then it would be bigger than 0."},{"Start":"03:33.740 ","End":"03:38.610","Text":"These 2 functions are increasing."},{"Start":"03:38.610 ","End":"03:40.540","Text":"If y were bigger than 0,"},{"Start":"03:40.540 ","End":"03:43.500","Text":"this would be bigger than 1,"},{"Start":"03:43.500 ","End":"03:45.210","Text":"and this would be bigger than 0,"},{"Start":"03:45.210 ","End":"03:47.645","Text":"so all together we\u0027d get bigger than 1."},{"Start":"03:47.645 ","End":"03:53.890","Text":"Anyway, perhaps it\u0027s not quite fair to have you solve an equation like this,"},{"Start":"03:53.890 ","End":"03:58.210","Text":"but under the circumstances it was easy to spot y equals 0."},{"Start":"03:58.210 ","End":"04:01.420","Text":"Now that we have x and y,"},{"Start":"04:01.420 ","End":"04:04.645","Text":"now we can substitute and say,"},{"Start":"04:04.645 ","End":"04:10.090","Text":"y-prime of x equals 0, y equals 1."},{"Start":"04:10.090 ","End":"04:14.050","Text":"This is, we just substitute here."},{"Start":"04:14.050 ","End":"04:18.680","Text":"We get minus something over something."},{"Start":"04:18.950 ","End":"04:24.165","Text":"Y equals 0, so this is 0 and this is 0,"},{"Start":"04:24.165 ","End":"04:27.660","Text":"and this is just minus 5."},{"Start":"04:27.660 ","End":"04:32.565","Text":"On the denominator, this is 0,"},{"Start":"04:32.565 ","End":"04:41.445","Text":"x is 1, 1e^0 is just 1,"},{"Start":"04:41.445 ","End":"04:48.970","Text":"so the answer is equal to just 5. We are done."}],"ID":8946},{"Watched":false,"Name":"Differentiation of Implicit Function example-3","Duration":"7m 55s","ChapterTopicVideoID":8666,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:07.845","Text":"Now, we come to example 3 out of 5 with the differentiation of implicit functions."},{"Start":"00:07.845 ","End":"00:13.440","Text":"This time, our example will be 2 natural log of"},{"Start":"00:13.440 ","End":"00:21.735","Text":"x plus natural log of y is equal to 1."},{"Start":"00:21.735 ","End":"00:29.099","Text":"What we have to find this time is y prime of"},{"Start":"00:29.099 ","End":"00:37.990","Text":"e and y double-prime of e. Both of them."},{"Start":"00:38.560 ","End":"00:43.250","Text":"Here we consider y to be an implicit function of x."},{"Start":"00:43.250 ","End":"00:47.130","Text":"It\u0027s given in this form."},{"Start":"00:47.620 ","End":"00:50.990","Text":"Now, it just so happens that in this case,"},{"Start":"00:50.990 ","End":"00:53.224","Text":"I could isolate y."},{"Start":"00:53.224 ","End":"00:59.060","Text":"You could bring this to the other side and take e to the power of, it\u0027s possible."},{"Start":"00:59.060 ","End":"01:00.395","Text":"I\u0027m not saying it isn\u0027t,"},{"Start":"01:00.395 ","End":"01:04.280","Text":"but I would prefer still to do it with implicit functions."},{"Start":"01:04.280 ","End":"01:07.130","Text":"It\u0027s more convenient and besides that\u0027s what we\u0027re learning now."},{"Start":"01:07.130 ","End":"01:10.160","Text":"Let\u0027s do our usual method."},{"Start":"01:10.160 ","End":"01:13.865","Text":"Bring everything to 1 side and leave 0 on the other."},{"Start":"01:13.865 ","End":"01:25.050","Text":"We have 2 natural log of x plus natural log of y minus 1 is equal to 0."},{"Start":"01:25.090 ","End":"01:28.385","Text":"What\u0027s here on the left-hand side,"},{"Start":"01:28.385 ","End":"01:31.510","Text":"we call f of x, y."},{"Start":"01:31.510 ","End":"01:34.720","Text":"We have a function of 2 variables."},{"Start":"01:35.060 ","End":"01:40.420","Text":"Now, we\u0027re going to use the implicit function theorem."},{"Start":"01:40.420 ","End":"01:44.319","Text":"I\u0027ve pasted it over here."},{"Start":"01:44.590 ","End":"01:52.430","Text":"What we get is that y prime is equal"},{"Start":"01:52.430 ","End":"02:01.310","Text":"to minus f with respect to x over derivative of f with respect to y,"},{"Start":"02:01.310 ","End":"02:03.545","Text":"partial derivatives, of course."},{"Start":"02:03.545 ","End":"02:07.055","Text":"This is equal to minus."},{"Start":"02:07.055 ","End":"02:11.540","Text":"On the numerator, I take this differentiate with respect to"},{"Start":"02:11.540 ","End":"02:17.615","Text":"x. I get from here 2 over x and the rest of it,"},{"Start":"02:17.615 ","End":"02:22.700","Text":"nothing over with respect to y, x is a constant."},{"Start":"02:22.700 ","End":"02:25.430","Text":"This is nothing. This is 1 over y."},{"Start":"02:25.430 ","End":"02:33.350","Text":"Basically, I get minus 2y over x."},{"Start":"02:33.350 ","End":"02:40.550","Text":"Now, I\u0027d like to substitute x equals e,"},{"Start":"02:40.550 ","End":"02:45.635","Text":"but I can\u0027t just substitute x equals e I need to know what y is as well."},{"Start":"02:45.635 ","End":"02:51.140","Text":"I mean, what I need is y prime when x"},{"Start":"02:51.140 ","End":"02:56.675","Text":"is e and y equals,"},{"Start":"02:56.675 ","End":"02:58.400","Text":"well, that\u0027s what I don\u0027t know."},{"Start":"02:58.400 ","End":"03:00.440","Text":"What I\u0027ll do figure out."},{"Start":"03:00.440 ","End":"03:03.050","Text":"What is y when x is e?"},{"Start":"03:03.050 ","End":"03:09.170","Text":"I take the original equation and put x equals e in here."},{"Start":"03:09.170 ","End":"03:16.270","Text":"I have 2 natural log of e plus natural log of y equals 1."},{"Start":"03:16.270 ","End":"03:19.920","Text":"Natural log of e is 1."},{"Start":"03:19.920 ","End":"03:26.715","Text":"I get the natural log of y is minus 1,"},{"Start":"03:26.715 ","End":"03:31.650","Text":"which means that y is e^minus 1,"},{"Start":"03:31.650 ","End":"03:36.420","Text":"which is 1 over e. Now,"},{"Start":"03:36.420 ","End":"03:38.055","Text":"I can put that in here."},{"Start":"03:38.055 ","End":"03:42.775","Text":"I\u0027m looking for y prime when x equals e and y equals 1 over"},{"Start":"03:42.775 ","End":"03:48.605","Text":"e. That means that I can put these 2 in here."},{"Start":"03:48.605 ","End":"03:52.320","Text":"I get minus 2y,"},{"Start":"03:52.320 ","End":"04:01.649","Text":"that\u0027s minus 2 over e over x,"},{"Start":"04:01.649 ","End":"04:11.860","Text":"which is e. This gives me minus 2 over e squared."},{"Start":"04:11.860 ","End":"04:15.670","Text":"That\u0027s the answer to this 1."},{"Start":"04:15.670 ","End":"04:20.525","Text":"The next thing is we need to find out what y double-prime is."},{"Start":"04:20.525 ","End":"04:23.585","Text":"We need to differentiate."},{"Start":"04:23.585 ","End":"04:29.195","Text":"Now, y prime was equal to from here"},{"Start":"04:29.195 ","End":"04:37.325","Text":"minus 2y over x. I want to differentiate this with respect to x,"},{"Start":"04:37.325 ","End":"04:41.945","Text":"but I have to remember that y also depends on x. I\u0027ll just"},{"Start":"04:41.945 ","End":"04:47.210","Text":"remind ourselves by putting y of x like this,"},{"Start":"04:47.210 ","End":"04:52.460","Text":"that if I had extracted y explicitly then it would have been a function of x."},{"Start":"04:52.460 ","End":"04:58.055","Text":"Then we\u0027ve got to still imagine that y is a function of x, just implicitly defined."},{"Start":"04:58.055 ","End":"05:05.660","Text":"Y double prime equals using the quotient rule, and you know what?"},{"Start":"05:05.660 ","End":"05:11.035","Text":"I want to leave the minus 2 outside the brackets."},{"Start":"05:11.035 ","End":"05:13.305","Text":"I\u0027ll leave the minus 2 separately."},{"Start":"05:13.305 ","End":"05:15.840","Text":"It\u0027s like y of x over x."},{"Start":"05:15.840 ","End":"05:22.325","Text":"It\u0027s the derivative of the numerator,"},{"Start":"05:22.325 ","End":"05:29.360","Text":"y prime of x times the denominator minus the numerator as"},{"Start":"05:29.360 ","End":"05:37.775","Text":"is y of x times derivative of the denominator times 1,"},{"Start":"05:37.775 ","End":"05:41.855","Text":"all over the denominator squared."},{"Start":"05:41.855 ","End":"05:43.820","Text":"Close brackets."},{"Start":"05:43.820 ","End":"05:51.800","Text":"Y prime is equal to minus 2y over x is what y prime is,"},{"Start":"05:51.800 ","End":"05:58.655","Text":"times x minus y."},{"Start":"05:58.655 ","End":"06:06.480","Text":"I\u0027ve dropped the parentheses x at this point over x squared."},{"Start":"06:06.640 ","End":"06:10.850","Text":"Now, let\u0027s see. On the top,"},{"Start":"06:10.850 ","End":"06:13.645","Text":"I\u0027ve got x with x cancels,"},{"Start":"06:13.645 ","End":"06:21.855","Text":"minus 2y minus y is minus 3y together with the minus 2 is plus 6y."},{"Start":"06:21.855 ","End":"06:32.100","Text":"I get 6y over x squared, y double prime."},{"Start":"06:32.100 ","End":"06:40.830","Text":"When x equals e and y equals 1 over e,"},{"Start":"06:40.830 ","End":"06:46.520","Text":"and this equals, I\u0027m just substituting in here 6"},{"Start":"06:46.520 ","End":"06:53.330","Text":"times 1 over e divided by e squared,"},{"Start":"06:53.330 ","End":"06:59.585","Text":"which gives me 6 over e cubed."},{"Start":"06:59.585 ","End":"07:04.520","Text":"That\u0027s the answer to y double prime when x is e."},{"Start":"07:04.520 ","End":"07:11.600","Text":"I can now put a checkmark over here as well."},{"Start":"07:11.600 ","End":"07:17.370","Text":"We are done with example 3."},{"Start":"07:17.660 ","End":"07:22.995","Text":"But if you\u0027d like to stay there is something else I want to show you."},{"Start":"07:22.995 ","End":"07:26.315","Text":"There is an alternative way to proceed."},{"Start":"07:26.315 ","End":"07:29.105","Text":"When we get to this line here,"},{"Start":"07:29.105 ","End":"07:33.220","Text":"we don\u0027t have to substitute what y prime is."},{"Start":"07:33.220 ","End":"07:36.180","Text":"We can keep it as y prime."},{"Start":"07:36.180 ","End":"07:40.010","Text":"Then we could just substitute 3 things."},{"Start":"07:40.010 ","End":"07:42.245","Text":"We could substitute x equals e,"},{"Start":"07:42.245 ","End":"07:44.975","Text":"y equals 1 over e,"},{"Start":"07:44.975 ","End":"07:50.435","Text":"and y prime is equal to 2 over e squared."},{"Start":"07:50.435 ","End":"07:53.270","Text":"That\u0027s just an alternative approach."},{"Start":"07:53.270 ","End":"07:56.220","Text":"Now, I really, I\u0027m done."}],"ID":8947},{"Watched":false,"Name":"Differentiation of Implicit Function example-4","Duration":"6m 50s","ChapterTopicVideoID":8667,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.435","Text":"This is example number 4 out of 5,"},{"Start":"00:03.435 ","End":"00:06.525","Text":"on the differentiation of implicit functions."},{"Start":"00:06.525 ","End":"00:14.070","Text":"Here\u0027s this example, z squared, etc."},{"Start":"00:14.070 ","End":"00:19.400","Text":"We\u0027re given an extra condition that z is bigger or equal to 0."},{"Start":"00:19.400 ","End":"00:22.815","Text":"What we\u0027re asking for is this and this,"},{"Start":"00:22.815 ","End":"00:24.790","Text":"and let me explain."},{"Start":"00:24.790 ","End":"00:30.680","Text":"Z is considered an implicit function of 2 variables, x and y."},{"Start":"00:30.680 ","End":"00:32.345","Text":"If things had been different,"},{"Start":"00:32.345 ","End":"00:36.205","Text":"I might have been able to extract z as a function of x and y,"},{"Start":"00:36.205 ","End":"00:43.820","Text":"but nevertheless, we imagine that we have done that and that z is a function of x and y."},{"Start":"00:43.820 ","End":"00:49.510","Text":"Therefore, it has 2 partial derivatives with respect to x and with respect to y."},{"Start":"00:49.510 ","End":"00:54.530","Text":"These 0, 0 means I substitute x equals 0,"},{"Start":"00:54.530 ","End":"00:57.870","Text":"y equals 0, and likewise here."},{"Start":"01:01.940 ","End":"01:05.150","Text":"This is already equal to 0,"},{"Start":"01:05.150 ","End":"01:10.415","Text":"so there\u0027s no need to put stuff from the right side to the left side."},{"Start":"01:10.415 ","End":"01:15.110","Text":"What I want to do is call this by some function name f."},{"Start":"01:15.110 ","End":"01:16.850","Text":"In other words,"},{"Start":"01:16.850 ","End":"01:22.330","Text":"we let f of x and y equal whatever is written here;"},{"Start":"01:22.330 ","End":"01:32.110","Text":"z squared minus e^x squared plus y squared plus x plus y times sine z."},{"Start":"01:32.110 ","End":"01:34.910","Text":"Back to the standard methods of doing this,"},{"Start":"01:34.910 ","End":"01:40.250","Text":"where we say that the derivative of z with respect to"},{"Start":"01:40.250 ","End":"01:45.975","Text":"x is equal to minus f something over f something."},{"Start":"01:45.975 ","End":"01:47.655","Text":"The x goes here,"},{"Start":"01:47.655 ","End":"01:49.910","Text":"the z goes here."},{"Start":"01:49.910 ","End":"01:59.625","Text":"Similarly, with z with respect to y minus f something, f something."},{"Start":"01:59.625 ","End":"02:06.660","Text":"The y goes here and the z goes here."},{"Start":"02:06.660 ","End":"02:08.790","Text":"Let\u0027s actually do it,"},{"Start":"02:08.790 ","End":"02:15.755","Text":"so we get that z with respect to x is equal to f with respect to x,"},{"Start":"02:15.755 ","End":"02:18.980","Text":"means that y and z are constants."},{"Start":"02:18.980 ","End":"02:23.525","Text":"I have something here, with respect to x,"},{"Start":"02:23.525 ","End":"02:32.480","Text":"I get minus e^x squared plus y squared,"},{"Start":"02:32.480 ","End":"02:37.860","Text":"times inner derivative, which is 2x."},{"Start":"02:38.830 ","End":"02:41.660","Text":"What else do I have?"},{"Start":"02:41.660 ","End":"02:44.615","Text":"X I have it here."},{"Start":"02:44.615 ","End":"02:46.730","Text":"This is a constant,"},{"Start":"02:46.730 ","End":"02:51.870","Text":"so I just have to differentiate x plus y and that comes out to be 1."},{"Start":"02:52.280 ","End":"02:59.280","Text":"This from here I get just the 1 times sine z,"},{"Start":"02:59.280 ","End":"03:01.660","Text":"which is the constant,"},{"Start":"03:03.530 ","End":"03:08.690","Text":"before I forget it, I\u0027ll put the minus here also, over,"},{"Start":"03:08.690 ","End":"03:14.400","Text":"f with respect to z. I get 2z,"},{"Start":"03:15.260 ","End":"03:25.695","Text":"nothing from here, and x plus y is a constant, cosine z,"},{"Start":"03:25.695 ","End":"03:34.380","Text":"derivative of sine z with respect to z. I need the other 1 also,"},{"Start":"03:34.380 ","End":"03:41.940","Text":"z with respect to y minus,"},{"Start":"03:41.940 ","End":"03:45.330","Text":"now let\u0027s see, with respect to y,"},{"Start":"03:45.330 ","End":"03:48.310","Text":"very similar,"},{"Start":"03:49.760 ","End":"03:59.800","Text":"minus e^x squared plus y squared, but times 2y."},{"Start":"04:00.740 ","End":"04:03.030","Text":"The same thing here,"},{"Start":"04:03.030 ","End":"04:09.910","Text":"we\u0027ll get plus 1 times sine z."},{"Start":"04:10.610 ","End":"04:16.430","Text":"On the denominator, I get the same thing because it\u0027s the same f with respect to z,"},{"Start":"04:16.430 ","End":"04:22.640","Text":"so I get 2z plus x"},{"Start":"04:22.640 ","End":"04:29.790","Text":"plus y cosine z."},{"Start":"04:29.790 ","End":"04:31.370","Text":"Now all I have to do,"},{"Start":"04:31.370 ","End":"04:36.700","Text":"is to substitute x equals 0, y equals 0."},{"Start":"04:36.700 ","End":"04:41.605","Text":"But not quite, because I have to know what z is also."},{"Start":"04:41.605 ","End":"04:51.010","Text":"If I look at this and I put x equals 0 and y equals 0 into this equation,"},{"Start":"04:51.010 ","End":"04:56.930","Text":"I get that z squared minus,"},{"Start":"04:56.930 ","End":"05:00.880","Text":"x is 0, y is 0, so that\u0027s 0,"},{"Start":"05:00.880 ","End":"05:04.935","Text":"e^0 is 1 minus 1, plus,"},{"Start":"05:04.935 ","End":"05:08.430","Text":"x plus y is also 0,"},{"Start":"05:08.430 ","End":"05:11.415","Text":"so there is nothing else."},{"Start":"05:11.415 ","End":"05:13.630","Text":"This is equal to 0,"},{"Start":"05:13.630 ","End":"05:17.960","Text":"that was the original equation."},{"Start":"05:18.110 ","End":"05:21.910","Text":"So z squared equals 1,"},{"Start":"05:21.910 ","End":"05:24.175","Text":"and because it\u0027s bigger or equal to 0,"},{"Start":"05:24.175 ","End":"05:27.230","Text":"I get z equals 1."},{"Start":"05:27.230 ","End":"05:31.630","Text":"I substitute x equals 0,"},{"Start":"05:31.630 ","End":"05:37.310","Text":"y equals 0, z equals 1."},{"Start":"05:37.310 ","End":"05:39.975","Text":"Here also I\u0027ll do that."},{"Start":"05:39.975 ","End":"05:43.710","Text":"So we get, let\u0027s see,"},{"Start":"05:43.710 ","End":"05:46.605","Text":"x is 0, y is 0."},{"Start":"05:46.605 ","End":"05:49.185","Text":"Well, x is 0 this makes this whole term 0."},{"Start":"05:49.185 ","End":"05:59.324","Text":"Sine z gives us sine 1 radian over,"},{"Start":"05:59.324 ","End":"06:01.980","Text":"z is 1, x and y are 0,"},{"Start":"06:01.980 ","End":"06:03.660","Text":"so this thing\u0027s 0, z is 1,"},{"Start":"06:03.660 ","End":"06:06.090","Text":"so it\u0027s over 2."},{"Start":"06:06.090 ","End":"06:10.705","Text":"That means that we have done this 1,"},{"Start":"06:10.705 ","End":"06:16.135","Text":"v. With respect to y,"},{"Start":"06:16.135 ","End":"06:22.885","Text":"y is 0 and so I get this time,"},{"Start":"06:22.885 ","End":"06:25.465","Text":"also sine of 1,"},{"Start":"06:25.465 ","End":"06:30.790","Text":"z is 1, on the numerator and on the denominator,"},{"Start":"06:30.790 ","End":"06:34.090","Text":"again, I get sine of 1 over 2,"},{"Start":"06:34.090 ","End":"06:36.985","Text":"and that\u0027s this 1 also."},{"Start":"06:36.985 ","End":"06:40.070","Text":"We\u0027re done. Wait, don\u0027t go."},{"Start":"06:40.070 ","End":"06:45.170","Text":"I just noticed that I forgot to copy the minus here and here."},{"Start":"06:45.170 ","End":"06:47.270","Text":"Otherwise, we\u0027d have the wrong answer."},{"Start":"06:47.270 ","End":"06:49.025","Text":"This is a minus and this is a minus,"},{"Start":"06:49.025 ","End":"06:51.210","Text":"and now we\u0027re really done."}],"ID":8948},{"Watched":false,"Name":"Differentiation of Implicit Function example-5","Duration":"6m 41s","ChapterTopicVideoID":8668,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.730","Text":"This is the last of 5 examples regarding differentiation of implicit functions"},{"Start":"00:05.730 ","End":"00:12.735","Text":"and this time we have this implicit function very similar to the previous example."},{"Start":"00:12.735 ","End":"00:16.395","Text":"We are given a condition that y is bigger or equal to 0."},{"Start":"00:16.395 ","End":"00:22.215","Text":"We\u0027re asked to find the partial derivatives of y with respect to x and z."},{"Start":"00:22.215 ","End":"00:29.670","Text":"It\u0027s clear that we are supposed to think of y as the function of x and z,"},{"Start":"00:29.670 ","End":"00:31.335","Text":"but it given implicitly."},{"Start":"00:31.335 ","End":"00:36.760","Text":"Nevertheless, y is the variable which is the function of the other 2."},{"Start":"00:37.070 ","End":"00:40.174","Text":"We proceed as usual."},{"Start":"00:40.174 ","End":"00:44.525","Text":"We bring this to the other side and let it equals 0 and we let that"},{"Start":"00:44.525 ","End":"00:50.360","Text":"be f(x, y, and z)."},{"Start":"00:50.360 ","End":"00:52.085","Text":"I\u0027ll do it all in 1."},{"Start":"00:52.085 ","End":"00:56.720","Text":"I just have to copy this with the e^4 as a plus over here."},{"Start":"00:56.720 ","End":"01:02.705","Text":"The squared minus e^x squared plus y squared plus x plus"},{"Start":"01:02.705 ","End":"01:10.415","Text":"y sine of z plus e^4 equals 0."},{"Start":"01:10.415 ","End":"01:14.405","Text":"We should be pretty adapted these by now."},{"Start":"01:14.405 ","End":"01:20.865","Text":"We can say straight away that y with respect to x,"},{"Start":"01:20.865 ","End":"01:29.810","Text":"the derivative is equal to minus f. The x goes here over fy."},{"Start":"01:30.240 ","End":"01:36.910","Text":"We also know that y with respect to z is"},{"Start":"01:36.910 ","End":"01:48.120","Text":"minus fz over fy."},{"Start":"01:48.120 ","End":"01:57.770","Text":"Let\u0027s expand and we get y partial derivative with respect to x is equal to."},{"Start":"01:58.170 ","End":"02:00.880","Text":"I\u0027ll put the minus at the side,"},{"Start":"02:00.880 ","End":"02:03.165","Text":"I\u0027ll put a dividing line,"},{"Start":"02:03.165 ","End":"02:04.800","Text":"f with respect to"},{"Start":"02:04.800 ","End":"02:12.680","Text":"x. I\u0027ll put it"},{"Start":"02:12.680 ","End":"02:20.300","Text":"in a derivative first 2x e^x squared plus y squared."},{"Start":"02:20.300 ","End":"02:27.350","Text":"We have this, the derivative of this is just 1 times sine z,"},{"Start":"02:27.350 ","End":"02:29.150","Text":"which is a constant."},{"Start":"02:29.150 ","End":"02:36.440","Text":"It\u0027s plus sine z."},{"Start":"02:36.440 ","End":"02:38.845","Text":"Extend this a bit."},{"Start":"02:38.845 ","End":"02:42.889","Text":"On the denominator with respect to y,"},{"Start":"02:42.889 ","End":"02:44.870","Text":"we get something very similar."},{"Start":"02:44.870 ","End":"02:49.580","Text":"We get minus 2y e^x squared"},{"Start":"02:49.580 ","End":"02:55.490","Text":"plus y squared also plus sine of z."},{"Start":"02:55.490 ","End":"03:04.085","Text":"To compute this, we have to substitute x equals 0 and z equals 0,"},{"Start":"03:04.085 ","End":"03:05.720","Text":"but we also need y."},{"Start":"03:05.720 ","End":"03:07.940","Text":"Let me do this at the side."},{"Start":"03:07.940 ","End":"03:14.360","Text":"If x equals 0 and z equals 0,"},{"Start":"03:14.360 ","End":"03:16.460","Text":"I need to find out what y is."},{"Start":"03:16.460 ","End":"03:18.425","Text":"I plug that into here."},{"Start":"03:18.425 ","End":"03:22.670","Text":"I get this 1 is 0, x is 0,"},{"Start":"03:22.670 ","End":"03:27.750","Text":"so I get minus e^y squared."},{"Start":"03:33.800 ","End":"03:37.000","Text":"Z is 0, so sine z is 0,"},{"Start":"03:37.000 ","End":"03:46.570","Text":"so this is also 0 and equals minus e^4."},{"Start":"03:46.570 ","End":"03:50.020","Text":"Easily we get that y squared equals 4,"},{"Start":"03:50.020 ","End":"03:52.810","Text":"which point, y equals 2."},{"Start":"03:52.810 ","End":"03:54.370","Text":"It\u0027s bigger or equal to 0,"},{"Start":"03:54.370 ","End":"03:56.485","Text":"if I didn\u0027t take the minus 2."},{"Start":"03:56.485 ","End":"04:05.405","Text":"Now I have to substitute here that x is 0."},{"Start":"04:05.405 ","End":"04:07.700","Text":"Let me just make a note of that."},{"Start":"04:07.700 ","End":"04:10.865","Text":"I have x equals 0,"},{"Start":"04:10.865 ","End":"04:15.050","Text":"y equals 2, z equals 0."},{"Start":"04:15.050 ","End":"04:17.630","Text":"That\u0027s what I\u0027m going to substitute."},{"Start":"04:17.630 ","End":"04:22.130","Text":"From here I get x is 0,"},{"Start":"04:22.130 ","End":"04:23.930","Text":"so this thing cancels."},{"Start":"04:23.930 ","End":"04:26.550","Text":"Z is 0."},{"Start":"04:27.620 ","End":"04:30.990","Text":"This thing also cancels."},{"Start":"04:30.990 ","End":"04:35.960","Text":"This is just equal to 0 because if the numerator is 0,"},{"Start":"04:35.960 ","End":"04:38.810","Text":"I don\u0027t have to bother with the denominator."},{"Start":"04:38.810 ","End":"04:44.200","Text":"The next 1, y with respect to z,"},{"Start":"04:46.060 ","End":"04:48.710","Text":"it\u0027s the same denominator,"},{"Start":"04:48.710 ","End":"04:53.975","Text":"it\u0027s minus, dividing sign that upon the numerator,"},{"Start":"04:53.975 ","End":"04:59.060","Text":"I need the derivative of f with respect to z."},{"Start":"04:59.060 ","End":"05:06.300","Text":"So that gives me 2z. Nothing from here."},{"Start":"05:10.430 ","End":"05:15.860","Text":"Sine z and x plus y are constants,"},{"Start":"05:15.860 ","End":"05:18.485","Text":"so it\u0027s x plus y cosine z."},{"Start":"05:18.485 ","End":"05:21.395","Text":"Let\u0027s see what this equals."},{"Start":"05:21.395 ","End":"05:23.780","Text":"Now I have to substitute x is 0,"},{"Start":"05:23.780 ","End":"05:26.450","Text":"y is 2z is 0 here,"},{"Start":"05:26.450 ","End":"05:33.050","Text":"a 0 and 0 plus 2 is 2,"},{"Start":"05:33.050 ","End":"05:35.465","Text":"cosine 0 is 1."},{"Start":"05:35.465 ","End":"05:40.020","Text":"This is 2 over,"},{"Start":"05:40.780 ","End":"05:44.720","Text":"let\u0027s see now, y is 2,"},{"Start":"05:44.720 ","End":"05:47.290","Text":"so that\u0027s minus 4."},{"Start":"05:47.290 ","End":"05:55.010","Text":"Minus 4e^4."},{"Start":"05:57.080 ","End":"06:04.540","Text":"Minus 4e^4 plus sine of 0 is also 0."},{"Start":"06:04.540 ","End":"06:06.945","Text":"This is what I get."},{"Start":"06:06.945 ","End":"06:16.900","Text":"I get minus 1/2 e^ minus 4."},{"Start":"06:16.990 ","End":"06:19.730","Text":"Oops, I forgot this minus."},{"Start":"06:19.730 ","End":"06:26.170","Text":"We have a minus here and all that does is change this minus to a plus."},{"Start":"06:26.170 ","End":"06:29.330","Text":"Here\u0027s the answer, or if you prefer it,"},{"Start":"06:29.330 ","End":"06:34.535","Text":"you could write it as 1 over 2 e^4."},{"Start":"06:34.535 ","End":"06:37.700","Text":"I\u0027m not sure if this is better than this, whatever."},{"Start":"06:37.700 ","End":"06:42.269","Text":"Anyway, we\u0027re done with this last example."}],"ID":8949},{"Watched":false,"Name":"Exercise 1","Duration":"4m 16s","ChapterTopicVideoID":8671,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.730","Text":"In this exercise, we\u0027re given an equation with x and y."},{"Start":"00:05.730 ","End":"00:09.780","Text":"We\u0027re assuming that this defines y as an implicit function of x."},{"Start":"00:09.780 ","End":"00:13.575","Text":"In other words, y is some function of x."},{"Start":"00:13.575 ","End":"00:18.510","Text":"We have to find y prime in general using implicit differentiation."},{"Start":"00:18.510 ","End":"00:22.945","Text":"Then to compute y prime at the point when x is 0,"},{"Start":"00:22.945 ","End":"00:25.520","Text":"we have to do an implicit differentiation because there\u0027s"},{"Start":"00:25.520 ","End":"00:28.670","Text":"no easy way to extract y in terms of x."},{"Start":"00:28.670 ","End":"00:31.880","Text":"It\u0027s a fifth degree equation in y and so on."},{"Start":"00:31.880 ","End":"00:33.875","Text":"Let\u0027s do the implicit."},{"Start":"00:33.875 ","End":"00:37.775","Text":"Usually, it\u0027s customary to take everything to the left and make it equal to 0."},{"Start":"00:37.775 ","End":"00:44.460","Text":"We get x squared plus y^5 minus x y minus 1 equals 0."},{"Start":"00:44.460 ","End":"00:46.395","Text":"Then we do the differentiation."},{"Start":"00:46.395 ","End":"00:49.155","Text":"With respect to x, we get 2x."},{"Start":"00:49.155 ","End":"00:51.120","Text":"Now y is a function of x."},{"Start":"00:51.120 ","End":"00:53.640","Text":"This is not just y,"},{"Start":"00:53.640 ","End":"00:58.225","Text":"5y^4, but we also have to multiply by the derivative of y."},{"Start":"00:58.225 ","End":"01:04.005","Text":"Here minus x y we need a product rule,"},{"Start":"01:04.005 ","End":"01:07.025","Text":"so the derivative of x y,"},{"Start":"01:07.025 ","End":"01:14.300","Text":"derivative of x times y plus x times the derivative of y."},{"Start":"01:14.300 ","End":"01:17.240","Text":"That\u0027s this; the derivative of minus 1 is"},{"Start":"01:17.240 ","End":"01:21.370","Text":"0 and the derivative of the other side is also 0."},{"Start":"01:21.370 ","End":"01:25.185","Text":"Let\u0027s collect terms with y prime."},{"Start":"01:25.185 ","End":"01:30.640","Text":"What I have is, let\u0027s see, 5y^4."},{"Start":"01:31.700 ","End":"01:38.010","Text":"From here, minus x, all this y prime."},{"Start":"01:38.010 ","End":"01:41.685","Text":"Then without the y prime, we have plus"},{"Start":"01:41.685 ","End":"01:49.860","Text":"2x and then minus y equals 0."},{"Start":"01:49.860 ","End":"01:53.310","Text":"Now what we do is we extract y prime,"},{"Start":"01:53.310 ","End":"01:55.630","Text":"we let y prime equal."},{"Start":"01:55.630 ","End":"01:58.009","Text":"I\u0027ll bring this over to the other side."},{"Start":"01:58.009 ","End":"01:59.630","Text":"If I bring it to the other side,"},{"Start":"01:59.630 ","End":"02:01.790","Text":"it becomes plus y and minus 2x,"},{"Start":"02:01.790 ","End":"02:04.400","Text":"I\u0027ll write it as y minus 2x,"},{"Start":"02:04.400 ","End":"02:06.800","Text":"but then I divide by this."},{"Start":"02:06.800 ","End":"02:10.640","Text":"Let\u0027s not worry about whether we\u0027re dividing by 0 or not."},{"Start":"02:10.640 ","End":"02:13.355","Text":"Don\u0027t go into the technical,"},{"Start":"02:13.355 ","End":"02:15.685","Text":"just assume everything is okay."},{"Start":"02:15.685 ","End":"02:21.320","Text":"Divided by 5y^4 minus x."},{"Start":"02:21.320 ","End":"02:24.330","Text":"Now we have y prime, but it\u0027s not in terms of x,"},{"Start":"02:24.330 ","End":"02:29.765","Text":"it\u0027s in terms of y and x is just causes that extra little bit of difficulty."},{"Start":"02:29.765 ","End":"02:35.250","Text":"Because although this is the answer for finding y prime,"},{"Start":"02:35.250 ","End":"02:37.954","Text":"what we need to find y prime of 0."},{"Start":"02:37.954 ","End":"02:40.685","Text":"It means we substitute x equals 0."},{"Start":"02:40.685 ","End":"02:44.570","Text":"Y prime is 0. I know what to substitute for x,"},{"Start":"02:44.570 ","End":"02:47.640","Text":"but what do I substitute for y?"},{"Start":"02:48.190 ","End":"02:51.845","Text":"You need to do this at the side when x equals 0,"},{"Start":"02:51.845 ","End":"02:54.215","Text":"I need to know what y equals."},{"Start":"02:54.215 ","End":"02:55.670","Text":"There might not be an answer,"},{"Start":"02:55.670 ","End":"02:58.490","Text":"there might be more than 1 answer because what we do is we let x"},{"Start":"02:58.490 ","End":"03:01.345","Text":"equals 0 in this equation or in this 1."},{"Start":"03:01.345 ","End":"03:03.125","Text":"Then we try to solve it for y."},{"Start":"03:03.125 ","End":"03:05.580","Text":"Let\u0027s say I plug it in here."},{"Start":"03:06.160 ","End":"03:16.795","Text":"Into here, I would get 0 squared plus y^5 minus 0 times y minus 1 equals 0."},{"Start":"03:16.795 ","End":"03:21.560","Text":"Well, obviously the 0 here and 0 here don\u0027t contribute anything."},{"Start":"03:21.560 ","End":"03:25.135","Text":"Then I get y^5 equals 1."},{"Start":"03:25.135 ","End":"03:27.760","Text":"There is only 1 number^5 is 1."},{"Start":"03:27.760 ","End":"03:30.185","Text":"There\u0027s no plus or minus because it\u0027s an odd number."},{"Start":"03:30.185 ","End":"03:32.900","Text":"It necessarily Y equals 1."},{"Start":"03:32.900 ","End":"03:39.305","Text":"What I have to do here is to substitute x equals 0 and y equals 1."},{"Start":"03:39.305 ","End":"03:42.575","Text":"If we got no solution or more than 1 solution,"},{"Start":"03:42.575 ","End":"03:45.600","Text":"or we\u0027ll see this in future exercises anyway."},{"Start":"03:46.690 ","End":"03:57.290","Text":"Y equals 1, so I get 1 minus twice 0 over 5 times 1^4 minus 0."},{"Start":"03:57.290 ","End":"04:00.665","Text":"This is 0, this is 0, it\u0027s 1 over 5."},{"Start":"04:00.665 ","End":"04:03.600","Text":"This is equal to 1/5."},{"Start":"04:03.800 ","End":"04:07.050","Text":"We found everything we needed."},{"Start":"04:07.050 ","End":"04:13.395","Text":"This is y prime and this is y prime of 0."},{"Start":"04:13.395 ","End":"04:16.270","Text":"That\u0027s what we had to compute, so we\u0027re done."}],"ID":8950},{"Watched":false,"Name":"Exercise 2","Duration":"5m 57s","ChapterTopicVideoID":8672,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we\u0027re given this equation which"},{"Start":"00:04.320 ","End":"00:09.210","Text":"relates y and x and we\u0027re assuming that this puts y as an implicit function of x,"},{"Start":"00:09.210 ","End":"00:11.205","Text":"y equals y of x."},{"Start":"00:11.205 ","End":"00:16.425","Text":"That being the case, we want to know what is y prime when x is 1."},{"Start":"00:16.425 ","End":"00:20.170","Text":"We can\u0027t directly extract y as a function of x."},{"Start":"00:20.170 ","End":"00:22.695","Text":"We\u0027ll have to do an implicit differentiation."},{"Start":"00:22.695 ","End":"00:24.585","Text":"Let me rewrite it."},{"Start":"00:24.585 ","End":"00:29.340","Text":"Usually what we do is we put everything on the left-hand side."},{"Start":"00:29.340 ","End":"00:34.065","Text":"That will be minus 5x plus 4 equals 0."},{"Start":"00:34.065 ","End":"00:36.230","Text":"Then we do the implicit differentiation,"},{"Start":"00:36.230 ","End":"00:38.330","Text":"remembering that y is a function of x."},{"Start":"00:38.330 ","End":"00:41.600","Text":"So e to the xy gives us e to the xy,"},{"Start":"00:41.600 ","End":"00:44.020","Text":"but it\u0027s an inner derivative,"},{"Start":"00:44.020 ","End":"00:48.279","Text":"and inner derivative, well, it\u0027s a product."},{"Start":"00:48.279 ","End":"00:57.740","Text":"The product we take the derivative of x times y plus x times the derivative of y,"},{"Start":"00:57.740 ","End":"00:59.765","Text":"which is y prime."},{"Start":"00:59.765 ","End":"01:05.419","Text":"Then here we also have a product."},{"Start":"01:05.419 ","End":"01:15.670","Text":"See we have 2x times y squared plus x squared times 2y is 2x squared y,"},{"Start":"01:15.670 ","End":"01:19.400","Text":"but that\u0027s not all because y is a function of x,"},{"Start":"01:19.400 ","End":"01:22.815","Text":"so that is y prime."},{"Start":"01:22.815 ","End":"01:27.920","Text":"All this was x squared times the derivative of y squared,"},{"Start":"01:27.920 ","End":"01:30.770","Text":"which is 2yy prime. That\u0027s this term."},{"Start":"01:30.770 ","End":"01:34.640","Text":"Next term, with respect to x is just minus"},{"Start":"01:34.640 ","End":"01:39.380","Text":"5 and then nothing else because the 4 is a constant, so this equals 0."},{"Start":"01:39.380 ","End":"01:42.155","Text":"What we do is we look where there is y prime,"},{"Start":"01:42.155 ","End":"01:44.420","Text":"and we take that outside the brackets."},{"Start":"01:44.420 ","End":"01:52.110","Text":"We have xe to the xy times y prime,"},{"Start":"01:52.110 ","End":"01:53.730","Text":"I\u0027ll leave the y prime for the moment,"},{"Start":"01:53.730 ","End":"01:58.230","Text":"plus 2x squared y,"},{"Start":"01:58.230 ","End":"02:00.455","Text":"all this y prime,"},{"Start":"02:00.455 ","End":"02:04.400","Text":"and now the terms without y prime,"},{"Start":"02:04.400 ","End":"02:06.040","Text":"that would be this one."},{"Start":"02:06.040 ","End":"02:13.549","Text":"That\u0027s ye to the xy"},{"Start":"02:13.549 ","End":"02:23.820","Text":"plus 2xy squared minus 5."},{"Start":"02:23.960 ","End":"02:29.585","Text":"Then all this equals 0. Then if I want y prime,"},{"Start":"02:29.585 ","End":"02:32.300","Text":"I\u0027ll just bring this over to the other side,"},{"Start":"02:32.300 ","End":"02:37.205","Text":"but instead of reversing everything let me just put minus of whatever this is,"},{"Start":"02:37.205 ","End":"02:45.860","Text":"which is ye to the xy plus 2xy squared minus 5 over this,"},{"Start":"02:45.860 ","End":"02:52.180","Text":"which is xe to the xy plus 2x squared y,"},{"Start":"02:52.180 ","End":"02:54.875","Text":"instead of reversing the signs."},{"Start":"02:54.875 ","End":"02:56.780","Text":"I also want to show you another way of getting to"},{"Start":"02:56.780 ","End":"03:00.725","Text":"this point that I might choose to use in future."},{"Start":"03:00.725 ","End":"03:06.870","Text":"If we think of this as an equation of the form f of xy equals 0,"},{"Start":"03:06.870 ","End":"03:08.685","Text":"where this is f of xy,"},{"Start":"03:08.685 ","End":"03:11.295","Text":"then the implicit differentiation,"},{"Start":"03:11.295 ","End":"03:17.210","Text":"instead of doing all this, we\u0027ve got a formula that y prime is minus f with"},{"Start":"03:17.210 ","End":"03:24.940","Text":"respect to x over f with respect to y."},{"Start":"03:27.110 ","End":"03:32.645","Text":"This is what we get here. If we take this and differentiate it with respect to x,"},{"Start":"03:32.645 ","End":"03:34.190","Text":"where y is a constant,"},{"Start":"03:34.190 ","End":"03:37.640","Text":"we get here ye to the xy."},{"Start":"03:37.640 ","End":"03:40.100","Text":"From here we get, with respect to x,"},{"Start":"03:40.100 ","End":"03:43.710","Text":"2xy squared and from here we get the minus"},{"Start":"03:43.710 ","End":"03:48.960","Text":"5 and so on with the denominator being this with respect to y."},{"Start":"03:50.050 ","End":"03:54.205","Text":"If we do it this way, it will give us the same result."},{"Start":"03:54.205 ","End":"03:57.600","Text":"Now the question asks for y prime of 1,"},{"Start":"03:57.600 ","End":"04:02.045","Text":"this is y prime for a general x in terms of x and y."},{"Start":"04:02.045 ","End":"04:04.790","Text":"So y prime of 1, for this,"},{"Start":"04:04.790 ","End":"04:08.895","Text":"we know that x is 1, but what is y?"},{"Start":"04:08.895 ","End":"04:14.325","Text":"We need to know what is y of 1 when x equals 1,"},{"Start":"04:14.325 ","End":"04:16.755","Text":"what does y equal?"},{"Start":"04:16.755 ","End":"04:23.565","Text":"We substitute x equals 1 in the original equation with this one probably better."},{"Start":"04:23.565 ","End":"04:27.020","Text":"If I let x equals 1 here,"},{"Start":"04:27.150 ","End":"04:32.210","Text":"then I get e to the y,"},{"Start":"04:32.930 ","End":"04:35.160","Text":"plus x is 1,"},{"Start":"04:35.160 ","End":"04:37.200","Text":"so it\u0027s just y squared,"},{"Start":"04:37.200 ","End":"04:39.945","Text":"from here minus 5,"},{"Start":"04:39.945 ","End":"04:46.750","Text":"from here plus 4 equals 0."},{"Start":"04:47.780 ","End":"04:54.215","Text":"I can rewrite this as e to the y plus y squared."},{"Start":"04:54.215 ","End":"04:56.660","Text":"So to the minus 5 plus 1 is minus 1,"},{"Start":"04:56.660 ","End":"04:59.255","Text":"so I bring it on the other side, equals 1."},{"Start":"04:59.255 ","End":"05:01.670","Text":"It\u0027s not clear how to solve this,"},{"Start":"05:01.670 ","End":"05:04.295","Text":"but guesswork actually, will do it."},{"Start":"05:04.295 ","End":"05:08.750","Text":"Just try y equals 0 and that does it."},{"Start":"05:08.750 ","End":"05:11.585","Text":"It turns out that this is the only solution."},{"Start":"05:11.585 ","End":"05:18.825","Text":"What I do is, I take this here and I let x equals 1,"},{"Start":"05:18.825 ","End":"05:28.410","Text":"y equals 0, wherever y is 0,"},{"Start":"05:28.410 ","End":"05:31.080","Text":"that makes this 0,"},{"Start":"05:31.080 ","End":"05:35.280","Text":"this will be 0 minus 5."},{"Start":"05:35.280 ","End":"05:39.885","Text":"Here, y is 0, e to the 0 is 1,"},{"Start":"05:39.885 ","End":"05:43.170","Text":"so this is 1, y is 0,"},{"Start":"05:43.170 ","End":"05:45.330","Text":"that\u0027s 0. What do I get?"},{"Start":"05:45.330 ","End":"05:50.280","Text":"Minus, minus 5/1, I make that 5."},{"Start":"05:50.280 ","End":"05:52.815","Text":"That\u0027s what we were asked for,"},{"Start":"05:52.815 ","End":"05:58.300","Text":"y prime of 1, and that\u0027s equal to 5, and we\u0027re done."}],"ID":8951},{"Watched":false,"Name":"Exercise 3","Duration":"9m 9s","ChapterTopicVideoID":8673,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"In this exercise, this actually is an equation in x and y,"},{"Start":"00:04.950 ","End":"00:10.420","Text":"but we wanted to represent y as an implicit function of x."},{"Start":"00:10.850 ","End":"00:15.300","Text":"We have to find the derivative"},{"Start":"00:15.300 ","End":"00:22.935","Text":"of y at the point where x equals e and also second derivative at the same point."},{"Start":"00:22.935 ","End":"00:28.200","Text":"Let\u0027s start by doing an implicit differentiation."},{"Start":"00:28.200 ","End":"00:34.200","Text":"I\u0027m going to do the method where we had everything go on the left and 0 on the right."},{"Start":"00:34.200 ","End":"00:41.720","Text":"So I\u0027ve got 2 natural log x plus natural log y minus 1 equals 0."},{"Start":"00:41.720 ","End":"00:46.580","Text":"I left space here because I want to call this a function of x and y."},{"Start":"00:46.580 ","End":"00:49.340","Text":"When it\u0027s in this form, the formula that"},{"Start":"00:49.340 ","End":"00:55.430","Text":"y-prime of this function y in terms of x is just equal to, in general,"},{"Start":"00:55.430 ","End":"00:58.685","Text":"minus fx over fy,"},{"Start":"00:58.685 ","End":"01:01.280","Text":"and in our case, let\u0027s see,"},{"Start":"01:01.280 ","End":"01:11.505","Text":"f with respect to x is just 2 over x because y and 1 are constants,"},{"Start":"01:11.505 ","End":"01:14.400","Text":"and with respect to y, this and this are constants,"},{"Start":"01:14.400 ","End":"01:16.860","Text":"so it\u0027s just 1 over y."},{"Start":"01:16.860 ","End":"01:19.800","Text":"In other words, we get that y-prime is,"},{"Start":"01:19.800 ","End":"01:21.060","Text":"let\u0027s see if we simplify this,"},{"Start":"01:21.060 ","End":"01:26.260","Text":"minus 2y over x."},{"Start":"01:26.300 ","End":"01:33.630","Text":"Okay. What is y-prime when x is e?"},{"Start":"01:33.630 ","End":"01:36.975","Text":"When x is e, we can substitute x, but what is y?"},{"Start":"01:36.975 ","End":"01:42.680","Text":"We have to do a little calculation to figure out what y of e is."},{"Start":"01:43.070 ","End":"01:47.305","Text":"We let x equals e in this equation."},{"Start":"01:47.305 ","End":"01:48.985","Text":"If x equals e,"},{"Start":"01:48.985 ","End":"01:59.875","Text":"we get that 2 natural log of e plus natural log of y equals 1."},{"Start":"01:59.875 ","End":"02:05.565","Text":"But natural log of e is 1."},{"Start":"02:05.565 ","End":"02:11.070","Text":"This is e^1."},{"Start":"02:11.070 ","End":"02:17.700","Text":"This gives us that natural log of y,"},{"Start":"02:17.700 ","End":"02:20.760","Text":"if I take 2 times 1 to the other side,"},{"Start":"02:20.760 ","End":"02:23.360","Text":"it\u0027s 1 minus 2 is minus 1."},{"Start":"02:23.360 ","End":"02:26.765","Text":"So y is e to the minus 1,"},{"Start":"02:26.765 ","End":"02:32.845","Text":"but I prefer to write it as 1 over e. When x is e,"},{"Start":"02:32.845 ","End":"02:41.645","Text":"y is 1 over e. What I do is I just substitute x equals e,"},{"Start":"02:41.645 ","End":"02:48.360","Text":"y equals 1 over e into here,"},{"Start":"02:48.360 ","End":"02:53.595","Text":"and so y-prime of e is minus 2 times 1 over"},{"Start":"02:53.595 ","End":"03:02.430","Text":"e divided by e, yeah."},{"Start":"03:02.430 ","End":"03:05.460","Text":"This gives us what?"},{"Start":"03:05.460 ","End":"03:08.770","Text":"Minus 2 over e squared."},{"Start":"03:08.960 ","End":"03:12.730","Text":"That\u0027s the first thing found."},{"Start":"03:12.920 ","End":"03:15.270","Text":"Let me just highlight that."},{"Start":"03:15.270 ","End":"03:17.660","Text":"That\u0027s y-prime of e that we wanted."},{"Start":"03:17.660 ","End":"03:20.270","Text":"Now we need to go for y double prime."},{"Start":"03:20.270 ","End":"03:24.665","Text":"I just want to emphasize y-prime as a function of y and x."},{"Start":"03:24.665 ","End":"03:28.550","Text":"This is what I want to differentiate down here."},{"Start":"03:28.550 ","End":"03:31.385","Text":"So I get y double prime equals,"},{"Start":"03:31.385 ","End":"03:34.070","Text":"just using the product rule."},{"Start":"03:34.070 ","End":"03:37.745","Text":"I\u0027ll leave the minus out front and then I have a quotient."},{"Start":"03:37.745 ","End":"03:46.470","Text":"In fact, I\u0027ll even leave the minus 2 out front and then I\u0027ll have y over x derivative."},{"Start":"03:46.470 ","End":"03:48.345","Text":"Now it\u0027s a quotient."},{"Start":"03:48.345 ","End":"03:51.365","Text":"What I do is use the quotient rule."},{"Start":"03:51.365 ","End":"03:55.190","Text":"It\u0027s the derivative of the numerator,"},{"Start":"03:55.190 ","End":"04:01.410","Text":"which is y-prime times the denominator as is,"},{"Start":"04:01.410 ","End":"04:04.500","Text":"minus the numerator as is,"},{"Start":"04:04.500 ","End":"04:09.180","Text":"times the derivative of the denominator,"},{"Start":"04:09.260 ","End":"04:14.570","Text":"which is 1, and all of this over the denominator squared."},{"Start":"04:14.570 ","End":"04:17.280","Text":"I\u0027m assuming you know the product rule."},{"Start":"04:17.810 ","End":"04:25.830","Text":"Now, we were asked to y double prime of e. That means that x is e. We have to substitute."},{"Start":"04:25.830 ","End":"04:28.385","Text":"I have 3 things to substitute;"},{"Start":"04:28.385 ","End":"04:30.275","Text":"x equals e, fine."},{"Start":"04:30.275 ","End":"04:38.600","Text":"We\u0027ve discovered that y equals 1 over e. But we also need y-prime when x is e,"},{"Start":"04:38.600 ","End":"04:42.205","Text":"and y-prime equals, I\u0027m just copying it from here,"},{"Start":"04:42.205 ","End":"04:46.370","Text":"and so this is minus 2 over e squared."},{"Start":"04:46.370 ","End":"04:49.880","Text":"All 3 of these, I substitute into here to get this."},{"Start":"04:49.880 ","End":"04:51.620","Text":"So we have minus 2."},{"Start":"04:51.620 ","End":"04:56.400","Text":"Then dividing line, y-prime minus 2 over e squared"},{"Start":"04:56.400 ","End":"05:02.265","Text":"x is e minus y 1 over e,"},{"Start":"05:02.265 ","End":"05:04.230","Text":"and then the denominator x squared,"},{"Start":"05:04.230 ","End":"05:06.265","Text":"which is e squared."},{"Start":"05:06.265 ","End":"05:11.305","Text":"Let\u0027s see. We can simplify this."},{"Start":"05:11.305 ","End":"05:13.630","Text":"This is equal to,"},{"Start":"05:13.630 ","End":"05:18.150","Text":"now I can take the minus,"},{"Start":"05:18.150 ","End":"05:21.300","Text":"this 2 minuses with this minus,"},{"Start":"05:21.300 ","End":"05:25.570","Text":"and basically just canceling all the minuses."},{"Start":"05:25.970 ","End":"05:31.240","Text":"I get, that\u0027s a 2 over e squared from this and this,"},{"Start":"05:31.240 ","End":"05:33.385","Text":"and what I\u0027m left with is the numerator."},{"Start":"05:33.385 ","End":"05:40.610","Text":"I have 2e over e squared."},{"Start":"05:42.720 ","End":"05:45.730","Text":"This cancels also."},{"Start":"05:45.730 ","End":"05:51.405","Text":"Because I can cancel the e in the numerator with 1 of the e\u0027s in the denominator."},{"Start":"05:51.405 ","End":"05:54.185","Text":"So this is just 2 over e,"},{"Start":"05:54.185 ","End":"06:00.770","Text":"and it\u0027s going to be plus 1 over e. This is"},{"Start":"06:00.770 ","End":"06:09.615","Text":"just 3 over e. It\u0027s 2 over e,"},{"Start":"06:09.615 ","End":"06:12.125","Text":"this is a plus 1 over e,"},{"Start":"06:12.125 ","End":"06:14.850","Text":"we\u0027ve got 3 over e. If I multiply out,"},{"Start":"06:14.850 ","End":"06:21.075","Text":"we get 6 over e cubed."},{"Start":"06:21.075 ","End":"06:26.970","Text":"That answers about y double prime of e. This is equal to this."},{"Start":"06:26.970 ","End":"06:31.390","Text":"We\u0027ve found both pieces of information we needed."},{"Start":"06:31.390 ","End":"06:33.495","Text":"I could say we\u0027re done."},{"Start":"06:33.495 ","End":"06:35.990","Text":"But I\u0027d just like to show you an alternative method."},{"Start":"06:35.990 ","End":"06:39.355","Text":"You can call it quits here if you like or you can stay."},{"Start":"06:39.355 ","End":"06:44.795","Text":"This is 1 of those cases where we actually can extract y in terms of x."},{"Start":"06:44.795 ","End":"06:48.035","Text":"If I took the original equation,"},{"Start":"06:48.035 ","End":"06:53.540","Text":"I could write natural log of y is equal"},{"Start":"06:53.540 ","End":"07:00.270","Text":"to 1 minus twice natural log of x."},{"Start":"07:00.270 ","End":"07:03.185","Text":"If the natural log of this is this,"},{"Start":"07:03.185 ","End":"07:09.845","Text":"that means that y is e to the power of 1 minus twice natural log of x."},{"Start":"07:09.845 ","End":"07:14.510","Text":"This can be simplified using the rules of exponents."},{"Start":"07:14.510 ","End":"07:24.480","Text":"First of all, I can write e^1 over e to the twice natural log of x."},{"Start":"07:25.390 ","End":"07:35.400","Text":"Well, this is e. This is e to power of natural log of x, all squared."},{"Start":"07:35.400 ","End":"07:37.175","Text":"Again, using rules of exponents."},{"Start":"07:37.175 ","End":"07:41.520","Text":"This bit here is just x using the rules."},{"Start":"07:41.520 ","End":"07:44.240","Text":"Exponents and logarithms cancel each other out."},{"Start":"07:44.240 ","End":"07:50.190","Text":"So what I get is y equals e over x squared,"},{"Start":"07:50.990 ","End":"07:54.110","Text":"and if that\u0027s the case,"},{"Start":"07:54.110 ","End":"07:59.490","Text":"then I can differentiate y-prime."},{"Start":"07:59.490 ","End":"08:01.170","Text":"This is actually the minus 2,"},{"Start":"08:01.170 ","End":"08:06.855","Text":"so I get minus 2e over x cubed."},{"Start":"08:06.855 ","End":"08:12.510","Text":"y double, since this is x to the minus 3 and minus 3 to the minus 2 is 6 ,"},{"Start":"08:12.510 ","End":"08:18.270","Text":"this is 6e over x^4."},{"Start":"08:18.270 ","End":"08:23.845","Text":"Now, we can substitute because this gives us,"},{"Start":"08:23.845 ","End":"08:26.340","Text":"let\u0027s see from here,"},{"Start":"08:26.340 ","End":"08:34.270","Text":"we get that y- prime of e is just minus 2e over e cubed,"},{"Start":"08:34.270 ","End":"08:40.325","Text":"which gives us minus 2 over e squared after we cancel an e from the top and bottom."},{"Start":"08:40.325 ","End":"08:49.275","Text":"Here, y double prime of e would be 6e over e^4,"},{"Start":"08:49.275 ","End":"08:53.395","Text":"which is to cancel e is 6 over e cubed."},{"Start":"08:53.395 ","End":"08:56.720","Text":"Notice that this is the same as this,"},{"Start":"08:56.720 ","End":"08:59.510","Text":"and that this is the same as this."},{"Start":"08:59.510 ","End":"09:02.495","Text":"Actually, it turned out easier."},{"Start":"09:02.495 ","End":"09:06.425","Text":"It\u0027s not always possible and you don\u0027t always know which is easier."},{"Start":"09:06.425 ","End":"09:09.720","Text":"We did it both ways. Now really done."}],"ID":8952},{"Watched":false,"Name":"Exercise 4","Duration":"7m 46s","ChapterTopicVideoID":8674,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.975","Text":"In this exercise, we\u0027re given z is a function of x,"},{"Start":"00:03.975 ","End":"00:05.880","Text":"y but it\u0027s not given explicitly,"},{"Start":"00:05.880 ","End":"00:10.215","Text":"it\u0027s given implicitly by this formula here."},{"Start":"00:10.215 ","End":"00:14.100","Text":"We\u0027re also told that z is bigger or equal to 0."},{"Start":"00:14.100 ","End":"00:16.020","Text":"We\u0027ll see where this comes in later."},{"Start":"00:16.020 ","End":"00:20.670","Text":"We have to compute the 2 partial derivatives of z with respect to x and with respect to"},{"Start":"00:20.670 ","End":"00:29.760","Text":"y. I might decide to call them z with respect to x and z with respect to y,"},{"Start":"00:29.760 ","End":"00:33.780","Text":"rather than the funny d notation."},{"Start":"00:33.780 ","End":"00:36.120","Text":"I\u0027ll choose to use these."},{"Start":"00:36.120 ","End":"00:42.885","Text":"Now, we happen to have this in the form F of x,"},{"Start":"00:42.885 ","End":"00:46.425","Text":"y, and z equals 0."},{"Start":"00:46.425 ","End":"00:50.195","Text":"We\u0027d bring everything to the left-hand side and leave 0 on the right."},{"Start":"00:50.195 ","End":"00:53.120","Text":"Now when the function is given in this form,"},{"Start":"00:53.120 ","End":"00:56.074","Text":"something with x, y, and z equals 0,"},{"Start":"00:56.074 ","End":"01:01.770","Text":"there is a formula that dz by dx, well,"},{"Start":"01:01.770 ","End":"01:03.495","Text":"let\u0027s just use these 2,"},{"Start":"01:03.495 ","End":"01:06.710","Text":"z with respect to x is equal to,"},{"Start":"01:06.710 ","End":"01:10.055","Text":"there\u0027s a minus, there\u0027s a dividing line."},{"Start":"01:10.055 ","End":"01:16.460","Text":"Now, what we do is we take F in both cases in top and bottom and the thing to"},{"Start":"01:16.460 ","End":"01:19.490","Text":"remember is that the 1 at the bottom goes on"},{"Start":"01:19.490 ","End":"01:22.955","Text":"the top and the 1 at the top goes on the bottom."},{"Start":"01:22.955 ","End":"01:25.225","Text":"This is the formula."},{"Start":"01:25.225 ","End":"01:28.230","Text":"We\u0027ll use it later for z, y,"},{"Start":"01:28.230 ","End":"01:38.480","Text":"and this is equal to minus dividing line F with respect to x. F is just this thing here."},{"Start":"01:38.480 ","End":"01:43.785","Text":"All this is what I called f of x, y, and z."},{"Start":"01:43.785 ","End":"01:48.260","Text":"We take this bit and with respect to x,"},{"Start":"01:48.260 ","End":"01:50.815","Text":"z squared is nothing."},{"Start":"01:50.815 ","End":"01:53.070","Text":"With respect to x,"},{"Start":"01:53.070 ","End":"02:00.095","Text":"I get still e^x squared plus y squared but the inner derivative of this thing is 2x,"},{"Start":"02:00.095 ","End":"02:06.450","Text":"and then longer, then x plus y,"},{"Start":"02:06.450 ","End":"02:08.930","Text":"the derivative of that is just 1."},{"Start":"02:08.930 ","End":"02:14.510","Text":"But the constant sine z just sticks, sine z."},{"Start":"02:14.510 ","End":"02:17.600","Text":"Then on the denominator with respect to z,"},{"Start":"02:17.600 ","End":"02:21.900","Text":"I\u0027ve got 2z and then minus,"},{"Start":"02:21.900 ","End":"02:24.300","Text":"well, this is nothing, it has no z in it."},{"Start":"02:24.300 ","End":"02:27.380","Text":"Here the sine becomes cosine,"},{"Start":"02:27.380 ","End":"02:29.810","Text":"but this constant sticks,"},{"Start":"02:29.810 ","End":"02:35.200","Text":"x plus y the constant, times cosine z."},{"Start":"02:36.790 ","End":"02:43.955","Text":"Now here we\u0027re going to substitute x equals 0 and y equals 0"},{"Start":"02:43.955 ","End":"02:47.930","Text":"but that\u0027s not enough because I also need to know"},{"Start":"02:47.930 ","End":"02:52.260","Text":"what z equals and I don\u0027t immediately have that."},{"Start":"02:52.260 ","End":"02:55.520","Text":"What we do is we put x equals 0,"},{"Start":"02:55.520 ","End":"02:56.930","Text":"y equals 0 here,"},{"Start":"02:56.930 ","End":"03:05.505","Text":"which is really here and we get a relationship or an equation where we have,"},{"Start":"03:05.505 ","End":"03:06.905","Text":"I\u0027ll do it at the side,"},{"Start":"03:06.905 ","End":"03:13.080","Text":"z squared minus e to the power, x and y are 0,"},{"Start":"03:13.080 ","End":"03:16.920","Text":"so that is 0 plus and x and y are 0,"},{"Start":"03:16.920 ","End":"03:21.270","Text":"so this is 0 times whatever is 0."},{"Start":"03:21.270 ","End":"03:25.290","Text":"All this equals 0."},{"Start":"03:25.290 ","End":"03:29.970","Text":"Well, 0 is nothing and e^0 is 1,"},{"Start":"03:29.970 ","End":"03:34.305","Text":"so we get that z squared equals 1."},{"Start":"03:34.305 ","End":"03:37.320","Text":"Now, you see where the bigger or equal to 0 comes"},{"Start":"03:37.320 ","End":"03:40.115","Text":"in because now I can say that z equals 1."},{"Start":"03:40.115 ","End":"03:45.590","Text":"I don\u0027t need the minus 1 because z is bigger or equal to 0."},{"Start":"03:45.590 ","End":"03:51.450","Text":"Now, I can write in here that z equals 1 and all these 3 values,"},{"Start":"03:51.450 ","End":"03:58.230","Text":"I plug in to here and this will give me what is zx of 0, 0."},{"Start":"03:58.230 ","End":"04:02.220","Text":"Let\u0027s see, x is 0 and y is 0."},{"Start":"04:02.220 ","End":"04:05.630","Text":"Well, this makes this thing 0 because x is 0."},{"Start":"04:05.630 ","End":"04:08.675","Text":"We\u0027re just left with here sine z."},{"Start":"04:08.675 ","End":"04:13.385","Text":"Sine z is the dividing line and a minus,"},{"Start":"04:13.385 ","End":"04:15.305","Text":"and at the bottom,"},{"Start":"04:15.305 ","End":"04:18.690","Text":"we are left with 2z,"},{"Start":"04:22.060 ","End":"04:28.125","Text":"sorry, I should have replaced z equals 1."},{"Start":"04:28.125 ","End":"04:32.770","Text":"Sorry, this erase, this erase,"},{"Start":"04:32.770 ","End":"04:36.530","Text":"here 1, here 1."},{"Start":"04:36.530 ","End":"04:39.260","Text":"Now, what else?"},{"Start":"04:39.260 ","End":"04:42.950","Text":"The bottom, x is 0 and y is 0,"},{"Start":"04:42.950 ","End":"04:46.600","Text":"this is just minus 0 plus 0,"},{"Start":"04:46.600 ","End":"04:49.270","Text":"I\u0027ll just plus 0, whatever."},{"Start":"04:49.270 ","End":"04:58.760","Text":"Anyway, this is equal to minus sine 1 over 2."},{"Start":"04:58.760 ","End":"05:00.765","Text":"That\u0027s the first bit,"},{"Start":"05:00.765 ","End":"05:02.995","Text":"we\u0027ve just wrote it in this notation,"},{"Start":"05:02.995 ","End":"05:05.110","Text":"minus sine 1 over 2."},{"Start":"05:05.110 ","End":"05:07.480","Text":"Now, let\u0027s do the other 1."},{"Start":"05:07.480 ","End":"05:10.660","Text":"The other 1\u0027s going to be easier because this bit"},{"Start":"05:10.660 ","End":"05:13.690","Text":"is going to be reusable for the next 1."},{"Start":"05:13.690 ","End":"05:20.375","Text":"In any event, z with respect to y in general equals minus a dividing line,"},{"Start":"05:20.375 ","End":"05:22.670","Text":"F at the top, F at the bottom."},{"Start":"05:22.670 ","End":"05:30.065","Text":"Now, this y goes up here and this z goes down here."},{"Start":"05:30.065 ","End":"05:33.350","Text":"Perhaps I\u0027ll use some coloring."},{"Start":"05:33.350 ","End":"05:36.095","Text":"The variable here goes here,"},{"Start":"05:36.095 ","End":"05:39.125","Text":"similarly, the variable here goes here."},{"Start":"05:39.125 ","End":"05:46.220","Text":"The variable here goes down here and the variable here goes down here."},{"Start":"05:46.220 ","End":"05:49.410","Text":"There\u0027s an F and an F and a minus in all cases."},{"Start":"05:50.930 ","End":"05:53.180","Text":"This time we get,"},{"Start":"05:53.180 ","End":"05:54.980","Text":"in fact, we\u0027ve got the same denominator."},{"Start":"05:54.980 ","End":"05:58.685","Text":"We have the minus the denominator I can just copy,"},{"Start":"05:58.685 ","End":"06:05.280","Text":"which is 2z plus x plus y cosine z."},{"Start":"06:05.280 ","End":"06:07.155","Text":"That bit is reusable also."},{"Start":"06:07.155 ","End":"06:10.580","Text":"I just need derivative with respect to y,"},{"Start":"06:10.580 ","End":"06:13.265","Text":"so I look over here,"},{"Start":"06:13.265 ","End":"06:17.225","Text":"with respect to y, z squared is still 0."},{"Start":"06:17.225 ","End":"06:19.760","Text":"This thing comes out just like this,"},{"Start":"06:19.760 ","End":"06:21.380","Text":"except as a y here instead,"},{"Start":"06:21.380 ","End":"06:27.830","Text":"minus e to the x squared plus y squared times 2y."},{"Start":"06:27.830 ","End":"06:33.280","Text":"Then the same thing here, plus sine z."},{"Start":"06:35.290 ","End":"06:39.465","Text":"When I substitute, I can just copy from here."},{"Start":"06:39.465 ","End":"06:42.315","Text":"I did the copy paste here."},{"Start":"06:42.315 ","End":"06:49.410","Text":"Now let\u0027s compute this at the point z,"},{"Start":"06:49.410 ","End":"06:51.045","Text":"y of 0, 0."},{"Start":"06:51.045 ","End":"06:55.325","Text":"The substitution of course is intended because of this."},{"Start":"06:55.325 ","End":"06:59.885","Text":"We get, let\u0027s see,"},{"Start":"06:59.885 ","End":"07:06.334","Text":"minus the denominator\u0027s going to come out the same."},{"Start":"07:06.334 ","End":"07:11.420","Text":"In fact, I didn\u0027t even need to copy this because is the same denominator,"},{"Start":"07:11.420 ","End":"07:13.865","Text":"so that bit comes out to be 2,"},{"Start":"07:13.865 ","End":"07:17.080","Text":"and the numerator, let\u0027s see,"},{"Start":"07:17.080 ","End":"07:19.840","Text":"y is 0, so this is 0 minus 0."},{"Start":"07:19.840 ","End":"07:21.800","Text":"I get sine of z,"},{"Start":"07:21.800 ","End":"07:27.330","Text":"which is same thing again as we had before, sine 1."},{"Start":"07:28.100 ","End":"07:30.255","Text":"We found the other 1,"},{"Start":"07:30.255 ","End":"07:32.250","Text":"z, y at the point 0,"},{"Start":"07:32.250 ","End":"07:37.125","Text":"0 is just, I\u0027ll put the minus up here,"},{"Start":"07:37.125 ","End":"07:40.665","Text":"here tidier, and that\u0027s equal to this."},{"Start":"07:40.665 ","End":"07:42.695","Text":"We found both of what we were looking for,"},{"Start":"07:42.695 ","End":"07:47.070","Text":"but we just wrote them in this form. We are done."}],"ID":8953},{"Watched":false,"Name":"Exercise 5","Duration":"9m ","ChapterTopicVideoID":8675,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.530 ","End":"00:05.460","Text":"This exercise is quite similar to a previous 1 we had."},{"Start":"00:05.460 ","End":"00:07.920","Text":"There, it was 0 on the right,"},{"Start":"00:07.920 ","End":"00:10.560","Text":"but the major difference was, there,"},{"Start":"00:10.560 ","End":"00:15.345","Text":"we had z as a function of x and y and here,"},{"Start":"00:15.345 ","End":"00:17.805","Text":"y is a function of x and z."},{"Start":"00:17.805 ","End":"00:20.490","Text":"You wouldn\u0027t know this from just looking at that,"},{"Start":"00:20.490 ","End":"00:23.610","Text":"which is a function of the other, but here it is y,"},{"Start":"00:23.610 ","End":"00:29.460","Text":"which is the dependent variable, you can say."},{"Start":"00:29.460 ","End":"00:34.080","Text":"We have to compute the 2 partial derivatives with respect to x and with respect to z"},{"Start":"00:34.080 ","End":"00:40.330","Text":"at the place where x is 0 and z is 0."},{"Start":"00:44.000 ","End":"00:51.710","Text":"I\u0027ll start by writing this as some F of x, y equals 0 basically,"},{"Start":"00:51.710 ","End":"00:54.140","Text":"sorry, F of x, y, z."},{"Start":"00:54.140 ","End":"00:58.205","Text":"If I just bring e^4 to the other side,"},{"Start":"00:58.205 ","End":"01:04.320","Text":"I have that this is equal to z squared minus e^x squared plus y squared,"},{"Start":"01:04.320 ","End":"01:12.245","Text":"I\u0027m basically just copying, x plus y sine z plus e^4,"},{"Start":"01:12.245 ","End":"01:13.775","Text":"just bring this over."},{"Start":"01:13.775 ","End":"01:17.989","Text":"Now, of course, this is F equals 0,"},{"Start":"01:17.989 ","End":"01:21.155","Text":"gives us our implicit function."},{"Start":"01:21.155 ","End":"01:23.780","Text":"In this case, y is a function of x and z."},{"Start":"01:23.780 ","End":"01:28.910","Text":"We\u0027re going to use the theorem about the partial derivatives of"},{"Start":"01:28.910 ","End":"01:38.040","Text":"implicit functions and say that y with respect to x,"},{"Start":"01:38.140 ","End":"01:42.680","Text":"it\u0027s a minus, there\u0027s a dividing line."},{"Start":"01:42.680 ","End":"01:45.290","Text":"Just remember the pattern F here and F here,"},{"Start":"01:45.290 ","End":"01:47.375","Text":"and the thing is that this"},{"Start":"01:47.375 ","End":"01:52.220","Text":"1 goes up here and this letter goes down here."},{"Start":"01:52.220 ","End":"01:59.850","Text":"So what we get is, with respect to,"},{"Start":"01:59.850 ","End":"02:02.490","Text":"well, minus here, before I forget."},{"Start":"02:02.490 ","End":"02:06.145","Text":"The derivative with respect to x of this thing,"},{"Start":"02:06.145 ","End":"02:07.750","Text":"this gives me nothing."},{"Start":"02:07.750 ","End":"02:16.050","Text":"This gives me minus e^x squared plus y squared,"},{"Start":"02:16.050 ","End":"02:26.125","Text":"but there\u0027s an inner derivative which is 2x plus derivative of this with respect to x."},{"Start":"02:26.125 ","End":"02:36.350","Text":"This derivative of x plus y is just 1 and sine z is a constant."},{"Start":"02:36.350 ","End":"02:37.860","Text":"I\u0027ll write the 1 in."},{"Start":"02:37.860 ","End":"02:43.530","Text":"1 sine z, and this is a constant,"},{"Start":"02:43.530 ","End":"02:46.310","Text":"no variables at all."},{"Start":"02:46.310 ","End":"02:52.400","Text":"Extend this line, and on the denominator we want, with respect to y,"},{"Start":"02:52.400 ","End":"02:53.750","Text":"the derivative of this."},{"Start":"02:53.750 ","End":"02:56.540","Text":"Again, z squared disappears."},{"Start":"02:56.540 ","End":"03:00.710","Text":"We get minus e^x squared plus y squared,"},{"Start":"03:00.710 ","End":"03:03.020","Text":"this time with a 2y."},{"Start":"03:03.020 ","End":"03:06.440","Text":"Here, very similarly with respect to y,"},{"Start":"03:06.440 ","End":"03:12.120","Text":"this thing is 1 and it\u0027s also sine z."},{"Start":"03:12.120 ","End":"03:21.750","Text":"What we want is this thing at the point."},{"Start":"03:21.750 ","End":"03:22.550","Text":"Just emphasize."},{"Start":"03:22.550 ","End":"03:25.400","Text":"This is x and this is z."},{"Start":"03:25.400 ","End":"03:26.870","Text":"What we want to do is,"},{"Start":"03:26.870 ","End":"03:30.590","Text":"in here, we want to substitute x equals 0,"},{"Start":"03:30.590 ","End":"03:37.200","Text":"and z equals 0, and y equals?"},{"Start":"03:37.200 ","End":"03:40.660","Text":"That\u0027s the question. What is y?"},{"Start":"03:41.540 ","End":"03:46.100","Text":"What we\u0027re going to do is have to evaluate it by taking"},{"Start":"03:46.100 ","End":"03:51.004","Text":"F of x, y equals 0 and letting x equals 0,"},{"Start":"03:51.004 ","End":"03:53.900","Text":"z equals 0, and get an equation in y."},{"Start":"03:53.900 ","End":"03:58.820","Text":"What we would get from here would be z squared,"},{"Start":"03:58.820 ","End":"04:08.585","Text":"which is 0 squared, minus e to the power of x squared is 0 squared plus y squared"},{"Start":"04:08.585 ","End":"04:21.650","Text":"plus 0 plus y sine 0 plus e^4 equals 0."},{"Start":"04:21.650 ","End":"04:25.820","Text":"Now, let\u0027s see what we\u0027re left with."},{"Start":"04:25.820 ","End":"04:33.945","Text":"This is 0 because sine 0 is 0, and this is 0."},{"Start":"04:33.945 ","End":"04:40.410","Text":"So all we\u0027re left with is, this disappears also,"},{"Start":"04:40.410 ","End":"04:46.040","Text":"we have minus e^y squared,"},{"Start":"04:46.040 ","End":"04:49.835","Text":"which I\u0027ll write on the other side as plus e^y squared."},{"Start":"04:49.835 ","End":"04:52.445","Text":"Here, we\u0027re left with e^4."},{"Start":"04:52.445 ","End":"04:59.840","Text":"From here, we can compare the exponents and get y squared is equal to 4."},{"Start":"04:59.840 ","End":"05:07.490","Text":"Notice that we were given that this is bigger or equal to 0."},{"Start":"05:07.490 ","End":"05:11.390","Text":"We can say that y equals square root of 4, which is 2."},{"Start":"05:11.390 ","End":"05:14.515","Text":"That\u0027s what I want to put now here."},{"Start":"05:14.515 ","End":"05:22.990","Text":"Now I can substitute and say that this equals minus dividing line,"},{"Start":"05:22.990 ","End":"05:32.290","Text":"x is 0 so this whole first term is 0, and z is 0."},{"Start":"05:32.570 ","End":"05:39.810","Text":"We get plus sine 0 over whatever."},{"Start":"05:39.810 ","End":"05:42.710","Text":"Already I see it 0 because sine 0 is 0,"},{"Start":"05:42.710 ","End":"05:45.560","Text":"so this thing is 0."},{"Start":"05:45.560 ","End":"05:48.425","Text":"Let me highlight this before we move on"},{"Start":"05:48.425 ","End":"05:51.710","Text":"because this is 1 of the things we were asked for,"},{"Start":"05:51.710 ","End":"05:56.830","Text":"this is equal to 0."},{"Start":"05:56.830 ","End":"06:00.480","Text":"Now let\u0027s do the other 1, y with respect to z."},{"Start":"06:00.480 ","End":"06:03.060","Text":"I\u0027ll do it a bit quicker,"},{"Start":"06:03.060 ","End":"06:06.030","Text":"y with respect to z following the examples."},{"Start":"06:06.030 ","End":"06:12.210","Text":"This time, the y on the denominator is still there,"},{"Start":"06:12.210 ","End":"06:20.440","Text":"but this is F with respect to z now."},{"Start":"06:20.530 ","End":"06:25.255","Text":"What I can do is I can copy the denominator,"},{"Start":"06:25.255 ","End":"06:28.695","Text":"but the numerators now, let\u0027s see."},{"Start":"06:28.695 ","End":"06:31.365","Text":"This gives me 2z."},{"Start":"06:31.365 ","End":"06:35.725","Text":"This doesn\u0027t contain z, so that\u0027s nothing."},{"Start":"06:35.725 ","End":"06:41.700","Text":"Derivative of sine z is cosine z,"},{"Start":"06:41.700 ","End":"06:45.005","Text":"and the x plus y is just a constant."},{"Start":"06:45.005 ","End":"06:49.780","Text":"So it stays and e^4 gives nothing."},{"Start":"06:49.780 ","End":"06:54.974","Text":"In order to get yz of 0, 0,"},{"Start":"06:54.974 ","End":"07:00.315","Text":"I just make the same substitutions."},{"Start":"07:00.315 ","End":"07:02.340","Text":"Just copy it from here."},{"Start":"07:02.340 ","End":"07:05.235","Text":"So now let\u0027s see what we get."},{"Start":"07:05.235 ","End":"07:23.200","Text":"2z is 0 minus 0 plus x plus y is 2 cosine of 0."},{"Start":"07:26.530 ","End":"07:28.760","Text":"Yeah, I didn\u0027t compute it here."},{"Start":"07:28.760 ","End":"07:29.720","Text":"I could have just copied it."},{"Start":"07:29.720 ","End":"07:35.070","Text":"Oh, well, this just substitute in here."},{"Start":"07:35.070 ","End":"07:40.520","Text":"We\u0027ve got minus e^x squared plus y squared is"},{"Start":"07:40.520 ","End":"07:51.340","Text":"4 times 2y is 4 plus 1 times sine z."},{"Start":"07:51.530 ","End":"07:55.030","Text":"Well, that\u0027s going to give me 0."},{"Start":"07:55.700 ","End":"07:58.360","Text":"Cosine 0 is 1,"},{"Start":"07:58.360 ","End":"08:01.520","Text":"so the whole numerator is just 2."},{"Start":"08:01.520 ","End":"08:04.595","Text":"In the denominator, sine 0 is 0,"},{"Start":"08:04.595 ","End":"08:17.515","Text":"so we have just minus 4e^fourth,"},{"Start":"08:17.515 ","End":"08:20.460","Text":"but I forgot the minus."},{"Start":"08:20.460 ","End":"08:23.340","Text":"So it\u0027s a minus."},{"Start":"08:23.340 ","End":"08:26.610","Text":"I can just make this a plus."},{"Start":"08:26.610 ","End":"08:33.515","Text":"So if I just divide by 2 top and bottom,"},{"Start":"08:33.515 ","End":"08:45.205","Text":"we will get, let\u0027s see, 1 over 2e^fourth."},{"Start":"08:45.205 ","End":"08:49.415","Text":"I\u0027ll just highlight that, that was the answer to the other bit,"},{"Start":"08:49.415 ","End":"08:55.640","Text":"the derivative with respect to z at 0, 0."},{"Start":"08:55.640 ","End":"08:59.480","Text":"That\u0027s it."}],"ID":8954},{"Watched":false,"Name":"Exercise 6","Duration":"9m 33s","ChapterTopicVideoID":8676,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.500","Text":"In this exercise, we have an implicit function z as a function of x and y."},{"Start":"00:07.500 ","End":"00:14.800","Text":"It\u0027s given implicitly by this equation where you see also that x and y participate."},{"Start":"00:14.930 ","End":"00:18.105","Text":"What we have to do is to find"},{"Start":"00:18.105 ","End":"00:24.195","Text":"the second-order derivative of z with respect to x twice at the point 1,1."},{"Start":"00:24.195 ","End":"00:29.760","Text":"We\u0027re also told that z is big or equal to 0,"},{"Start":"00:29.760 ","End":"00:32.340","Text":"we\u0027ll see where this comes in later."},{"Start":"00:32.340 ","End":"00:35.870","Text":"Now, this is already given in the form of something equals 0,"},{"Start":"00:35.870 ","End":"00:39.030","Text":"so we\u0027ll let our auxiliary function,"},{"Start":"00:39.030 ","End":"00:41.525","Text":"F of x, y,"},{"Start":"00:41.525 ","End":"00:49.130","Text":"and z be equal to this z cubed minus 2xz plus y."},{"Start":"00:49.130 ","End":"00:52.940","Text":"Then this is like F equals 0,"},{"Start":"00:52.940 ","End":"00:56.750","Text":"so that means that F is called the auxiliary function."},{"Start":"00:56.750 ","End":"01:02.990","Text":"Now we can use the theorem on partial derivatives that z with"},{"Start":"01:02.990 ","End":"01:11.145","Text":"respect to x is equal to minus,"},{"Start":"01:11.145 ","End":"01:12.845","Text":"remember it\u0027s always a minus,"},{"Start":"01:12.845 ","End":"01:15.575","Text":"there\u0027s always an F at the top and an F at the bottom."},{"Start":"01:15.575 ","End":"01:22.460","Text":"The letter here goes at the top and the bigger one goes at the bottom."},{"Start":"01:22.460 ","End":"01:26.955","Text":"What we get is minus,"},{"Start":"01:26.955 ","End":"01:31.065","Text":"dividing line, with respect to x,"},{"Start":"01:31.065 ","End":"01:33.735","Text":"this z is a constant,"},{"Start":"01:33.735 ","End":"01:38.950","Text":"so we just get minus 2z,"},{"Start":"01:38.950 ","End":"01:42.295","Text":"it\u0027s some constant times x, it\u0027s minus 2z."},{"Start":"01:42.295 ","End":"01:45.650","Text":"This doesn\u0027t contribute at all either."},{"Start":"01:45.650 ","End":"01:52.170","Text":"With respect to z, we have from here 3z squared from here,"},{"Start":"01:52.170 ","End":"01:55.780","Text":"minus 2x and nothing from here."},{"Start":"01:56.240 ","End":"01:59.175","Text":"Then we just rewrite it."},{"Start":"01:59.175 ","End":"02:02.805","Text":"I\u0027ll combine the minuses so I have"},{"Start":"02:02.805 ","End":"02:12.400","Text":"2z over 3z squared minus 2x."},{"Start":"02:12.400 ","End":"02:19.875","Text":"Now we need the second derivative also with respect to x, the second partial."},{"Start":"02:19.875 ","End":"02:28.620","Text":"We need zxx, and that\u0027s going to be the derivative of this with respect to x."},{"Start":"02:28.620 ","End":"02:32.280","Text":"In other words, 2z over"},{"Start":"02:32.280 ","End":"02:39.675","Text":"3z squared minus 2x with respect to x. I\u0027m going to use the quotient rule."},{"Start":"02:39.675 ","End":"02:42.200","Text":"I\u0027ll just write it over here in case you\u0027ve forgotten it,"},{"Start":"02:42.200 ","End":"02:44.555","Text":"u over v prime,"},{"Start":"02:44.555 ","End":"02:47.690","Text":"well, in this case, prime means with respect to x,"},{"Start":"02:47.690 ","End":"02:52.640","Text":"would be the derivative of this times the other one,"},{"Start":"02:52.640 ","End":"02:58.940","Text":"minus this times the derivative of that and this one squared."},{"Start":"02:58.940 ","End":"03:02.180","Text":"Here, we will get,"},{"Start":"03:02.180 ","End":"03:04.760","Text":"let\u0027s start with the denominator squared."},{"Start":"03:04.760 ","End":"03:10.640","Text":"I\u0027ve got 3z squared minus 2x squared,"},{"Start":"03:10.640 ","End":"03:19.260","Text":"and the derivative of 2z is 2z with respect"},{"Start":"03:19.260 ","End":"03:25.820","Text":"to x times the denominator"},{"Start":"03:25.820 ","End":"03:30.245","Text":"as is 3z squared minus 2x,"},{"Start":"03:30.245 ","End":"03:35.315","Text":"and then minus, let\u0027s make this longer."},{"Start":"03:35.315 ","End":"03:40.270","Text":"We have this one as is 2z,"},{"Start":"03:40.270 ","End":"03:43.065","Text":"and then the derivative of this."},{"Start":"03:43.065 ","End":"03:44.970","Text":"Derivative of this, let\u0027s see,"},{"Start":"03:44.970 ","End":"03:51.680","Text":"3z squared gives us 6z but times the inner derivative,"},{"Start":"03:51.680 ","End":"04:00.120","Text":"which is z with respect to x minus this one with respect to x is just 2."},{"Start":"04:02.860 ","End":"04:06.600","Text":"Let\u0027s see if we can simplify it a bit."},{"Start":"04:07.210 ","End":"04:13.910","Text":"I\u0027m going to collect zx from the numerator."},{"Start":"04:13.910 ","End":"04:16.670","Text":"Let\u0027s see, how many zx do I have?"},{"Start":"04:16.670 ","End":"04:22.715","Text":"I have 2 times 3z squared is 6z squared."},{"Start":"04:22.715 ","End":"04:31.470","Text":"From here, I\u0027m going to get minus 6z squared."},{"Start":"04:38.710 ","End":"04:45.080","Text":"What did I say? 6z squared times zx and here minus"},{"Start":"04:45.080 ","End":"04:51.515","Text":"12z squared zx altogether minus 6z squared,"},{"Start":"04:51.515 ","End":"04:55.100","Text":"z partial with respect to x."},{"Start":"04:55.100 ","End":"05:03.380","Text":"Let\u0027s see now, expanding,"},{"Start":"05:03.380 ","End":"05:09.330","Text":"we\u0027ve got 6z squared zx,"},{"Start":"05:10.090 ","End":"05:19.220","Text":"2 times 2 is 4 minus 4xzx,"},{"Start":"05:19.220 ","End":"05:25.385","Text":"and then minus 12z squared zx,"},{"Start":"05:25.385 ","End":"05:29.400","Text":"and then plus 4z,"},{"Start":"05:29.980 ","End":"05:34.820","Text":"over the same thing,"},{"Start":"05:34.820 ","End":"05:39.950","Text":"3z squared minus 2x all squared."},{"Start":"05:39.950 ","End":"05:45.150","Text":"Let\u0027s see, there\u0027s something to cancel,"},{"Start":"05:45.150 ","End":"05:48.670","Text":"this and this can be combined, I mean."},{"Start":"05:50.480 ","End":"05:53.250","Text":"6 minus 12 is minus 6,"},{"Start":"05:53.250 ","End":"06:00.410","Text":"so how about I just eliminate this term and write this as a minus,"},{"Start":"06:00.410 ","End":"06:02.959","Text":"and that will be okay."},{"Start":"06:02.959 ","End":"06:05.540","Text":"Now, here\u0027s the difficulty."},{"Start":"06:05.540 ","End":"06:13.460","Text":"When we want to do zxx of 1,1,"},{"Start":"06:13.460 ","End":"06:17.785","Text":"we know that the x is 1 and the y is 1,"},{"Start":"06:17.785 ","End":"06:21.270","Text":"but we also have here 2 other quantity,"},{"Start":"06:21.270 ","End":"06:25.670","Text":"z and zx, which we don\u0027t know,"},{"Start":"06:25.670 ","End":"06:27.500","Text":"but we actually can figure out."},{"Start":"06:27.500 ","End":"06:29.675","Text":"Let\u0027s go and take care of that."},{"Start":"06:29.675 ","End":"06:32.285","Text":"What I\u0027m saying is that in here,"},{"Start":"06:32.285 ","End":"06:37.085","Text":"we\u0027re going to be substituting x equals 1,"},{"Start":"06:37.085 ","End":"06:39.785","Text":"y equals 1, z,"},{"Start":"06:39.785 ","End":"06:41.195","Text":"we don\u0027t know yet,"},{"Start":"06:41.195 ","End":"06:43.745","Text":"and z with respect to x,"},{"Start":"06:43.745 ","End":"06:45.200","Text":"we also don\u0027t know yet."},{"Start":"06:45.200 ","End":"06:50.120","Text":"Let\u0027s start with z. Now we have this implicit function,"},{"Start":"06:50.120 ","End":"06:52.205","Text":"if we let x equals 1,"},{"Start":"06:52.205 ","End":"06:55.460","Text":"y equals 1 here we can figure out z."},{"Start":"06:55.460 ","End":"07:03.330","Text":"We would get z cubed minus 2,"},{"Start":"07:03.330 ","End":"07:12.185","Text":"x is 1, and then z plus y is also 1 equals 0."},{"Start":"07:12.185 ","End":"07:13.580","Text":"Instead of 2 times 1,"},{"Start":"07:13.580 ","End":"07:15.875","Text":"I\u0027ll just leave the 1 out."},{"Start":"07:15.875 ","End":"07:19.805","Text":"This is a cubic equation and we don\u0027t know how to solve it,"},{"Start":"07:19.805 ","End":"07:25.910","Text":"but there is a way of just guessing whole numbers and we"},{"Start":"07:25.910 ","End":"07:32.720","Text":"can see that z equals 1 is actually a solution and this certainly is big or equal to 0."},{"Start":"07:32.720 ","End":"07:38.795","Text":"Then we can find what z with respect to x is at"},{"Start":"07:38.795 ","End":"07:47.250","Text":"the point 1,1 because now we do have what we need to substitute."},{"Start":"07:47.290 ","End":"07:50.990","Text":"Here, we\u0027re letting x equals 1,"},{"Start":"07:50.990 ","End":"07:53.945","Text":"y equals 1 as before,"},{"Start":"07:53.945 ","End":"07:57.380","Text":"and we also have z equals 1."},{"Start":"07:57.380 ","End":"08:01.445","Text":"I\u0027m going to substitute that in here, we\u0027ll get zx."},{"Start":"08:01.445 ","End":"08:03.050","Text":"Let\u0027s see 2z, well,"},{"Start":"08:03.050 ","End":"08:04.250","Text":"all the variables are 1,"},{"Start":"08:04.250 ","End":"08:06.920","Text":"so it\u0027s going to be easy to follow."},{"Start":"08:06.920 ","End":"08:12.350","Text":"2 times 1, 3 times 1 squared minus 2 times 1."},{"Start":"08:12.350 ","End":"08:17.540","Text":"Let\u0027s see. Numerator is 2 denominator 3 minus 2 is 1."},{"Start":"08:17.540 ","End":"08:19.700","Text":"I make that 2."},{"Start":"08:19.700 ","End":"08:28.025","Text":"Now we have z equals 1 and partial of z with respect to x is 2."},{"Start":"08:28.025 ","End":"08:31.460","Text":"Now we have everything we need to substitute here."},{"Start":"08:31.460 ","End":"08:37.550","Text":"We get on the numerator minus 6 times 1"},{"Start":"08:37.550 ","End":"08:44.630","Text":"squared times 2 minus 4 times 1 times 2 plus 4 times 1,"},{"Start":"08:44.630 ","End":"08:47.205","Text":"and on the denominator,"},{"Start":"08:47.205 ","End":"08:55.499","Text":"3 times 1 squared minus 2 times 1,"},{"Start":"08:55.499 ","End":"08:57.810","Text":"all of this squared."},{"Start":"08:57.810 ","End":"09:00.765","Text":"Let\u0027s see what we end up with."},{"Start":"09:00.765 ","End":"09:08.940","Text":"Minus 12, minus 8 is minus 20,"},{"Start":"09:08.940 ","End":"09:13.980","Text":"plus 4 is minus 16."},{"Start":"09:13.980 ","End":"09:17.640","Text":"Here we have 3 minus 2 is 1,"},{"Start":"09:17.640 ","End":"09:23.745","Text":"1 squared is 1, so just minus 16."},{"Start":"09:23.745 ","End":"09:32.680","Text":"So zxx at the point 1,1 is minus 16. We are done."}],"ID":8955},{"Watched":false,"Name":"Exercise 7","Duration":"19m 53s","ChapterTopicVideoID":8677,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.180","Text":"In this exercise, we are told that z is a function of x,"},{"Start":"00:06.180 ","End":"00:12.390","Text":"y, but it\u0027s given implicitly in this implicit equation."},{"Start":"00:12.390 ","End":"00:19.035","Text":"Now, we\u0027re also given that z of 2,"},{"Start":"00:19.035 ","End":"00:24.225","Text":"1, which means when x is 2 and y is 1 is minus 2."},{"Start":"00:24.225 ","End":"00:35.110","Text":"Then we have to find the partial derivatives of the second-order at the point, 2,1."},{"Start":"00:36.490 ","End":"00:45.510","Text":"Now, I want to just verify something because this is given that when x is 2 and y is 1,"},{"Start":"00:45.510 ","End":"00:46.720","Text":"that z is minus 2."},{"Start":"00:46.720 ","End":"00:51.975","Text":"In other words, we are assuming that the point 2,"},{"Start":"00:51.975 ","End":"00:55.110","Text":"1 minus 2 for x, y,"},{"Start":"00:55.110 ","End":"01:00.520","Text":"z satisfies this equation."},{"Start":"01:00.600 ","End":"01:02.740","Text":"Let\u0027s just verify this."},{"Start":"01:02.740 ","End":"01:06.295","Text":"I just want to make sure that there\u0027s no mistake here."},{"Start":"01:06.295 ","End":"01:13.960","Text":"Z cubed minus 3xyz, 3,"},{"Start":"01:13.960 ","End":"01:18.820","Text":"x, y,"},{"Start":"01:18.820 ","End":"01:25.620","Text":"z, is this equal to 4?"},{"Start":"01:25.620 ","End":"01:28.070","Text":"Let\u0027s see, this is 2 cubed is 8."},{"Start":"01:28.070 ","End":"01:34.780","Text":"Here we have minus and minus is plus."},{"Start":"01:34.780 ","End":"01:37.190","Text":"Something\u0027s wrong here."},{"Start":"01:37.370 ","End":"01:39.540","Text":"Let\u0027s check,"},{"Start":"01:39.540 ","End":"01:44.655","Text":"z cubed minus 2 cubed minus 3xyz,"},{"Start":"01:44.655 ","End":"01:49.965","Text":"minus 3, x, y, z."},{"Start":"01:49.965 ","End":"01:53.795","Text":"Does this equal 4?"},{"Start":"01:53.795 ","End":"01:56.154","Text":"I claim that, yes,"},{"Start":"01:56.154 ","End":"02:00.130","Text":"because this is minus 8 and this is minus,"},{"Start":"02:00.130 ","End":"02:04.240","Text":"minus is plus, 3 times 2 times 2 is 12,"},{"Start":"02:04.240 ","End":"02:06.470","Text":"and so this does equal 4."},{"Start":"02:06.470 ","End":"02:10.384","Text":"Yes. This is actually correct."},{"Start":"02:10.384 ","End":"02:12.665","Text":"But if you just got any value here,"},{"Start":"02:12.665 ","End":"02:16.200","Text":"it could be wrong it should be verified."},{"Start":"02:16.200 ","End":"02:18.170","Text":"Now let\u0027s get to it."},{"Start":"02:18.170 ","End":"02:20.525","Text":"We have this implicit equation."},{"Start":"02:20.525 ","End":"02:23.299","Text":"What we wanna do is find the auxiliary function,"},{"Start":"02:23.299 ","End":"02:28.670","Text":"which is what we get when we set everything equal to 0."},{"Start":"02:28.670 ","End":"02:30.845","Text":"In other words, I bring the 4 to the left,"},{"Start":"02:30.845 ","End":"02:37.455","Text":"so that\u0027s z cubed minus 3xyz minus 4."},{"Start":"02:37.455 ","End":"02:41.525","Text":"Then this is equivalent to saying that f equals 0."},{"Start":"02:41.525 ","End":"02:43.819","Text":"But this is called the auxiliary function."},{"Start":"02:43.819 ","End":"02:52.080","Text":"There is a formula that we can get the derivative\u0027s first-order for example,"},{"Start":"02:52.080 ","End":"02:58.740","Text":"Z_x is equal to minus fraction line,"},{"Start":"02:58.740 ","End":"03:01.745","Text":"here I put f here I put f. Now,"},{"Start":"03:01.745 ","End":"03:03.759","Text":"here\u0027s the thing not to get backwards."},{"Start":"03:03.759 ","End":"03:05.530","Text":"The smaller one at the bottom,"},{"Start":"03:05.530 ","End":"03:07.130","Text":"this goes at the top,"},{"Start":"03:07.130 ","End":"03:08.770","Text":"and this letter here,"},{"Start":"03:08.770 ","End":"03:11.615","Text":"the big one goes at the bottom."},{"Start":"03:11.615 ","End":"03:13.975","Text":"This is a nice formula."},{"Start":"03:13.975 ","End":"03:17.080","Text":"You might say, why are we computing Z_x?"},{"Start":"03:17.080 ","End":"03:20.710","Text":"Well we have to get to Z_xx in a couple of steps."},{"Start":"03:20.710 ","End":"03:22.225","Text":"First of all, have to go through this,"},{"Start":"03:22.225 ","End":"03:24.650","Text":"and later we\u0027ll also need Z_y."},{"Start":"03:24.750 ","End":"03:29.635","Text":"This is equal to minus dividing line,"},{"Start":"03:29.635 ","End":"03:32.765","Text":"with respect to x,"},{"Start":"03:32.765 ","End":"03:35.850","Text":"we only get x in this middle term,"},{"Start":"03:35.850 ","End":"03:41.060","Text":"and so we just get the coefficient which is minus 3yz,"},{"Start":"03:41.340 ","End":"03:44.765","Text":"this constant times x."},{"Start":"03:44.765 ","End":"03:47.460","Text":"On the bottom with respect to z,"},{"Start":"03:47.460 ","End":"03:52.090","Text":"well we have a 3z squared here."},{"Start":"03:52.430 ","End":"03:55.600","Text":"From here, again, it\u0027s a constant times z,"},{"Start":"03:55.600 ","End":"03:58.760","Text":"so it\u0027s just minus 3xy."},{"Start":"04:00.680 ","End":"04:05.460","Text":"Actually, we can cancel."},{"Start":"04:05.460 ","End":"04:10.770","Text":"Let me just say that Z_x is equal to the minus,"},{"Start":"04:10.770 ","End":"04:13.425","Text":"we\u0027ll cancel with the minus,"},{"Start":"04:13.425 ","End":"04:17.325","Text":"also the 3 top and bottom."},{"Start":"04:17.325 ","End":"04:26.520","Text":"This is just yz over z squared minus xy."},{"Start":"04:26.520 ","End":"04:28.630","Text":"Not too bad."},{"Start":"04:28.760 ","End":"04:35.285","Text":"Now, we want Z_x at the point 2,"},{"Start":"04:35.285 ","End":"04:38.645","Text":"1, so we have to substitute."},{"Start":"04:38.645 ","End":"04:40.880","Text":"Now it\u0027s not enough to substitute x and y."},{"Start":"04:40.880 ","End":"04:42.205","Text":"We also need the z."},{"Start":"04:42.205 ","End":"04:46.355","Text":"Let me just rewrite them so that is 2, 1, minus 2,"},{"Start":"04:46.355 ","End":"04:50.270","Text":"let me just write specifically that x is 2 at our point,"},{"Start":"04:50.270 ","End":"04:55.100","Text":"y equals 1, and z is minus 2."},{"Start":"04:55.100 ","End":"05:00.500","Text":"If I take these values and plug them in here and say yz,"},{"Start":"05:00.500 ","End":"05:06.224","Text":"1 times minus 2/z squared"},{"Start":"05:06.224 ","End":"05:13.060","Text":"minus 2 squared minus, xy, 2, 1."},{"Start":"05:13.640 ","End":"05:16.020","Text":"What do we get? Let\u0027s see,"},{"Start":"05:16.020 ","End":"05:18.005","Text":"our numerator\u0027s minus 2."},{"Start":"05:18.005 ","End":"05:26.030","Text":"The denominator is 4 minus 2 is 2, minus 2/2."},{"Start":"05:26.030 ","End":"05:28.940","Text":"I make that minus 1."},{"Start":"05:28.940 ","End":"05:30.920","Text":"Let me add this to my list."},{"Start":"05:30.920 ","End":"05:39.570","Text":"I know I\u0027m going to use it later though at this point z with respect to x is minus 1."},{"Start":"05:39.570 ","End":"05:45.390","Text":"Now I\u0027m also going to need this when I come to it, which is next."},{"Start":"05:45.470 ","End":"05:48.990","Text":"Z_y, by the same principle,"},{"Start":"05:48.990 ","End":"05:50.145","Text":"I put a minus,"},{"Start":"05:50.145 ","End":"05:52.560","Text":"a dividing line, f here,"},{"Start":"05:52.560 ","End":"05:56.585","Text":"f here, and I need the one on the bottom,"},{"Start":"05:56.585 ","End":"06:00.870","Text":"goes on the top and the big one goes on the bottom."},{"Start":"06:01.000 ","End":"06:06.455","Text":"This time we get minus dividing line."},{"Start":"06:06.455 ","End":"06:10.235","Text":"Now with respect to z, I can just copy it."},{"Start":"06:10.235 ","End":"06:15.440","Text":"It\u0027s 3z squared minus 3xy, right from here."},{"Start":"06:15.440 ","End":"06:17.990","Text":"Now all I need is with respect to y."},{"Start":"06:17.990 ","End":"06:20.360","Text":"This is nothing and this gives nothing."},{"Start":"06:20.360 ","End":"06:22.920","Text":"I\u0027m just left with the minus 3xz."},{"Start":"06:24.500 ","End":"06:30.970","Text":"As before, the 3\u0027s cancel and the minus cancels."},{"Start":"06:31.250 ","End":"06:35.280","Text":"So Z_y is equal"},{"Start":"06:35.280 ","End":"06:43.300","Text":"to xz/z squared minus xy."},{"Start":"06:43.300 ","End":"06:44.675","Text":"Very similar to this,"},{"Start":"06:44.675 ","End":"06:46.955","Text":"but with an x instead of a y."},{"Start":"06:46.955 ","End":"06:51.290","Text":"Once again, I want Z_y at the point 2,"},{"Start":"06:51.290 ","End":"06:55.415","Text":"1, it will come in handy later with the second derivatives."},{"Start":"06:55.415 ","End":"06:57.410","Text":"This is going to equal,"},{"Start":"06:57.410 ","End":"07:00.665","Text":"taking this formula and making the same substitutions,"},{"Start":"07:00.665 ","End":"07:06.305","Text":"x is 2, z is minus 2."},{"Start":"07:06.305 ","End":"07:08.810","Text":"It\u0027s going to be the same denominator,"},{"Start":"07:08.810 ","End":"07:11.735","Text":"which we already said was 2,"},{"Start":"07:11.735 ","End":"07:16.130","Text":"and so the 2\u0027s cancel."},{"Start":"07:16.130 ","End":"07:18.755","Text":"I\u0027m just left with minus 2,"},{"Start":"07:18.755 ","End":"07:21.410","Text":"and I\u0027m adding it to my list, minus 2."},{"Start":"07:21.410 ","End":"07:24.800","Text":"You\u0027ll see where I\u0027m going to use all these later."},{"Start":"07:24.800 ","End":"07:30.365","Text":"In fact, I\u0027m even going to shift them down because later I\u0027m going to use them,"},{"Start":"07:30.365 ","End":"07:33.360","Text":"and I don\u0027t want to scroll off-screen."},{"Start":"07:34.030 ","End":"07:42.410","Text":"This is where we start with the second derivatives rather."},{"Start":"07:42.410 ","End":"07:49.800","Text":"I need Z_xx, so that\u0027s going to be differentiating Z_x with respect to x."},{"Start":"07:50.350 ","End":"07:56.690","Text":"Back to x is going to equal the derivative of this I\u0027ll just copy,"},{"Start":"07:56.690 ","End":"08:04.595","Text":"yz/z squared minus xy,"},{"Start":"08:04.595 ","End":"08:07.040","Text":"derivative with respect to x,"},{"Start":"08:07.040 ","End":"08:08.780","Text":"I\u0027ll indicate it this way."},{"Start":"08:08.780 ","End":"08:12.320","Text":"Perhaps for reference I\u0027ll write the product and quotient rules."},{"Start":"08:12.320 ","End":"08:14.330","Text":"I can see we\u0027re going to need them,"},{"Start":"08:14.330 ","End":"08:15.470","Text":"unless you memorize them,"},{"Start":"08:15.470 ","End":"08:16.565","Text":"which would be a good idea."},{"Start":"08:16.565 ","End":"08:23.225","Text":"Uv prime is u prime v plus uv prime,"},{"Start":"08:23.225 ","End":"08:28.155","Text":"and u/v prime with respect to whatever,"},{"Start":"08:28.155 ","End":"08:32.160","Text":"in this case with respect to x would be,"},{"Start":"08:32.160 ","End":"08:35.980","Text":"this one derived, this one not,"},{"Start":"08:35.980 ","End":"08:38.090","Text":"and then this one as is,"},{"Start":"08:38.090 ","End":"08:39.935","Text":"and the other one derived,"},{"Start":"08:39.935 ","End":"08:42.935","Text":"all over the denominator squared."},{"Start":"08:42.935 ","End":"08:45.720","Text":"Let\u0027s get to this here."},{"Start":"08:45.760 ","End":"08:50.390","Text":"Easiest to start with the denominator because that\u0027s"},{"Start":"08:50.390 ","End":"08:55.770","Text":"just copying it and putting a squared."},{"Start":"08:55.770 ","End":"08:59.445","Text":"Now, u is the top and v is the bottom."},{"Start":"08:59.445 ","End":"09:03.270","Text":"U prime, but u itself,"},{"Start":"09:03.270 ","End":"09:05.280","Text":"we just going to say it\u0027s a product,"},{"Start":"09:05.280 ","End":"09:06.900","Text":"it\u0027s not really a product."},{"Start":"09:06.900 ","End":"09:08.610","Text":"Y is a constant,"},{"Start":"09:08.610 ","End":"09:12.440","Text":"but z is not a constant because z is a function of x and y,"},{"Start":"09:12.440 ","End":"09:15.180","Text":"what we need is with respect to x."},{"Start":"09:15.180 ","End":"09:21.900","Text":"The constant y stays and this is differentiated with respect to x, so it\u0027s Z_x."},{"Start":"09:22.960 ","End":"09:25.745","Text":"That\u0027s just the u prime part."},{"Start":"09:25.745 ","End":"09:28.615","Text":"Now I need all of the denominator,"},{"Start":"09:28.615 ","End":"09:32.055","Text":"z squared minus xy."},{"Start":"09:32.055 ","End":"09:35.385","Text":"Now we\u0027re up to here, the minus sign,"},{"Start":"09:35.385 ","End":"09:38.505","Text":"u as the numerator as is,"},{"Start":"09:38.505 ","End":"09:42.850","Text":"yz, and then I need the derivative of the denominator."},{"Start":"09:42.850 ","End":"09:45.725","Text":"If I differentiate the denominator,"},{"Start":"09:45.725 ","End":"09:49.485","Text":"z squared doesn\u0027t just give me 2z,"},{"Start":"09:49.485 ","End":"09:54.100","Text":"but I also need z with respect to x because z is a function of x,"},{"Start":"09:54.100 ","End":"09:59.045","Text":"partial derivative, z external is to z internal this."},{"Start":"09:59.045 ","End":"10:01.850","Text":"The next one with respect to x,"},{"Start":"10:01.850 ","End":"10:06.410","Text":"that\u0027s straightforward because it\u0027s just a constant times x,"},{"Start":"10:06.410 ","End":"10:10.260","Text":"so it becomes minus y."},{"Start":"10:10.500 ","End":"10:14.800","Text":"It could be simplified a bit. I\u0027m not going to bother."},{"Start":"10:14.800 ","End":"10:23.050","Text":"What we want is zxx at the 2,1."},{"Start":"10:23.050 ","End":"10:26.920","Text":"What we need to do is to make substitutions here,"},{"Start":"10:26.920 ","End":"10:28.525","Text":"according to this table."},{"Start":"10:28.525 ","End":"10:32.710","Text":"Basically, we need everything except zy."},{"Start":"10:32.710 ","End":"10:37.435","Text":"I need the first 4 things to substitute wherever appropriate."},{"Start":"10:37.435 ","End":"10:41.200","Text":"I won\u0027t bother writing the y because it\u0027s 1,"},{"Start":"10:41.200 ","End":"10:47.860","Text":"zx we get is minus 1 and then z-squared."},{"Start":"10:47.860 ","End":"10:54.589","Text":"This right here already is 4 minus xy is 2,"},{"Start":"10:55.230 ","End":"10:57.820","Text":"minus y is 1,"},{"Start":"10:57.820 ","End":"11:00.925","Text":"z is minus 2,"},{"Start":"11:00.925 ","End":"11:06.520","Text":"2 times z times"},{"Start":"11:06.520 ","End":"11:15.670","Text":"zx minus 1 around these brackets,"},{"Start":"11:15.670 ","End":"11:19.990","Text":"and all this over, well,"},{"Start":"11:19.990 ","End":"11:23.590","Text":"we already computed the denominator here without the squared,"},{"Start":"11:23.590 ","End":"11:25.390","Text":"so it\u0027s just the same thing squared."},{"Start":"11:25.390 ","End":"11:27.310","Text":"This part was computed already."},{"Start":"11:27.310 ","End":"11:30.490","Text":"Let\u0027s see what this comes down to."},{"Start":"11:30.490 ","End":"11:37.510","Text":"This becomes 4 minus 2 is 2, that\u0027s minus 2."},{"Start":"11:37.510 ","End":"11:42.984","Text":"I have minus 2 and then here,"},{"Start":"11:42.984 ","End":"11:46.645","Text":"2 times minus 2 times minus 1 is 4,"},{"Start":"11:46.645 ","End":"11:54.280","Text":"4 minus 1 is 3 but it\u0027s plus 2 times 3,"},{"Start":"11:54.280 ","End":"11:57.490","Text":"all this over 4."},{"Start":"11:57.490 ","End":"12:00.655","Text":"6 minus 2 is 4,"},{"Start":"12:00.655 ","End":"12:05.320","Text":"4 over 4 is equal to 1."},{"Start":"12:05.320 ","End":"12:07.240","Text":"This is one of the results."},{"Start":"12:07.240 ","End":"12:09.445","Text":"This is part a, I\u0027m going to highlight it."},{"Start":"12:09.445 ","End":"12:13.420","Text":"This is equal to 1."},{"Start":"12:13.420 ","End":"12:15.445","Text":"1 down 2 to go."},{"Start":"12:15.445 ","End":"12:18.070","Text":"Next is zxy."},{"Start":"12:18.070 ","End":"12:24.520","Text":"I\u0027ll scroll a bit, but I\u0027ll leave z with respect to x up here,"},{"Start":"12:24.520 ","End":"12:32.560","Text":"and now let\u0027s compute zxy which is similar to what we did before when we did zxx."},{"Start":"12:32.560 ","End":"12:34.525","Text":"We just take the same thing."},{"Start":"12:34.525 ","End":"12:41.485","Text":"I\u0027m going to just copy paste that and I have to replace x by y,"},{"Start":"12:41.485 ","End":"12:47.650","Text":"I still have the product and quotient rule here in case I need them."},{"Start":"12:47.650 ","End":"12:57.840","Text":"This thing is going"},{"Start":"12:57.840 ","End":"13:01.370","Text":"to be the same only with respect to y."},{"Start":"13:01.370 ","End":"13:05.500","Text":"No, perhaps not. No, it\u0027s different."},{"Start":"13:05.500 ","End":"13:07.075","Text":"Here we really have a product."},{"Start":"13:07.075 ","End":"13:09.040","Text":"Both of these depend on y."},{"Start":"13:09.040 ","End":"13:13.765","Text":"For the product, the derivative would be derivative of y,"},{"Start":"13:13.765 ","End":"13:21.430","Text":"which is 1 times z plus y as is times the derivative of y, which is zy."},{"Start":"13:21.430 ","End":"13:25.825","Text":"That\u0027s just the u prime."},{"Start":"13:25.825 ","End":"13:28.630","Text":"All this times v,"},{"Start":"13:28.630 ","End":"13:30.415","Text":"which is the denominator,"},{"Start":"13:30.415 ","End":"13:33.850","Text":"z squared minus xy,"},{"Start":"13:33.850 ","End":"13:36.010","Text":"then minus, we\u0027re up to here,"},{"Start":"13:36.010 ","End":"13:39.505","Text":"u which is the yz at the top,"},{"Start":"13:39.505 ","End":"13:42.970","Text":"and then v prime,"},{"Start":"13:42.970 ","End":"13:47.695","Text":"which is denominator prime."},{"Start":"13:47.695 ","End":"13:55.790","Text":"Denominator prime will be the derivative of the denominator is 2z."},{"Start":"13:56.100 ","End":"14:03.880","Text":"It looks a bit like what we did here."},{"Start":"14:03.880 ","End":"14:06.590","Text":"Here 2z, only zy,"},{"Start":"14:09.660 ","End":"14:12.220","Text":"this time instead of minus y,"},{"Start":"14:12.220 ","End":"14:15.620","Text":"it\u0027s going to be minus x."},{"Start":"14:16.980 ","End":"14:23.110","Text":"All this over the denominator squared,"},{"Start":"14:23.110 ","End":"14:25.015","Text":"I just copied it from here."},{"Start":"14:25.015 ","End":"14:28.435","Text":"Now once again, we want it not in general,"},{"Start":"14:28.435 ","End":"14:34.015","Text":"but at this specific point 2,1."},{"Start":"14:34.015 ","End":"14:39.400","Text":"X is 2 and y is 1 but we need also z and zy."},{"Start":"14:39.400 ","End":"14:41.215","Text":"Well, we have everything in this table,"},{"Start":"14:41.215 ","End":"14:43.480","Text":"so we just have to substitute."},{"Start":"14:43.480 ","End":"14:50.185","Text":"We get 1 times z is just minus 2"},{"Start":"14:50.185 ","End":"14:58.645","Text":"plus yzy is 1 times minus 2, is minus 2."},{"Start":"14:58.645 ","End":"15:02.125","Text":"Change the plus to a minus 2."},{"Start":"15:02.125 ","End":"15:06.220","Text":"Next, z squared is 4,"},{"Start":"15:06.220 ","End":"15:10.510","Text":"xy is 2 minus."},{"Start":"15:10.510 ","End":"15:15.835","Text":"Yz is minus 2,"},{"Start":"15:15.835 ","End":"15:22.730","Text":"let\u0027s just make it a plus and a 2 and then 2zzy."},{"Start":"15:24.000 ","End":"15:30.980","Text":"2 minus 2, minus 2 is plus 8,"},{"Start":"15:30.980 ","End":"15:36.990","Text":"minus x is 2 and all"},{"Start":"15:36.990 ","End":"15:45.069","Text":"this is over what we got before,"},{"Start":"15:45.069 ","End":"15:48.410","Text":"which was 2 squared, which was 4."},{"Start":"15:49.110 ","End":"15:56.589","Text":"This is minus 4 times 2 is minus"},{"Start":"15:56.589 ","End":"16:04.390","Text":"8 plus twice 6 is 12 over 4."},{"Start":"16:04.390 ","End":"16:09.355","Text":"It came out to be 1 again."},{"Start":"16:09.355 ","End":"16:12.910","Text":"I\u0027ll highlight this. This is another thing that we were requested."},{"Start":"16:12.910 ","End":"16:18.835","Text":"The second of the 3 second-order partial derivatives."},{"Start":"16:18.835 ","End":"16:20.650","Text":"It\u0027s not 3, it\u0027s actually 4 of them,"},{"Start":"16:20.650 ","End":"16:24.160","Text":"but zyx almost always comes out the same as zxy."},{"Start":"16:24.160 ","End":"16:26.710","Text":"It\u0027s got one more to go, which will be the zyy."},{"Start":"16:26.710 ","End":"16:29.200","Text":"I\u0027ll scroll a bit."},{"Start":"16:29.200 ","End":"16:32.230","Text":"I just want to keep our table up here."},{"Start":"16:32.230 ","End":"16:39.490","Text":"Zyy in general is going"},{"Start":"16:39.490 ","End":"16:47.470","Text":"to equal the derivative of zy with respect to y."},{"Start":"16:47.470 ","End":"16:57.520","Text":"Let me just copy this and then prime derived according to y."},{"Start":"16:57.520 ","End":"17:02.080","Text":"At this time you should have memorized the product and quotient rules."},{"Start":"17:02.080 ","End":"17:07.105","Text":"I\u0027m going to take the derivative of the numerator,"},{"Start":"17:07.105 ","End":"17:10.270","Text":"which is just x,"},{"Start":"17:10.270 ","End":"17:12.430","Text":"which is a constant here,"},{"Start":"17:12.430 ","End":"17:17.980","Text":"zy times the denominator as is,"},{"Start":"17:17.980 ","End":"17:22.630","Text":"z-squared minus xy minus numerator,"},{"Start":"17:22.630 ","End":"17:27.505","Text":"which is xz times derivative of denominator."},{"Start":"17:27.505 ","End":"17:37.300","Text":"This is 2z but times zy and with respect to y, that\u0027s minus x."},{"Start":"17:37.300 ","End":"17:44.200","Text":"All this, I copied it from above over the same thing, denominator squared."},{"Start":"17:44.200 ","End":"17:51.505","Text":"What we are looking for is zyy at the point 2,1,"},{"Start":"17:51.505 ","End":"17:55.030","Text":"which means that we take this and substitute"},{"Start":"17:55.030 ","End":"17:59.350","Text":"all the values from our little table here into here."},{"Start":"17:59.350 ","End":"18:02.845","Text":"What do we get? X times"},{"Start":"18:02.845 ","End":"18:12.920","Text":"zy is 2 times minus 2 is minus 4."},{"Start":"18:13.470 ","End":"18:19.270","Text":"Z squared minus xy is the same as here."},{"Start":"18:19.270 ","End":"18:25.520","Text":"It\u0027s 4 minus 2 as before,"},{"Start":"18:26.850 ","End":"18:35.695","Text":"minus xz is minus 4,"},{"Start":"18:35.695 ","End":"18:39.190","Text":"let\u0027s make it plus 4,"},{"Start":"18:39.190 ","End":"18:44.620","Text":"and then 2zzy twice this,"},{"Start":"18:44.620 ","End":"18:48.100","Text":"times this, we had that before."},{"Start":"18:48.100 ","End":"19:08.570","Text":"That is equal to 8 minus x."},{"Start":"19:08.570 ","End":"19:10.155","Text":"We also had that,"},{"Start":"19:10.155 ","End":"19:18.210","Text":"x is 2 and all this over the denominator squared always came out 4."},{"Start":"19:18.210 ","End":"19:20.860","Text":"It\u0027s the same each time."},{"Start":"19:20.870 ","End":"19:29.800","Text":"We get minus 4 times 2 is minus 8."},{"Start":"19:29.800 ","End":"19:34.885","Text":"Here we have 8 minus 2 is 6,"},{"Start":"19:34.885 ","End":"19:40.060","Text":"6 times 4 is 24 over 4,"},{"Start":"19:40.060 ","End":"19:45.310","Text":"that\u0027s 16 over 4 and it equals 4."},{"Start":"19:45.310 ","End":"19:48.910","Text":"Let me highlight this is equal to this,"},{"Start":"19:48.910 ","End":"19:54.080","Text":"that\u0027s the third bit information we\u0027re looking for and actually we\u0027re done."}],"ID":8956},{"Watched":false,"Name":"Exercise 8","Duration":"15m 17s","ChapterTopicVideoID":8678,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.660","Text":"In this exercise, I\u0027ve got 2 equations in x,"},{"Start":"00:06.660 ","End":"00:09.585","Text":"y, u, and v,"},{"Start":"00:09.585 ","End":"00:13.935","Text":"so we have a system of implicit functions."},{"Start":"00:13.935 ","End":"00:19.395","Text":"Now, the important thing is to see how many equations?"},{"Start":"00:19.395 ","End":"00:22.570","Text":"I have 2 equations,"},{"Start":"00:23.090 ","End":"00:26.670","Text":"and how many variables?"},{"Start":"00:26.670 ","End":"00:29.595","Text":"I have 4 variables."},{"Start":"00:29.595 ","End":"00:31.830","Text":"That would be u,"},{"Start":"00:31.830 ","End":"00:36.750","Text":"v, x, and y."},{"Start":"00:36.750 ","End":"00:42.090","Text":"Now, if you do the subtraction,"},{"Start":"00:42.090 ","End":"00:44.825","Text":"we don\u0027t really have 4 variables."},{"Start":"00:44.825 ","End":"00:48.095","Text":"We have to subtract the 2 constraints and that leaves us with 2."},{"Start":"00:48.095 ","End":"00:55.550","Text":"This means that we basically have 2 free variables and the others are functions of this."},{"Start":"00:55.550 ","End":"00:58.910","Text":"Now the way the question is set up is if you look at it,"},{"Start":"00:58.910 ","End":"01:02.780","Text":"we will looking for derivatives with respect to x and y."},{"Start":"01:02.780 ","End":"01:08.220","Text":"We\u0027ll let x and y be the free variables,"},{"Start":"01:08.990 ","End":"01:16.280","Text":"and then we consider u to be like u of x and y,"},{"Start":"01:16.280 ","End":"01:22.050","Text":"and v to be v of x and y implicitly."},{"Start":"01:22.780 ","End":"01:29.240","Text":"We\u0027re going to find these partial derivatives of u and v with respect to x and"},{"Start":"01:29.240 ","End":"01:35.590","Text":"y using certain theorems, Jacobian etc."},{"Start":"01:35.590 ","End":"01:38.105","Text":"Let\u0027s just proceed."},{"Start":"01:38.105 ","End":"01:43.790","Text":"According to the system is where we write these 2 in terms of something equals 0."},{"Start":"01:43.790 ","End":"01:48.380","Text":"The first 1 I can write as f of u, v, x,"},{"Start":"01:48.380 ","End":"01:54.620","Text":"y is equal to u squared minus v. I Just bring everything to"},{"Start":"01:54.620 ","End":"02:02.325","Text":"the left and leaves 0 on the right minus 3x minus y,"},{"Start":"02:02.325 ","End":"02:04.710","Text":"and that would be equal to 0."},{"Start":"02:04.710 ","End":"02:07.655","Text":"For the other equation,"},{"Start":"02:07.655 ","End":"02:12.495","Text":"that would be where G of u,"},{"Start":"02:12.495 ","End":"02:22.365","Text":"v, x, and y is u minus 2v squared minus x plus 2y."},{"Start":"02:22.365 ","End":"02:25.790","Text":"As I said, this first equation is equivalent to saying"},{"Start":"02:25.790 ","End":"02:29.390","Text":"that F is 0 and the other one is equivalent to saying G is 0."},{"Start":"02:29.390 ","End":"02:31.745","Text":"That\u0027s how we set it up."},{"Start":"02:31.745 ","End":"02:36.110","Text":"Now the way the theorem works is that if we want,"},{"Start":"02:36.110 ","End":"02:37.520","Text":"let\u0027s say this one,"},{"Start":"02:37.520 ","End":"02:41.809","Text":"which is I\u0027ll write it in the other notation du by dx."},{"Start":"02:41.809 ","End":"02:43.864","Text":"What we do is as follows."},{"Start":"02:43.864 ","End":"02:50.120","Text":"We put a minus sign and then a dividing line and we put 2 different Jacobians,"},{"Start":"02:50.120 ","End":"02:52.130","Text":"one on top and one on the bottom."},{"Start":"02:52.130 ","End":"02:56.010","Text":"On the bottom, it\u0027s always the same."},{"Start":"03:02.930 ","End":"03:05.220","Text":"Take 2 on that last bit,"},{"Start":"03:05.220 ","End":"03:12.915","Text":"it\u0027s the Jacobian and here we put the 2 functions, F,"},{"Start":"03:12.915 ","End":"03:21.675","Text":"and G. Here we put the 2 that are the dependent variables,"},{"Start":"03:21.675 ","End":"03:25.460","Text":"u and v, the ones that are functions of x and y."},{"Start":"03:25.460 ","End":"03:33.380","Text":"That\u0027s u and v and then initially I copy from the denominator,"},{"Start":"03:33.380 ","End":"03:34.940","Text":"I just did a copy-paste here."},{"Start":"03:34.940 ","End":"03:37.400","Text":"But then, because is du by dx,"},{"Start":"03:37.400 ","End":"03:40.370","Text":"we take u, the one from here,"},{"Start":"03:40.370 ","End":"03:45.310","Text":"and replace it with x and"},{"Start":"03:45.310 ","End":"03:50.670","Text":"then we expand the Jacobian so we get a determinant on the numerator."},{"Start":"03:52.580 ","End":"03:58.380","Text":"Also, a determinant here and there\u0027s the minus."},{"Start":"03:58.380 ","End":"04:04.790","Text":"Now, you know I\u0027m going to go back to the other notation and not the G notation."},{"Start":"04:04.790 ","End":"04:09.185","Text":"So I do F with respect to x,"},{"Start":"04:09.185 ","End":"04:12.744","Text":"and then F with respect to v,"},{"Start":"04:12.744 ","End":"04:15.515","Text":"G with respect to x,"},{"Start":"04:15.515 ","End":"04:19.880","Text":"all combinations G with respect to v. On the bottom,"},{"Start":"04:19.880 ","End":"04:23.220","Text":"F with respect to u,"},{"Start":"04:23.220 ","End":"04:25.320","Text":"F with respect to v,"},{"Start":"04:25.320 ","End":"04:27.704","Text":"G with respect to u,"},{"Start":"04:27.704 ","End":"04:35.840","Text":"G with respect to v. We have these 6 things to compute."},{"Start":"04:35.840 ","End":"04:38.330","Text":"The first 2-column, this column, and this column are the same,"},{"Start":"04:38.330 ","End":"04:40.070","Text":"but I still have 1, 2, 3, 4,"},{"Start":"04:40.070 ","End":"04:42.930","Text":"5, 6 things to compute."},{"Start":"04:45.080 ","End":"04:49.860","Text":"Let\u0027s start doing that we have a minus."},{"Start":"04:49.860 ","End":"04:58.585","Text":"Now, here we have F with respect to x. I\u0027ve lost, oh, there it is."},{"Start":"04:58.585 ","End":"05:02.884","Text":"F with respect to x is,"},{"Start":"05:02.884 ","End":"05:04.370","Text":"this is a constant,"},{"Start":"05:04.370 ","End":"05:05.900","Text":"this is a constant."},{"Start":"05:05.900 ","End":"05:09.210","Text":"This is just minus 3."},{"Start":"05:09.730 ","End":"05:16.245","Text":"F with respect to v is minus 1,"},{"Start":"05:16.245 ","End":"05:19.695","Text":"G with respect to x,"},{"Start":"05:19.695 ","End":"05:24.285","Text":"that\u0027s this one with respect to x is minus 1,"},{"Start":"05:24.285 ","End":"05:32.140","Text":"and G with respect to v would be minus 4v."},{"Start":"05:33.440 ","End":"05:40.420","Text":"That\u0027s one determinant divided by the other determinant."},{"Start":"05:40.420 ","End":"05:43.700","Text":"Let\u0027s still see this,"},{"Start":"05:43.700 ","End":"05:53.050","Text":"here F with respect to u 2u."},{"Start":"05:56.420 ","End":"06:00.850","Text":"The second column I said we can copy that\u0027s minus 1 minus 4."},{"Start":"06:00.850 ","End":"06:03.265","Text":"V equals this is the same as this."},{"Start":"06:03.265 ","End":"06:10.550","Text":"The other one I need is G with respect to u and that just comes out to be 1."},{"Start":"06:10.550 ","End":"06:14.590","Text":"Now, hope you remember how to compute a determinant."},{"Start":"06:14.590 ","End":"06:15.610","Text":"Just to remind you,"},{"Start":"06:15.610 ","End":"06:18.789","Text":"if we have a, b, c, d,"},{"Start":"06:18.789 ","End":"06:25.915","Text":"this is ad minus bc and so we get"},{"Start":"06:25.915 ","End":"06:34.930","Text":"minus 3 times minus 4v is 12v less bc,"},{"Start":"06:34.930 ","End":"06:42.240","Text":"which is 1 over 2u times minus"},{"Start":"06:42.240 ","End":"06:52.340","Text":"4v is minus 8uv minus minus 1 is plus 1."},{"Start":"06:52.340 ","End":"06:56.410","Text":"But there\u0027s still a minus upfront from here"},{"Start":"06:56.410 ","End":"07:03.415","Text":"and sure I could put the minus in and reverse,I just leave it like that for now."},{"Start":"07:03.415 ","End":"07:06.195","Text":"Perhaps I\u0027ll keep track of them,"},{"Start":"07:06.195 ","End":"07:13.275","Text":"so far we have that U_x is minus 12v minus 1."},{"Start":"07:13.275 ","End":"07:22.210","Text":"I\u0027ll just reverse the order of this I\u0027ll make it 1 minus 8uv."},{"Start":"07:22.210 ","End":"07:28.770","Text":"Next, I want U_y and see what this is."},{"Start":"07:28.770 ","End":"07:39.220","Text":"Actually, I\u0027ll leave place for all 4 of them and then I need V_x and I need the V_y."},{"Start":"07:40.220 ","End":"07:47.440","Text":"I\u0027m going to erase this bit and the next one,"},{"Start":"07:47.440 ","End":"07:49.270","Text":"u with respect to y,"},{"Start":"07:49.270 ","End":"07:52.700","Text":"is du by dy."},{"Start":"07:55.560 ","End":"08:04.105","Text":"The Jacobian, F, G with respect to u,"},{"Start":"08:04.105 ","End":"08:07.525","Text":"v, as before, this is always it."},{"Start":"08:07.525 ","End":"08:10.315","Text":"This time in the numerator,"},{"Start":"08:10.315 ","End":"08:12.085","Text":"I\u0027ll just copy it myself."},{"Start":"08:12.085 ","End":"08:17.410","Text":"F, G over u, v for starters."},{"Start":"08:17.410 ","End":"08:21.800","Text":"Now this tells me to replace u with y."},{"Start":"08:21.960 ","End":"08:27.354","Text":"Out with the u and in with the y,"},{"Start":"08:27.354 ","End":"08:30.235","Text":"and there\u0027s a minus in front."},{"Start":"08:30.235 ","End":"08:33.835","Text":"What do we get this time?"},{"Start":"08:33.835 ","End":"08:42.955","Text":"We get minus, this is a determinant and this 1 is also a determinant."},{"Start":"08:42.955 ","End":"08:48.990","Text":"F_u, F_v,"},{"Start":"08:48.990 ","End":"08:53.980","Text":"G_u, G_v."},{"Start":"08:53.980 ","End":"09:00.745","Text":"Here, we have F with respect to y,"},{"Start":"09:00.745 ","End":"09:03.985","Text":"and here it\u0027s going to be G with respect to y,"},{"Start":"09:03.985 ","End":"09:11.800","Text":"F_v and G_v."},{"Start":"09:11.800 ","End":"09:13.750","Text":"Now let\u0027s see what we get."},{"Start":"09:13.750 ","End":"09:20.929","Text":"F with respect to y is minus 1,"},{"Start":"09:21.690 ","End":"09:27.310","Text":"let\u0027s see, do this 1, G with respect to y is 2."},{"Start":"09:27.310 ","End":"09:34.030","Text":"F with respect to v is minus 1."},{"Start":"09:34.030 ","End":"09:44.590","Text":"G with respect to v is minus 4v, determinant, quotient, minus."},{"Start":"09:44.590 ","End":"09:49.855","Text":"Then these 2 we had before."},{"Start":"09:49.855 ","End":"09:57.190","Text":"That\u0027s got to be this 1 minus 8uv."},{"Start":"09:57.190 ","End":"10:00.205","Text":"That bit comes out the same each time."},{"Start":"10:00.205 ","End":"10:06.790","Text":"If I multiply, this is plus 4v and then subtract this diagonal,"},{"Start":"10:06.790 ","End":"10:10.120","Text":"plus 4v minus minus 2 is going to"},{"Start":"10:10.120 ","End":"10:18.880","Text":"be 4v plus 2 over 1 minus 8uv."},{"Start":"10:18.880 ","End":"10:21.640","Text":"I\u0027ll just copy it to my table,"},{"Start":"10:21.640 ","End":"10:28.670","Text":"4v plus 2 over 1 minus 8uv."},{"Start":"10:28.860 ","End":"10:32.555","Text":"In fact, in all of them,"},{"Start":"10:32.555 ","End":"10:36.340","Text":"it\u0027s going to be 1 minus 8uv."},{"Start":"10:39.230 ","End":"10:45.970","Text":"That\u0027s always the minus with the determinant on the denominator."},{"Start":"10:45.970 ","End":"10:50.090","Text":"All we just need to do the determinants and the numerator."},{"Start":"10:50.250 ","End":"10:59.320","Text":"Again, I\u0027m going to erase and this time we want dv by dx,"},{"Start":"10:59.320 ","End":"11:02.300","Text":"same formula as before."},{"Start":"11:06.270 ","End":"11:08.650","Text":"No, I won\u0027t make a shortcut,"},{"Start":"11:08.650 ","End":"11:10.225","Text":"no sorry, I\u0027ll just do it."},{"Start":"11:10.225 ","End":"11:17.180","Text":"Minus, just because the denominator keeps coming out the same thing."},{"Start":"11:18.060 ","End":"11:21.340","Text":"Here we have d(F,"},{"Start":"11:21.340 ","End":"11:25.555","Text":"G) by d(u, v)."},{"Start":"11:25.555 ","End":"11:28.765","Text":"Here we start off with the same thing."},{"Start":"11:28.765 ","End":"11:36.739","Text":"F, G u, v but we replace v with x,"},{"Start":"11:36.990 ","End":"11:42.890","Text":"this goes x in its place."},{"Start":"11:43.020 ","End":"11:45.385","Text":"This is equal to,"},{"Start":"11:45.385 ","End":"11:50.260","Text":"now we know that minus on the denominator just comes out 1 minus 8uv,"},{"Start":"11:50.260 ","End":"11:52.345","Text":"I\u0027m not going to do it again."},{"Start":"11:52.345 ","End":"11:57.970","Text":"This bit here, just the numerator is going to"},{"Start":"11:57.970 ","End":"12:05.245","Text":"be F with respect to u,"},{"Start":"12:05.245 ","End":"12:09.475","Text":"F with respect to x. F is the first, so it\u0027s the top,"},{"Start":"12:09.475 ","End":"12:12.100","Text":"G is at the bottom, and then it\u0027s F_u and then x,"},{"Start":"12:12.100 ","End":"12:15.820","Text":"then G_u and x."},{"Start":"12:15.820 ","End":"12:19.435","Text":"That\u0027s just the numerator and that\u0027s what we need."},{"Start":"12:19.435 ","End":"12:27.445","Text":"Now this is equal to F with respect to u is 2u."},{"Start":"12:27.445 ","End":"12:35.770","Text":"This bit here, this 2u with respect to x, it\u0027s minus 3."},{"Start":"12:35.770 ","End":"12:39.340","Text":"Here with respect to u, it\u0027s just 1."},{"Start":"12:39.340 ","End":"12:47.860","Text":"Here with respect to x is minus 1."},{"Start":"12:47.860 ","End":"12:56.360","Text":"This bit comes out as minus 2u plus 3."},{"Start":"12:57.800 ","End":"13:01.095","Text":"I\u0027ll just copy that to here."},{"Start":"13:01.095 ","End":"13:05.830","Text":"Minus 2u plus 3."},{"Start":"13:06.720 ","End":"13:13.495","Text":"For the last 1, we\u0027ll just compute the Jacobian in the numerator."},{"Start":"13:13.495 ","End":"13:19.030","Text":"Just this bit would be,"},{"Start":"13:19.030 ","End":"13:21.310","Text":"we start off with the same thing, F,"},{"Start":"13:21.310 ","End":"13:28.705","Text":"G over u, v but in this 1 we need to replace v by y,"},{"Start":"13:28.705 ","End":"13:34.134","Text":"bye-bye v, hello y."},{"Start":"13:34.134 ","End":"13:40.840","Text":"That is the determinant of F with respect to u,"},{"Start":"13:40.840 ","End":"13:43.240","Text":"F with respect to y,"},{"Start":"13:43.240 ","End":"13:45.579","Text":"G with respect to u,"},{"Start":"13:45.579 ","End":"13:48.325","Text":"G with respect to y."},{"Start":"13:48.325 ","End":"13:52.210","Text":"This comes out to be, let\u0027s see,"},{"Start":"13:52.210 ","End":"13:56.184","Text":"F_u, we probably had it before."},{"Start":"13:56.184 ","End":"13:58.960","Text":"Yeah, sure."},{"Start":"13:58.960 ","End":"14:04.160","Text":"F_u was minus 2u or we could just do it again each time."},{"Start":"14:04.530 ","End":"14:14.770","Text":"F with respect to u is 2u and g with respect to y,"},{"Start":"14:14.770 ","End":"14:16.165","Text":"I better write it."},{"Start":"14:16.165 ","End":"14:21.970","Text":"It\u0027s 2u, then let\u0027s see,"},{"Start":"14:21.970 ","End":"14:26.920","Text":"F with respect to y is minus 1,"},{"Start":"14:26.920 ","End":"14:30.340","Text":"G with respect to u is 1,"},{"Start":"14:30.340 ","End":"14:33.895","Text":"G with respect to y is 2,"},{"Start":"14:33.895 ","End":"14:36.760","Text":"and this comes out to be 4u,"},{"Start":"14:36.760 ","End":"14:41.150","Text":"minus minus 1 is 4u plus 1."},{"Start":"14:42.420 ","End":"14:47.965","Text":"I\u0027ll put that here 4u plus 1."},{"Start":"14:47.965 ","End":"14:52.105","Text":"That\u0027s basically it except that I\u0027d like to just get rid of"},{"Start":"14:52.105 ","End":"14:57.865","Text":"the minuses in front and reverse the numerator."},{"Start":"14:57.865 ","End":"15:01.705","Text":"Here it\u0027s minus plus,"},{"Start":"15:01.705 ","End":"15:05.980","Text":"here it\u0027s minus minus,"},{"Start":"15:05.980 ","End":"15:09.610","Text":"here it\u0027s plus minus,"},{"Start":"15:09.610 ","End":"15:14.530","Text":"and here it will be minus and minus,"},{"Start":"15:14.530 ","End":"15:18.710","Text":"and these are the 4 quantities we\u0027re looking for and we\u0027re done."}],"ID":8957},{"Watched":false,"Name":"Exercise 9","Duration":"16m 13s","ChapterTopicVideoID":8679,"CourseChapterTopicPlaylistID":4961,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we\u0027re given a bunch of equations."},{"Start":"00:03.480 ","End":"00:07.005","Text":"Let\u0027s see, we have 3 equations."},{"Start":"00:07.005 ","End":"00:09.735","Text":"We also like to know how many variables there are,"},{"Start":"00:09.735 ","End":"00:14.145","Text":"we have x, y, u,"},{"Start":"00:14.145 ","End":"00:23.620","Text":"w, and v, that\u0027s 5 variables."},{"Start":"00:24.080 ","End":"00:26.520","Text":"If we think about it,"},{"Start":"00:26.520 ","End":"00:29.460","Text":"it means that we have 2 free variables."},{"Start":"00:29.460 ","End":"00:32.630","Text":"You take a number of variables minus the number of"},{"Start":"00:32.630 ","End":"00:36.015","Text":"constraints or equations, that gives us 2."},{"Start":"00:36.015 ","End":"00:43.915","Text":"We can fix 2 of them and then compute the other 3 from the 3 equations."},{"Start":"00:43.915 ","End":"00:47.330","Text":"Now, because of the nature of the question that we\u0027re asked to"},{"Start":"00:47.330 ","End":"00:52.055","Text":"find the partial derivatives of w with respect to x and y,"},{"Start":"00:52.055 ","End":"00:56.600","Text":"we assume that the 2 variables are going to be x and"},{"Start":"00:56.600 ","End":"01:03.689","Text":"y and that u is going to be u of x and y,"},{"Start":"01:03.689 ","End":"01:06.474","Text":"we\u0027re going to have v of x and y,"},{"Start":"01:06.474 ","End":"01:08.300","Text":"and w of x and y."},{"Start":"01:08.300 ","End":"01:11.385","Text":"The u, v, w will be functions of x and y."},{"Start":"01:11.385 ","End":"01:16.385","Text":"Now, the 3, we write these in,"},{"Start":"01:16.385 ","End":"01:20.360","Text":"I think it\u0027s called the auxiliary system or auxiliary equations, it doesn\u0027t matter."},{"Start":"01:20.360 ","End":"01:22.250","Text":"What we do is we want to define"},{"Start":"01:22.250 ","End":"01:25.640","Text":"functions which basically are the same as these equations,"},{"Start":"01:25.640 ","End":"01:28.070","Text":"but in the form something equals 0."},{"Start":"01:28.070 ","End":"01:30.590","Text":"In the first case, it doesn\u0027t matter which way I subtract,"},{"Start":"01:30.590 ","End":"01:31.970","Text":"I\u0027ll throw the x over,"},{"Start":"01:31.970 ","End":"01:38.660","Text":"I can take f. Now I want the function to be of all 5 variables."},{"Start":"01:38.660 ","End":"01:43.485","Text":"I want it to be let\u0027s say u, v, w,"},{"Start":"01:43.485 ","End":"01:46.190","Text":"x, y, even if they don\u0027t all appear,"},{"Start":"01:46.190 ","End":"01:50.100","Text":"it\u0027s going to be u plus v minus x."},{"Start":"01:51.710 ","End":"01:56.165","Text":"The first one basically just says F equals 0."},{"Start":"01:56.165 ","End":"02:00.860","Text":"The second one, I\u0027m going to call let\u0027s say capital G,"},{"Start":"02:00.860 ","End":"02:03.530","Text":"also of u, v, w,"},{"Start":"02:03.530 ","End":"02:06.940","Text":"x, y is from the second equation."},{"Start":"02:06.940 ","End":"02:15.270","Text":"To make it equal to 0, I\u0027ll take it as u squared plus v squared minus y."},{"Start":"02:15.270 ","End":"02:20.695","Text":"This equation is equivalent to saying that G equals 0."},{"Start":"02:20.695 ","End":"02:24.270","Text":"The last one, I\u0027ll call it H of u,"},{"Start":"02:24.270 ","End":"02:27.859","Text":"v, w, x, y."},{"Start":"02:27.859 ","End":"02:30.470","Text":"That will be from the last equation,"},{"Start":"02:30.470 ","End":"02:37.670","Text":"u cubed plus v cubed minus w. The last equation essentially says that"},{"Start":"02:37.670 ","End":"02:46.440","Text":"this thing is equal to 0 and these are the auxiliary equations system."},{"Start":"02:46.660 ","End":"02:52.445","Text":"Now, we\u0027re going to use the concept of the Jacobian."},{"Start":"02:52.445 ","End":"02:57.620","Text":"There is a theorem that tells us how to find each of these."},{"Start":"02:57.620 ","End":"03:04.000","Text":"If I want to find w with respect to x,"},{"Start":"03:04.850 ","End":"03:08.300","Text":"yeah, we\u0027ll go specifically for w with respect to x,"},{"Start":"03:08.300 ","End":"03:10.715","Text":"but it could apply to all 6 combinations."},{"Start":"03:10.715 ","End":"03:14.800","Text":"I have u, v, or w with respect to x or y,"},{"Start":"03:14.800 ","End":"03:17.030","Text":"so the 6 possible partial derivatives,"},{"Start":"03:17.030 ","End":"03:20.240","Text":"we just have to find those belonging to w. Now,"},{"Start":"03:20.240 ","End":"03:24.785","Text":"it always starts with a minus and there\u0027s always a dividing line."},{"Start":"03:24.785 ","End":"03:26.845","Text":"I\u0027ll need more space."},{"Start":"03:26.845 ","End":"03:30.020","Text":"On the denominator, it\u0027s always the same thing."},{"Start":"03:30.020 ","End":"03:34.699","Text":"It\u0027s the Jacobian of functions F, G,"},{"Start":"03:34.699 ","End":"03:39.320","Text":"H with respect to variables u, v,"},{"Start":"03:39.320 ","End":"03:43.940","Text":"and w, not the independent ones,"},{"Start":"03:43.940 ","End":"03:46.505","Text":"the other ones that are functions."},{"Start":"03:46.505 ","End":"03:52.500","Text":"Now, the trick is on the numerator,"},{"Start":"03:53.480 ","End":"03:57.885","Text":"to start off with the same thing I\u0027ll just copy it."},{"Start":"03:57.885 ","End":"04:00.955","Text":"Now here\u0027s the thing, you see W_x,"},{"Start":"04:00.955 ","End":"04:06.230","Text":"you look for w on the top and erase it and replace"},{"Start":"04:06.230 ","End":"04:12.330","Text":"it by the letter x. I\u0027ll just highlight those."},{"Start":"04:12.330 ","End":"04:17.910","Text":"Replace the w with x and that\u0027s the only difference."},{"Start":"04:17.910 ","End":"04:20.510","Text":"Now all the partial derivatives,"},{"Start":"04:20.510 ","End":"04:22.590","Text":"even if I had ux uy, vx,"},{"Start":"04:22.590 ","End":"04:25.385","Text":"vy, they all start off with the same denominator,"},{"Start":"04:25.385 ","End":"04:27.710","Text":"but each time I make a replacement."},{"Start":"04:27.710 ","End":"04:30.395","Text":"For example, in the next 1,"},{"Start":"04:30.395 ","End":"04:32.735","Text":"I\u0027ll do it in parallel."},{"Start":"04:32.735 ","End":"04:37.580","Text":"If I wanted to compute W with respect to y,"},{"Start":"04:37.580 ","End":"04:41.040","Text":"then this is equal to,"},{"Start":"04:41.380 ","End":"04:43.595","Text":"I just wrote the stuff out."},{"Start":"04:43.595 ","End":"04:46.690","Text":"We started off by this same thing, this F, G,"},{"Start":"04:46.690 ","End":"04:50.410","Text":"H over u, v, w Jacobian top and bottom."},{"Start":"04:50.410 ","End":"04:52.990","Text":"Now, we replace w by y on the top,"},{"Start":"04:52.990 ","End":"04:59.460","Text":"so with w in with y and this w,"},{"Start":"04:59.460 ","End":"05:07.380","Text":"I\u0027ll just highlight, w was replaced by y on the top."},{"Start":"05:07.910 ","End":"05:10.790","Text":"We have 3 computations to make."},{"Start":"05:10.790 ","End":"05:12.349","Text":"We have to compute this Jacobian."},{"Start":"05:12.349 ","End":"05:17.950","Text":"Let me just call this one a,"},{"Start":"05:17.950 ","End":"05:21.135","Text":"and I\u0027ll call this one b."},{"Start":"05:21.135 ","End":"05:26.245","Text":"This one is also a and this 1 I\u0027ll call it c,"},{"Start":"05:26.245 ","End":"05:27.710","Text":"and I\u0027ll do each side."},{"Start":"05:27.710 ","End":"05:29.000","Text":"Let\u0027s start off with a,"},{"Start":"05:29.000 ","End":"05:32.180","Text":"which is the common 1 in the denominator."},{"Start":"05:32.180 ","End":"05:36.350","Text":"If you expand it and if we go over to the other notation,"},{"Start":"05:36.350 ","End":"05:39.785","Text":"it\u0027s the determinant 3 by 3."},{"Start":"05:39.785 ","End":"05:42.465","Text":"Top row is F with respect to u,"},{"Start":"05:42.465 ","End":"05:44.235","Text":"F with respect to v,"},{"Start":"05:44.235 ","End":"05:48.380","Text":"F with respect to w. I\u0027m writing it in this notation."},{"Start":"05:48.380 ","End":"05:52.639","Text":"I\u0027m not going to write df by du because it\u0027s messier."},{"Start":"05:52.639 ","End":"05:58.320","Text":"Next row is G with respect to u, v, and w,"},{"Start":"05:58.320 ","End":"06:00.930","Text":"and then H with respect to u,"},{"Start":"06:00.930 ","End":"06:06.270","Text":"v and w. 3 by 3 determinant."},{"Start":"06:06.270 ","End":"06:09.620","Text":"Let\u0027s see, let\u0027s evaluate each of these."},{"Start":"06:09.620 ","End":"06:13.760","Text":"We\u0027ve got the functions F and so get the top row,"},{"Start":"06:13.760 ","End":"06:18.125","Text":"I just differentiate this with respect to u, v,"},{"Start":"06:18.125 ","End":"06:25.320","Text":"and w. This gives us respect to u it\u0027s 1,"},{"Start":"06:25.320 ","End":"06:26.985","Text":"respect to v it\u0027s 1,"},{"Start":"06:26.985 ","End":"06:35.370","Text":"there is no w and so it\u0027s 0."},{"Start":"06:35.370 ","End":"06:38.430","Text":"Then G with respect to u, v, and w,"},{"Start":"06:38.430 ","End":"06:41.475","Text":"that\u0027s G with respect to u,"},{"Start":"06:41.475 ","End":"06:45.210","Text":"it\u0027s 2u, everything else is constant."},{"Start":"06:45.210 ","End":"06:47.309","Text":"With respect to v, it\u0027s 2v,"},{"Start":"06:47.309 ","End":"06:48.884","Text":"everything else is constant,"},{"Start":"06:48.884 ","End":"06:54.855","Text":"and with respect to w, nothing because there is no w, it\u0027s all constants."},{"Start":"06:54.855 ","End":"06:59.220","Text":"Then as far as H goes with respect to u,"},{"Start":"06:59.220 ","End":"07:03.140","Text":"it\u0027s 3u squared and then 3v squared."},{"Start":"07:03.140 ","End":"07:05.405","Text":"But we do have a w here,"},{"Start":"07:05.405 ","End":"07:10.990","Text":"and its derivative is minus 1 and the rest of it is a constant."},{"Start":"07:10.990 ","End":"07:14.480","Text":"Now we have a 3 by 3 determinant,"},{"Start":"07:14.480 ","End":"07:16.130","Text":"and we\u0027ll get an expression in u and"},{"Start":"07:16.130 ","End":"07:21.590","Text":"v. I\u0027m assuming you know how to compute determinants."},{"Start":"07:21.590 ","End":"07:23.660","Text":"I\u0027m going to do it 1 way, there are several ways."},{"Start":"07:23.660 ","End":"07:25.655","Text":"If you know at least 1 way you should do it,"},{"Start":"07:25.655 ","End":"07:28.025","Text":"you don\u0027t know determinants, you should look them up."},{"Start":"07:28.025 ","End":"07:31.160","Text":"I\u0027m going to expand"},{"Start":"07:31.160 ","End":"07:36.875","Text":"this determinant by the last column because I\u0027ve got a lot of 0s there."},{"Start":"07:36.875 ","End":"07:38.750","Text":"I\u0027m going to expand by this element,"},{"Start":"07:38.750 ","End":"07:43.745","Text":"which means that what we do is we erase the row and column containing it."},{"Start":"07:43.745 ","End":"07:48.320","Text":"We figure out whether it\u0027s a plus or minus checkerboard style plus, minus, plus,"},{"Start":"07:48.320 ","End":"07:53.860","Text":"minus, plus or anyway, it\u0027s a plus."},{"Start":"07:53.860 ","End":"07:56.100","Text":"Just don\u0027t write anything."},{"Start":"07:56.100 ","End":"08:00.000","Text":"Then we take the co-factor,"},{"Start":"08:00.000 ","End":"08:03.240","Text":"which is like what\u0027s remaining,"},{"Start":"08:03.240 ","End":"08:07.560","Text":"this part here, and take its determinant."},{"Start":"08:07.560 ","End":"08:12.740","Text":"We have minus this element times this determinant."},{"Start":"08:12.740 ","End":"08:20.020","Text":"We have minus 1 times 1, 1, 2u, 2v."},{"Start":"08:20.020 ","End":"08:24.065","Text":"We would\u0027ve had an extra plus or minus if we hadn\u0027t landed on plus,"},{"Start":"08:24.065 ","End":"08:25.700","Text":"plus, minus, plus, minus,"},{"Start":"08:25.700 ","End":"08:27.755","Text":"plus, just go checkerboard."},{"Start":"08:27.755 ","End":"08:30.395","Text":"This one, you start with a plus in the upper left,"},{"Start":"08:30.395 ","End":"08:34.670","Text":"and every alternate on is minus."},{"Start":"08:34.670 ","End":"08:38.210","Text":"This is a 2 by 2 determinant,"},{"Start":"08:38.210 ","End":"08:41.465","Text":"we do this diagonal minus this diagonal,"},{"Start":"08:41.465 ","End":"08:42.680","Text":"because of the minus,"},{"Start":"08:42.680 ","End":"08:45.145","Text":"I\u0027m going to do this diagonal minus this diagonal,"},{"Start":"08:45.145 ","End":"08:51.385","Text":"is going to be 2v minus 2u."},{"Start":"08:51.385 ","End":"08:53.635","Text":"That\u0027s part a done."},{"Start":"08:53.635 ","End":"08:59.615","Text":"Now let\u0027s do determinant number b. I\u0027ll scroll,"},{"Start":"08:59.615 ","End":"09:02.400","Text":"l have room here."},{"Start":"09:03.450 ","End":"09:08.860","Text":"For b, we need the determinant."},{"Start":"09:08.860 ","End":"09:11.725","Text":"Starts out the same,"},{"Start":"09:11.725 ","End":"09:15.400","Text":"except that w is replaced by x,"},{"Start":"09:15.400 ","End":"09:20.275","Text":"so why don\u0027t I just copy this here,"},{"Start":"09:20.275 ","End":"09:22.180","Text":"but notice, we replace w by x,"},{"Start":"09:22.180 ","End":"09:27.155","Text":"so I\u0027m going to erase the w\u0027s and add x\u0027s."},{"Start":"09:27.155 ","End":"09:32.785","Text":"Let\u0027s see what we get this time, well,"},{"Start":"09:32.785 ","End":"09:37.285","Text":"the first 2 columns are the same, so it\u0027s going to be 1,"},{"Start":"09:37.285 ","End":"09:43.510","Text":"2u, 3u squared, 1, 2v, 3v squared."},{"Start":"09:43.510 ","End":"09:47.620","Text":"Then all I need is, the derivatives with respect to x of F,"},{"Start":"09:47.620 ","End":"09:52.330","Text":"G, and H, I\u0027ll need to go back up, they\u0027re still here."},{"Start":"09:52.330 ","End":"09:55.105","Text":"With respect to x,"},{"Start":"09:55.105 ","End":"09:58.735","Text":"the first 1 gives me minus 1,"},{"Start":"09:58.735 ","End":"10:01.750","Text":"the second 1 doesn\u0027t have any x\u0027s, so that\u0027s a 0,"},{"Start":"10:01.750 ","End":"10:03.670","Text":"the third 1 doesn\u0027t have x\u0027s,"},{"Start":"10:03.670 ","End":"10:08.080","Text":"so that\u0027s a 0, and so this determinant."},{"Start":"10:08.080 ","End":"10:13.470","Text":"This time, I\u0027ll expand using this element."},{"Start":"10:13.470 ","End":"10:16.350","Text":"We, again, do the checkerboard, plus, minus,"},{"Start":"10:16.350 ","End":"10:18.000","Text":"plus, it\u0027s also a plus,"},{"Start":"10:18.000 ","End":"10:19.845","Text":"so no need for extra adjustment."},{"Start":"10:19.845 ","End":"10:24.400","Text":"We take the value of this times the determinant of the cofactor,"},{"Start":"10:24.400 ","End":"10:28.040","Text":"which we would get by erasing row and column."},{"Start":"10:28.620 ","End":"10:39.130","Text":"We get minus 1 times determinant of 2u, 2v, 3u squared,"},{"Start":"10:39.130 ","End":"10:42.790","Text":"3v squared, and normally,"},{"Start":"10:42.790 ","End":"10:44.530","Text":"this diagonal product minus"},{"Start":"10:44.530 ","End":"10:47.410","Text":"this diagonal product has the minus or do it in the reverse order,"},{"Start":"10:47.410 ","End":"10:52.070","Text":"so this is going to be the plus 6u squared v,"},{"Start":"10:52.380 ","End":"10:57.620","Text":"minus this is 6uv squared."},{"Start":"10:58.920 ","End":"11:09.565","Text":"That\u0027s b, now we need c. I need to scroll a bit and lose some of the stuff, never mind."},{"Start":"11:09.565 ","End":"11:15.640","Text":"C, I just copied this which is also this,"},{"Start":"11:15.640 ","End":"11:18.250","Text":"and I\u0027m going to replace w with y,"},{"Start":"11:18.250 ","End":"11:22.015","Text":"so erase the w\u0027s,"},{"Start":"11:22.015 ","End":"11:25.810","Text":"and put y instead,"},{"Start":"11:25.810 ","End":"11:28.760","Text":"and let\u0027s see what we get."},{"Start":"11:29.940 ","End":"11:38.170","Text":"Well, again, these 2 are the same as above so 1, 1, 2u, 2v,"},{"Start":"11:38.170 ","End":"11:41.515","Text":"3u squared, 3v squared,"},{"Start":"11:41.515 ","End":"11:46.495","Text":"and here I have to have the derivatives of the 3 auxiliaries with respect to y,"},{"Start":"11:46.495 ","End":"11:50.365","Text":"I\u0027ll just go back up and see what we get with respect to y."},{"Start":"11:50.365 ","End":"11:55.120","Text":"Here we get 0, here we get minus 1,"},{"Start":"11:55.120 ","End":"11:57.130","Text":"and here we get 0."},{"Start":"11:57.130 ","End":"12:00.010","Text":"If I remember correctly,"},{"Start":"12:00.010 ","End":"12:05.690","Text":"it was 0, minus 1, 0."},{"Start":"12:06.030 ","End":"12:10.464","Text":"Now I\u0027m going to expand by this column,"},{"Start":"12:10.464 ","End":"12:13.555","Text":"which just only has this element."},{"Start":"12:13.555 ","End":"12:15.910","Text":"This time plus, minus, plus,"},{"Start":"12:15.910 ","End":"12:17.410","Text":"this 1 is a minus,"},{"Start":"12:17.410 ","End":"12:19.645","Text":"so I need an extra minus in here,"},{"Start":"12:19.645 ","End":"12:24.805","Text":"and then I erase the column and the row that remains,"},{"Start":"12:24.805 ","End":"12:29.740","Text":"and we get this thing times what\u0027s left is"},{"Start":"12:29.740 ","End":"12:37.105","Text":"the co-factor so we get,"},{"Start":"12:37.105 ","End":"12:41.020","Text":"first of all, the minus that I wrote here to indicate plus, minus,"},{"Start":"12:41.020 ","End":"12:45.565","Text":"plus, minus, then this element which is minus 1,"},{"Start":"12:45.565 ","End":"12:51.205","Text":"and then the determinant of what\u0027s left, which is 1,"},{"Start":"12:51.205 ","End":"12:58.495","Text":"1, 3u squared, 3v squared."},{"Start":"12:58.495 ","End":"13:00.895","Text":"This minus, minus is a plus,"},{"Start":"13:00.895 ","End":"13:02.710","Text":"so it\u0027s this diagonal,"},{"Start":"13:02.710 ","End":"13:08.230","Text":"3v squared minus the other diagonal 3u squared."},{"Start":"13:08.230 ","End":"13:11.425","Text":"Now, each of these can be factored,"},{"Start":"13:11.425 ","End":"13:15.745","Text":"this 1, I can do as 2u minus v,"},{"Start":"13:15.745 ","End":"13:19.525","Text":"here I\u0027ve got 6uv in common,"},{"Start":"13:19.525 ","End":"13:24.640","Text":"and again, u minus v. It turns out here also,"},{"Start":"13:24.640 ","End":"13:32.335","Text":"I can do 3 v squared minus u squared,"},{"Start":"13:32.335 ","End":"13:36.800","Text":"but v squared minus u squared"},{"Start":"13:36.920 ","End":"13:44.380","Text":"is just v minus u, v plus u."},{"Start":"13:45.090 ","End":"13:47.680","Text":"The difference of square is formula."},{"Start":"13:47.680 ","End":"13:50.140","Text":"In general, a squared minus b squared is a minus b,"},{"Start":"13:50.140 ","End":"13:52.855","Text":"a plus b, or a plus b, a minus b."},{"Start":"13:52.855 ","End":"13:59.215","Text":"Now notice that we have u minus v in all of them."},{"Start":"13:59.215 ","End":"14:03.715","Text":"Well, u minus v is backwards of v minus u, we\u0027ll handle that."},{"Start":"14:03.715 ","End":"14:06.355","Text":"Let\u0027s just see what we get."},{"Start":"14:06.355 ","End":"14:12.805","Text":"For W_x, we get minus this which is b,"},{"Start":"14:12.805 ","End":"14:21.625","Text":"which is 6uv u minus v over a,"},{"Start":"14:21.625 ","End":"14:32.035","Text":"which is twice u minus v. I need a little bit more room here to squash it."},{"Start":"14:32.035 ","End":"14:41.500","Text":"Here we have this equals minus part c,"},{"Start":"14:41.500 ","End":"14:47.950","Text":"which is 3v minus u, v plus u."},{"Start":"14:47.950 ","End":"14:50.020","Text":"Leave it the way,"},{"Start":"14:50.020 ","End":"14:57.430","Text":"it was originally v squared minus u squared over a"},{"Start":"14:57.430 ","End":"15:05.800","Text":"is twice u minus v. Now cancellation,"},{"Start":"15:05.800 ","End":"15:09.670","Text":"u minus v goes into u minus v evenly,"},{"Start":"15:09.670 ","End":"15:13.730","Text":"2 goes into 6, 3 times."},{"Start":"15:15.090 ","End":"15:18.280","Text":"Let me just record that somewhere."},{"Start":"15:18.280 ","End":"15:23.720","Text":"We have W_x is equal to minus 3uv,"},{"Start":"15:23.970 ","End":"15:28.225","Text":"and in a moment, we\u0027ll see what W_y is."},{"Start":"15:28.225 ","End":"15:33.970","Text":"W_y will be, I\u0027m back to canceling here,"},{"Start":"15:33.970 ","End":"15:39.979","Text":"u minus v goes into v squared minus u squared,"},{"Start":"15:40.320 ","End":"15:48.385","Text":"not exactly u minus v times v minus u times."},{"Start":"15:48.385 ","End":"15:50.905","Text":"But If I also take it with a minus,"},{"Start":"15:50.905 ","End":"15:52.825","Text":"then it will cancel,"},{"Start":"15:52.825 ","End":"15:58.435","Text":"and we\u0027ll be just left with u plus v. So we\u0027re left here with no minus,"},{"Start":"15:58.435 ","End":"16:02.965","Text":"just 3 over 2 times"},{"Start":"16:02.965 ","End":"16:10.269","Text":"u plus v. This is our answer,"},{"Start":"16:10.269 ","End":"16:12.470","Text":"and we are done."}],"ID":8958}],"Thumbnail":null,"ID":4961}]

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1.1

1