Introduction to Partial Derivative
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- Partial Derivatives
- Partial Derivatives of the Second Order
- Exercise 1 part a
- Exercise 1 part b
- Exercise 1 part c
- Exercise 1 part d
- Exercise 1 part e
- Exercise 1 part f
- Exercise 1 part g
- Exercise 1 part h
- Exercise 1 part i
- Exercise 1 part j
- Exercise 2 part a
- Exercise 2 part b
- Exercise 2 part c
- Exercise 2 part d
- Exercise 3
- Exercise 4
- Exercise 5 part a
- Exercise 5 part b
- Exercise 6

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[{"Name":"Introduction to Partial Derivative","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Partial Derivatives","Duration":"19m 43s","ChapterTopicVideoID":8576,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8576.jpeg","UploadDate":"2020-02-23T14:57:51.9270000","DurationForVideoObject":"PT19M43S","Description":null,"MetaTitle":"Partial Derivatives: Video + Workbook | Proprep","MetaDescription":"Partial Derivative - Introduction to Partial Derivative. 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/partial-derivative/introduction-to-partial-derivative/vid20871","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.800","Text":"This clip is titled partial derivatives but actually,"},{"Start":"00:04.800 ","End":"00:10.260","Text":"what it\u0027s about is how to differentiate a function of 2 variables."},{"Start":"00:10.260 ","End":"00:12.675","Text":"Let\u0027s start with an example."},{"Start":"00:12.675 ","End":"00:17.400","Text":"Let\u0027s take the function f of x,"},{"Start":"00:17.400 ","End":"00:23.655","Text":"y is equal to x squared plus y squared."},{"Start":"00:23.655 ","End":"00:29.670","Text":"Now, if I was to ask you to take a guess of what the derivative of this is,"},{"Start":"00:29.670 ","End":"00:40.590","Text":"you might say that we would take the derivative is equal to 2x from x squared plus 2y,"},{"Start":"00:40.590 ","End":"00:43.185","Text":"and you\u0027d be wrong."},{"Start":"00:43.185 ","End":"00:46.100","Text":"Although it\u0027s a reasonable guess and in fact,"},{"Start":"00:46.100 ","End":"00:48.005","Text":"we could define it this way,"},{"Start":"00:48.005 ","End":"00:49.910","Text":"but the trouble is there\u0027s nothing useful that"},{"Start":"00:49.910 ","End":"00:51.860","Text":"can be done with it if you define it this way."},{"Start":"00:51.860 ","End":"00:57.769","Text":"You can\u0027t for example find extrema and all the useful things we do with derivatives."},{"Start":"00:57.769 ","End":"01:00.440","Text":"This is not the way to go about it and in fact,"},{"Start":"01:00.440 ","End":"01:02.630","Text":"because it\u0027s wrong, I want to erase it."},{"Start":"01:02.630 ","End":"01:06.350","Text":"It turns out that there is no 1 single derivative and"},{"Start":"01:06.350 ","End":"01:09.950","Text":"what we have to do is to define 2 derivatives,"},{"Start":"01:09.950 ","End":"01:14.375","Text":"1 with respect to x and the other with respect to y."},{"Start":"01:14.375 ","End":"01:24.425","Text":"These are called f with respect to x and f with respect to y."},{"Start":"01:24.425 ","End":"01:31.535","Text":"What we do when we take the derivative of f with respect to x,"},{"Start":"01:31.535 ","End":"01:34.655","Text":"of x and y."},{"Start":"01:34.655 ","End":"01:39.180","Text":"This is 1 and this is the other."},{"Start":"01:41.240 ","End":"01:47.060","Text":"The derivative with respect to x means that we take x"},{"Start":"01:47.060 ","End":"01:52.085","Text":"as our variable and y is a constant or parameter."},{"Start":"01:52.085 ","End":"01:53.480","Text":"I\u0027ll write it as parameter,"},{"Start":"01:53.480 ","End":"01:55.300","Text":"but parameter basically means the constant,"},{"Start":"01:55.300 ","End":"01:57.200","Text":"we just don\u0027t know what the constant is."},{"Start":"01:57.200 ","End":"01:58.910","Text":"Whereas in the other case, of course,"},{"Start":"01:58.910 ","End":"02:01.955","Text":"we want x as a parameter."},{"Start":"02:01.955 ","End":"02:03.890","Text":"Read that x as a constant,"},{"Start":"02:03.890 ","End":"02:08.345","Text":"if you like, and y is the variable."},{"Start":"02:08.345 ","End":"02:11.585","Text":"If I look at the first example,"},{"Start":"02:11.585 ","End":"02:17.300","Text":"I take x squared plus y squared and I take x as the variable."},{"Start":"02:17.300 ","End":"02:22.715","Text":"y is just some constant like x squared plus a squared or x squared plus 3 squared."},{"Start":"02:22.715 ","End":"02:31.205","Text":"The derivative of x squared is 2x and this being a constant, goes to 0."},{"Start":"02:31.205 ","End":"02:34.540","Text":"I\u0027ll emphasize that by writing the 0 in."},{"Start":"02:34.540 ","End":"02:37.085","Text":"Likewise in this case,"},{"Start":"02:37.085 ","End":"02:38.360","Text":"x is the parameter,"},{"Start":"02:38.360 ","End":"02:40.145","Text":"so you differentiate it,"},{"Start":"02:40.145 ","End":"02:45.095","Text":"you get 0 and the derivative of y squared is 2y."},{"Start":"02:45.095 ","End":"02:48.745","Text":"Sometimes we write it with a prime,"},{"Start":"02:48.745 ","End":"02:52.580","Text":"I\u0027ll just put it in gray to say that this is optional."},{"Start":"02:52.580 ","End":"02:58.190","Text":"If you like to remind yourself that it\u0027s a derivative with a prime here,"},{"Start":"02:58.190 ","End":"03:01.580","Text":"that\u0027s okay, some books do it that way."},{"Start":"03:01.580 ","End":"03:06.950","Text":"Actually, there are other notations for partial derivatives."},{"Start":"03:06.950 ","End":"03:12.340","Text":"Another 1 that\u0027s very acceptable is to write it like this."},{"Start":"03:12.340 ","End":"03:20.045","Text":"This letter is some old-fashioned d. It reminds me of the functions in 1 variable."},{"Start":"03:20.045 ","End":"03:22.580","Text":"When we have a function f of x,"},{"Start":"03:22.580 ","End":"03:26.285","Text":"let\u0027s say f of x is equal to x squared,"},{"Start":"03:26.285 ","End":"03:31.970","Text":"then we usually write f-prime of x as the derivative is 2x,"},{"Start":"03:31.970 ","End":"03:39.260","Text":"but some use an alternative notation and write df over dx is equal to 2x."},{"Start":"03:39.260 ","End":"03:41.245","Text":"This is a regular d,"},{"Start":"03:41.245 ","End":"03:42.650","Text":"but with partial derivatives,"},{"Start":"03:42.650 ","End":"03:49.025","Text":"we use this fancy or old-fashion d. This is equal to 2x and the same thing here."},{"Start":"03:49.025 ","End":"03:52.010","Text":"For y, we can say that df,"},{"Start":"03:52.010 ","End":"03:56.400","Text":"I pronounce it d but it\u0027s written this way,"},{"Start":"03:56.400 ","End":"04:02.125","Text":"not exactly sure, over dy would equal 2y."},{"Start":"04:02.125 ","End":"04:05.075","Text":"In economics, they use yet another system."},{"Start":"04:05.075 ","End":"04:11.400","Text":"You would write that f_1 of x,y is equal to 2x."},{"Start":"04:11.400 ","End":"04:17.130","Text":"The 1 indicates the derivative according to the 1st variable and likewise,"},{"Start":"04:17.130 ","End":"04:20.270","Text":"f_2 is the derivative according to the 2nd variable."},{"Start":"04:20.270 ","End":"04:21.710","Text":"That\u0027s mostly in economics,"},{"Start":"04:21.710 ","End":"04:24.400","Text":"we won\u0027t be using it in this clip,"},{"Start":"04:24.400 ","End":"04:26.470","Text":"just so you\u0027re familiar with it."},{"Start":"04:26.470 ","End":"04:30.455","Text":"Enough with notation, let\u0027s get onto some more examples."},{"Start":"04:30.455 ","End":"04:32.990","Text":"Before that, I forget if I mentioned it,"},{"Start":"04:32.990 ","End":"04:38.150","Text":"but this derivative is called the partial derivative of f with respect to x,"},{"Start":"04:38.150 ","End":"04:43.085","Text":"and this 1 or these are the partial derivative of f with respect to y."},{"Start":"04:43.085 ","End":"04:45.835","Text":"That\u0027s where the partial derivatives come in."},{"Start":"04:45.835 ","End":"04:48.570","Text":"Let\u0027s get on to an example."},{"Start":"04:48.570 ","End":"04:54.020","Text":"Let\u0027s take the example f of x and"},{"Start":"04:54.020 ","End":"04:59.375","Text":"y is equal to x squared times y^4,"},{"Start":"04:59.375 ","End":"05:02.090","Text":"I want both partial derivatives."},{"Start":"05:02.090 ","End":"05:05.975","Text":"First, let\u0027s take the partial derivative with respect to x."},{"Start":"05:05.975 ","End":"05:08.665","Text":"When I take it with respect to x,"},{"Start":"05:08.665 ","End":"05:16.205","Text":"remember it means that x is the variable and y is the constant or a parameter."},{"Start":"05:16.205 ","End":"05:20.630","Text":"Now, what does it mean that y is the parameter or a constant?"},{"Start":"05:20.630 ","End":"05:26.630","Text":"I\u0027ll do it at the side. It\u0027s as if I had x squared times 9."},{"Start":"05:26.630 ","End":"05:30.005","Text":"If I differentiate this with respect to x,"},{"Start":"05:30.005 ","End":"05:32.565","Text":"the 9 is a constant,"},{"Start":"05:32.565 ","End":"05:39.225","Text":"so I\u0027d get 2x from this and times the 9."},{"Start":"05:39.225 ","End":"05:42.765","Text":"The 9 just stays if it\u0027s a constant."},{"Start":"05:42.765 ","End":"05:45.330","Text":"In the end I\u0027d write 18x."},{"Start":"05:45.330 ","End":"05:47.775","Text":"Well, very similar here."},{"Start":"05:47.775 ","End":"05:51.630","Text":"I have x squared times some constant."},{"Start":"05:51.630 ","End":"05:58.084","Text":"The derivative is 2x times this constant."},{"Start":"05:58.084 ","End":"05:59.780","Text":"Often we put the constant in front,"},{"Start":"05:59.780 ","End":"06:03.935","Text":"but in this case, we can leave it at the 2nd place, 2xy^4."},{"Start":"06:03.935 ","End":"06:07.789","Text":"Then if I want the derivative with respect to y,"},{"Start":"06:07.789 ","End":"06:09.230","Text":"then it\u0027s the opposite."},{"Start":"06:09.230 ","End":"06:11.000","Text":"Then in this case,"},{"Start":"06:11.000 ","End":"06:17.510","Text":"we have that y is the variable and x is the constant or parameter."},{"Start":"06:17.510 ","End":"06:21.990","Text":"Suppose we had 10y^4,"},{"Start":"06:24.890 ","End":"06:27.780","Text":"and we differentiated this,"},{"Start":"06:27.780 ","End":"06:34.635","Text":"we would get 10 times 4y cubed and ultimately 40y cubed."},{"Start":"06:34.635 ","End":"06:39.580","Text":"Here, same thing, the 2x is just the constant parameter."},{"Start":"06:39.770 ","End":"06:44.170","Text":"Then multiply by 4y cubed."},{"Start":"06:44.170 ","End":"06:50.990","Text":"I\u0027ll just collect the constants together and this will be 8xy cubed."},{"Start":"06:50.990 ","End":"06:54.420","Text":"Similar to this, instead of the 10,"},{"Start":"06:54.420 ","End":"06:56.505","Text":"I have the x squared,"},{"Start":"06:56.505 ","End":"06:59.655","Text":"and that\u0027s like a constant parameter,"},{"Start":"06:59.655 ","End":"07:04.230","Text":"so I get x squared times"},{"Start":"07:04.230 ","End":"07:11.720","Text":"4y cubed which I just rewrite as 4x squared y cubed, and that\u0027s it."},{"Start":"07:11.720 ","End":"07:15.260","Text":"The derivative with respect to x is this,"},{"Start":"07:15.260 ","End":"07:18.305","Text":"and the derivative with respect to y is this."},{"Start":"07:18.305 ","End":"07:22.190","Text":"We don\u0027t have 1 derivative now like in 1 variable,"},{"Start":"07:22.190 ","End":"07:23.420","Text":"we have 2 variables,"},{"Start":"07:23.420 ","End":"07:25.535","Text":"so we have 2 derivatives."},{"Start":"07:25.535 ","End":"07:28.410","Text":"Next example, f of x,"},{"Start":"07:28.410 ","End":"07:33.315","Text":"y is x squared plus y times 4x minus 10y."},{"Start":"07:33.315 ","End":"07:37.875","Text":"Let\u0027s go first according to x."},{"Start":"07:37.875 ","End":"07:40.550","Text":"We\u0027ll take the derivative with respect to x,"},{"Start":"07:40.550 ","End":"07:43.820","Text":"and remember this means that y is just a parameter,"},{"Start":"07:43.820 ","End":"07:45.830","Text":"that x is the variable,"},{"Start":"07:45.830 ","End":"07:47.975","Text":"but we have a product."},{"Start":"07:47.975 ","End":"07:53.285","Text":"Let\u0027s use the product rule and I want to remind you what the product rule is."},{"Start":"07:53.285 ","End":"08:01.490","Text":"That uv-derivative is derivative of u times v plus"},{"Start":"08:01.490 ","End":"08:11.000","Text":"u times the derivative of v. I want to derive this and take this as is."},{"Start":"08:11.000 ","End":"08:13.570","Text":"I derive this, I\u0027ll just indicate"},{"Start":"08:13.570 ","End":"08:17.540","Text":"the derivative but I want to actually differentiate yet."},{"Start":"08:17.540 ","End":"08:21.995","Text":"It\u0027s the derivative with respect to x."},{"Start":"08:21.995 ","End":"08:25.510","Text":"That\u0027s how we write when we have an expression,"},{"Start":"08:25.510 ","End":"08:26.600","Text":"derivative with respect to x."},{"Start":"08:26.600 ","End":"08:29.735","Text":"You do include this prime sign."},{"Start":"08:29.735 ","End":"08:39.570","Text":"Times 4x minus 10y plus x squared plus y,"},{"Start":"08:39.570 ","End":"08:48.500","Text":"as is, times the derivative of 4x minus 10y also according to x."},{"Start":"08:48.500 ","End":"08:50.930","Text":"Because it\u0027s according to x here,"},{"Start":"08:50.930 ","End":"08:55.690","Text":"here also with respect to x, and let\u0027s expand."},{"Start":"08:55.690 ","End":"08:57.635","Text":"We get, let\u0027s see,"},{"Start":"08:57.635 ","End":"09:03.455","Text":"the derivative of x squared plus y with respect to x is just 2x because y is a constant,"},{"Start":"09:03.455 ","End":"09:11.865","Text":"so it\u0027s 2x times 4x minus 10y plus x squared plus y as it is."},{"Start":"09:11.865 ","End":"09:14.915","Text":"The derivative of this 10y again,"},{"Start":"09:14.915 ","End":"09:22.400","Text":"is a constant as far as x is concerned so we just get the 4 from 4x."},{"Start":"09:22.400 ","End":"09:28.865","Text":"Let\u0027s expand, we get 8x squared minus"},{"Start":"09:28.865 ","End":"09:35.280","Text":"20xy plus 4x squared plus 4y."},{"Start":"09:35.280 ","End":"09:38.385","Text":"I guess that\u0027s it as far as x goes."},{"Start":"09:38.385 ","End":"09:42.439","Text":"Now, the derivative of f according to y,"},{"Start":"09:42.439 ","End":"09:48.340","Text":"same product rule, but this time y is the variable and x as the parameter."},{"Start":"09:48.340 ","End":"09:51.080","Text":"The derivative of x squared plus"},{"Start":"09:51.080 ","End":"09:58.920","Text":"y with respect to y this time and the other 1,"},{"Start":"09:58.920 ","End":"10:08.860","Text":"just as is 4x minus 10y plus x squared plus y as is times the derivative of this,"},{"Start":"10:08.860 ","End":"10:13.240","Text":"also derivative with respect to y."},{"Start":"10:13.240 ","End":"10:19.825","Text":"Notice the notation, because it\u0027s with respect to y here,"},{"Start":"10:19.825 ","End":"10:22.930","Text":"we also take this with respect to y and this with respect to"},{"Start":"10:22.930 ","End":"10:26.815","Text":"y and customary to leave the prime sign here."},{"Start":"10:26.815 ","End":"10:30.160","Text":"What we get is the derivative of this with respect"},{"Start":"10:30.160 ","End":"10:33.640","Text":"to y is just 1 because the x squared is a constant."},{"Start":"10:33.640 ","End":"10:44.764","Text":"It\u0027s 1 times 4x minus 10y plus x squared plus y,"},{"Start":"10:44.764 ","End":"10:49.310","Text":"and the derivative of this is minus 10 with"},{"Start":"10:49.310 ","End":"10:53.615","Text":"respect to y. I\u0027m not going to bother expanding this."},{"Start":"10:53.615 ","End":"10:55.700","Text":"The purpose is to just get the derivative,"},{"Start":"10:55.700 ","End":"10:57.530","Text":"I don\u0027t have to tidy up."},{"Start":"10:57.530 ","End":"11:00.570","Text":"That\u0027s this example."},{"Start":"11:00.570 ","End":"11:07.035","Text":"I\u0027ve just highlighted this just so we remember that with 2 variables,"},{"Start":"11:07.035 ","End":"11:09.250","Text":"we want 2 derivatives."},{"Start":"11:09.250 ","End":"11:15.485","Text":"Note also that the rules such as the product rule from 1 variable,"},{"Start":"11:15.485 ","End":"11:19.100","Text":"also apply in the case of partial derivatives,"},{"Start":"11:19.100 ","End":"11:23.540","Text":"because a partial derivative really is 1 variable."},{"Start":"11:23.540 ","End":"11:27.420","Text":"We\u0027ve did an example of the product rule,"},{"Start":"11:27.420 ","End":"11:31.620","Text":"let\u0027s do another example this time with the quotient rule."},{"Start":"11:32.970 ","End":"11:37.900","Text":"This time, we\u0027ll take f of x,"},{"Start":"11:37.900 ","End":"11:42.610","Text":"y equals a quotient x squared"},{"Start":"11:42.610 ","End":"11:49.480","Text":"plus y squared over 4x plus y."},{"Start":"11:49.480 ","End":"11:52.915","Text":"First, the derivative with respect to x,"},{"Start":"11:52.915 ","End":"11:56.710","Text":"and I\u0027ll remind you of the quotient rule in 1 variable,"},{"Start":"11:56.710 ","End":"12:00.820","Text":"the derivative of u over v is u"},{"Start":"12:00.820 ","End":"12:08.815","Text":"prime v minus uv prime over v squared."},{"Start":"12:08.815 ","End":"12:14.470","Text":"We get derivative of this,"},{"Start":"12:14.470 ","End":"12:24.310","Text":"x squared plus y derivative with respect to x times 4x plus y"},{"Start":"12:24.310 ","End":"12:30.370","Text":"minus this as is x"},{"Start":"12:30.370 ","End":"12:37.610","Text":"squared plus y times the derivative of 4x plus y."},{"Start":"12:38.010 ","End":"12:44.365","Text":"Notice again that we have derivative according to x here."},{"Start":"12:44.365 ","End":"12:49.450","Text":"Derivative according to x here and here over v squared,"},{"Start":"12:49.450 ","End":"12:54.835","Text":"so here that\u0027s 4x plus y all squared."},{"Start":"12:54.835 ","End":"13:04.120","Text":"We get over 4x plus y squared. That\u0027s the easy bit."},{"Start":"13:04.120 ","End":"13:10.374","Text":"Now, derivative of x squared plus y with respect to x is just 2x because y is a constant,"},{"Start":"13:10.374 ","End":"13:17.200","Text":"times 4x plus y minus x squared plus y."},{"Start":"13:17.200 ","End":"13:22.465","Text":"The derivative of this with respect to x would be just 4,"},{"Start":"13:22.465 ","End":"13:25.540","Text":"because 4x plus a constant."},{"Start":"13:25.540 ","End":"13:29.500","Text":"It\u0027s just 4 and I\u0027ll leave it at that."},{"Start":"13:29.500 ","End":"13:32.770","Text":"That\u0027s the derivative, simplification is another matter."},{"Start":"13:32.770 ","End":"13:36.475","Text":"We see that the product rule"},{"Start":"13:36.475 ","End":"13:41.005","Text":"and the quotient rule also apply in the case of partial derivatives."},{"Start":"13:41.005 ","End":"13:45.384","Text":"Next, I\u0027d like to show you that the chain rule also works."},{"Start":"13:45.384 ","End":"13:50.005","Text":"Let\u0027s take the example, f of x,"},{"Start":"13:50.005 ","End":"13:57.715","Text":"y equals x squared plus y cubed,"},{"Start":"13:57.715 ","End":"13:59.845","Text":"all to the power of 4."},{"Start":"13:59.845 ","End":"14:01.345","Text":"Now the chain rule,"},{"Start":"14:01.345 ","End":"14:03.399","Text":"at least in 1 variable,"},{"Start":"14:03.399 ","End":"14:09.385","Text":"says that if we have something to the power of 4 and we take the derivative,"},{"Start":"14:09.385 ","End":"14:18.534","Text":"then it\u0027s 4 times that something to the power of 3 times the inner derivative, box prime."},{"Start":"14:18.534 ","End":"14:23.440","Text":"Well, the same thing works here with partial derivatives."},{"Start":"14:23.440 ","End":"14:28.760","Text":"If I take the derivative with respect to x,"},{"Start":"14:34.140 ","End":"14:37.315","Text":"this thing in the brackets is box,"},{"Start":"14:37.315 ","End":"14:41.920","Text":"4 times x squared plus y cubed^3,"},{"Start":"14:41.920 ","End":"14:47.275","Text":"the derivative of x squared plus y cubed,"},{"Start":"14:47.275 ","End":"14:50.570","Text":"but with respect to x."},{"Start":"14:51.720 ","End":"14:57.040","Text":"This equals 4x squared plus y cubed."},{"Start":"14:57.040 ","End":"15:01.240","Text":"Cubed times the derivative with respect to x is just 2x,"},{"Start":"15:01.240 ","End":"15:02.875","Text":"y is like a constant."},{"Start":"15:02.875 ","End":"15:07.000","Text":"Similarly, the derivative of f,"},{"Start":"15:07.000 ","End":"15:11.020","Text":"the partial derivative with respect to y is also equal to,"},{"Start":"15:11.020 ","End":"15:15.985","Text":"it starts out the same with the 4x squared plus y, cubed cubed."},{"Start":"15:15.985 ","End":"15:21.445","Text":"But this time, we have to take the inner derivative with respect to y"},{"Start":"15:21.445 ","End":"15:28.090","Text":"and we get 4x squared plus y cubed, cubed."},{"Start":"15:28.090 ","End":"15:33.115","Text":"The derivative of this with respect to y is 3y squared."},{"Start":"15:33.115 ","End":"15:36.475","Text":"I won\u0027t simplify this. We\u0027ll leave it as this."},{"Start":"15:36.475 ","End":"15:39.355","Text":"All the rules, the product,"},{"Start":"15:39.355 ","End":"15:41.680","Text":"the quotient, and the chain rule,"},{"Start":"15:41.680 ","End":"15:43.960","Text":"they all work with partial derivatives because like I said,"},{"Start":"15:43.960 ","End":"15:46.855","Text":"a partial derivative is like a derivative in 1 variable,"},{"Start":"15:46.855 ","End":"15:49.239","Text":"the other variables become like constants."},{"Start":"15:49.239 ","End":"15:51.730","Text":"Let\u0027s do another example with the chain rule."},{"Start":"15:51.730 ","End":"15:55.960","Text":"This time, we\u0027ll use the natural logarithm."},{"Start":"15:55.960 ","End":"15:58.000","Text":"So f of x,"},{"Start":"15:58.000 ","End":"16:06.560","Text":"y equals the log of x squared plus y^4 plus 1."},{"Start":"16:13.170 ","End":"16:18.415","Text":"First, let\u0027s remember the template for natural log."},{"Start":"16:18.415 ","End":"16:26.905","Text":"If I have natural log of something box and I want to differentiate it,"},{"Start":"16:26.905 ","End":"16:33.775","Text":"I get 1 over the box because of the natural logarithm and because of the chain rule,"},{"Start":"16:33.775 ","End":"16:35.200","Text":"I get the inner derivative,"},{"Start":"16:35.200 ","End":"16:36.850","Text":"which is box prime."},{"Start":"16:36.850 ","End":"16:43.420","Text":"Applying that here, I get the derivative according to x is 1"},{"Start":"16:43.420 ","End":"16:50.950","Text":"over x squared plus y^4 plus 1 times the derivative of this."},{"Start":"16:50.950 ","End":"16:55.795","Text":"I\u0027ll just mark it for differentiation."},{"Start":"16:55.795 ","End":"16:58.120","Text":"I\u0027ll mark it as prime. Then the next step,"},{"Start":"16:58.120 ","End":"17:00.730","Text":"we\u0027ll actually do the differentiation."},{"Start":"17:00.730 ","End":"17:03.340","Text":"There\u0027s no prime within 2 variables."},{"Start":"17:03.340 ","End":"17:05.035","Text":"I have to say which variable."},{"Start":"17:05.035 ","End":"17:10.300","Text":"It\u0027s x because this is x and this equals the derivative of"},{"Start":"17:10.300 ","End":"17:17.425","Text":"this is just 2x because y^4 plus 1 is a constant as far as x goes."},{"Start":"17:17.425 ","End":"17:20.140","Text":"This is 2x over the same denominator,"},{"Start":"17:20.140 ","End":"17:27.280","Text":"2x over x squared plus y^4 plus 1."},{"Start":"17:27.280 ","End":"17:30.160","Text":"I didn\u0027t quite align this because I\u0027m running out of space."},{"Start":"17:30.160 ","End":"17:35.485","Text":"The derivative with respect to y starts out the same,"},{"Start":"17:35.485 ","End":"17:41.635","Text":"1 over x squared plus y^4 plus 1."},{"Start":"17:41.635 ","End":"17:48.340","Text":"But this time, multiplied by the same x squared plus y^4 plus 1 but this time,"},{"Start":"17:48.340 ","End":"17:51.805","Text":"the derivative according to y."},{"Start":"17:51.805 ","End":"17:59.710","Text":"This equals the derivative with respect to y of this is 4y cubed."},{"Start":"17:59.710 ","End":"18:09.985","Text":"It\u0027s 4y cubed over"},{"Start":"18:09.985 ","End":"18:15.685","Text":"the same x squared plus y^4 plus 1."},{"Start":"18:15.685 ","End":"18:19.240","Text":"We tried it out on something ^4,"},{"Start":"18:19.240 ","End":"18:20.785","Text":"natural log of something."},{"Start":"18:20.785 ","End":"18:22.870","Text":"It works with the chain rule also."},{"Start":"18:22.870 ","End":"18:27.580","Text":"Now I want to give you 1 more partial derivative for us to do."},{"Start":"18:27.580 ","End":"18:29.875","Text":"I want to make a point in it."},{"Start":"18:29.875 ","End":"18:32.514","Text":"Here\u0027s this nasty looking example,"},{"Start":"18:32.514 ","End":"18:37.390","Text":"but I just want the partial derivative of x,"},{"Start":"18:37.390 ","End":"18:39.805","Text":"y with respect to x."},{"Start":"18:39.805 ","End":"18:42.250","Text":"Now what do you think this is?"},{"Start":"18:42.250 ","End":"18:45.685","Text":"At first sight, it looks a mess and you might start"},{"Start":"18:45.685 ","End":"18:49.659","Text":"using all sorts of derivation rules until you realize,"},{"Start":"18:49.659 ","End":"18:53.245","Text":"wait a minute, this derivative is with respect to x."},{"Start":"18:53.245 ","End":"18:56.425","Text":"So y is just like a constant."},{"Start":"18:56.425 ","End":"18:58.660","Text":"In fact, all this thing is just a constant."},{"Start":"18:58.660 ","End":"19:00.475","Text":"It\u0027s as if I\u0027d written,"},{"Start":"19:00.475 ","End":"19:04.690","Text":"instead of all this, I\u0027d written 7x."},{"Start":"19:04.690 ","End":"19:12.430","Text":"Then the derivative of this with respect to x would be just the 7."},{"Start":"19:12.430 ","End":"19:16.060","Text":"Similarly here, this thing times x,"},{"Start":"19:16.060 ","End":"19:17.620","Text":"the derivative is just this thing,"},{"Start":"19:17.620 ","End":"19:21.050","Text":"all I have to do is copy and paste it."},{"Start":"19:21.210 ","End":"19:24.820","Text":"There\u0027s the answer just without the"},{"Start":"19:24.820 ","End":"19:30.700","Text":"x. I wonder how many of you would have started to do all sorts of difficult procedures."},{"Start":"19:30.700 ","End":"19:32.245","Text":"This is just a warning."},{"Start":"19:32.245 ","End":"19:35.140","Text":"Realize that the derivative is with respect to x,"},{"Start":"19:35.140 ","End":"19:37.195","Text":"y is just like a constant,"},{"Start":"19:37.195 ","End":"19:40.180","Text":"even if it appears like this in a total mess,"},{"Start":"19:40.180 ","End":"19:44.540","Text":"it\u0027s just a constant. That\u0027s all for this clip."}],"ID":20871},{"Watched":false,"Name":"Partial Derivatives of the Second Order","Duration":"6m 16s","ChapterTopicVideoID":8577,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this clip, I\u0027ll be talking about partial derivatives of the second order."},{"Start":"00:05.070 ","End":"00:09.450","Text":"This is similar to the second derivative in 1 variable."},{"Start":"00:09.450 ","End":"00:11.460","Text":"I\u0027ll show you what I mean."},{"Start":"00:11.460 ","End":"00:16.710","Text":"Let\u0027s indeed start off with the case of 1 variable."},{"Start":"00:16.710 ","End":"00:24.765","Text":"Say we have a function f of x is equal to x cubed."},{"Start":"00:24.765 ","End":"00:27.655","Text":"Then we have a first derivative,"},{"Start":"00:27.655 ","End":"00:29.345","Text":"f prime of x,"},{"Start":"00:29.345 ","End":"00:31.950","Text":"which is 3x squared."},{"Start":"00:31.950 ","End":"00:36.050","Text":"This is called the derivative of the first order also."},{"Start":"00:36.050 ","End":"00:40.800","Text":"The derivative of the second-order is what we call the second derivative,"},{"Start":"00:40.800 ","End":"00:42.470","Text":"so it\u0027s just a different name for it,"},{"Start":"00:42.470 ","End":"00:44.315","Text":"derivative of the second order,"},{"Start":"00:44.315 ","End":"00:47.695","Text":"and this one happens to be 6x."},{"Start":"00:47.695 ","End":"00:52.880","Text":"Now we saw that in the case of a function of 2 variables, we have a function,"},{"Start":"00:52.880 ","End":"00:59.400","Text":"but we have 2 derivatives of the first order with respect to x and y."},{"Start":"00:59.400 ","End":"01:04.215","Text":"In fact, we will get 4 derivatives of the second order."},{"Start":"01:04.215 ","End":"01:06.880","Text":"I\u0027ll show you what I mean and how we write this."},{"Start":"01:06.880 ","End":"01:11.785","Text":"Let\u0027s start right away with an example with 2 variables."},{"Start":"01:11.785 ","End":"01:15.430","Text":"Let\u0027s take a function f of x,"},{"Start":"01:15.430 ","End":"01:22.185","Text":"y equals x squared times y cubed."},{"Start":"01:22.185 ","End":"01:24.465","Text":"Now as we saw,"},{"Start":"01:24.465 ","End":"01:31.110","Text":"it has 2 derivatives of the first order, 2 partial derivatives."},{"Start":"01:31.110 ","End":"01:37.065","Text":"We have according to x and according to y,"},{"Start":"01:37.065 ","End":"01:45.100","Text":"so here we call it f_x and here we\u0027ll call it f_y."},{"Start":"01:45.410 ","End":"01:49.220","Text":"This equals, let\u0027s see,"},{"Start":"01:49.220 ","End":"01:50.780","Text":"if it was with respect to x,"},{"Start":"01:50.780 ","End":"01:53.540","Text":"then we get 2x and y is like a constant,"},{"Start":"01:53.540 ","End":"01:56.585","Text":"so it\u0027s 2xy cubed."},{"Start":"01:56.585 ","End":"02:00.040","Text":"With respect to y, the x squared is a constant,"},{"Start":"02:00.040 ","End":"02:05.125","Text":"and we differentiate y cubed and we get 3y squared."},{"Start":"02:05.125 ","End":"02:09.800","Text":"Now, each of these is a function of x and y and could be"},{"Start":"02:09.800 ","End":"02:17.770","Text":"differentiated partially with respect to x and with respect to y."},{"Start":"02:17.770 ","End":"02:22.850","Text":"This one could be differentiated with respect to x and with respect to y."},{"Start":"02:22.850 ","End":"02:25.320","Text":"You can already see where the 4 comes in,"},{"Start":"02:25.320 ","End":"02:27.820","Text":"there\u0027s 4 of these. Let\u0027s see."},{"Start":"02:27.820 ","End":"02:30.185","Text":"This one would be,"},{"Start":"02:30.185 ","End":"02:34.980","Text":"we call it f_xx, which means first x,"},{"Start":"02:34.980 ","End":"02:36.390","Text":"then by x again,"},{"Start":"02:36.390 ","End":"02:39.135","Text":"and this one will be called f_xy,"},{"Start":"02:39.135 ","End":"02:42.705","Text":"which means first x and then y."},{"Start":"02:42.705 ","End":"02:47.425","Text":"Then we have f_yx and f_yy."},{"Start":"02:47.425 ","End":"02:51.755","Text":"All these 4 are partial derivatives of the second order,"},{"Start":"02:51.755 ","End":"02:54.250","Text":"because I have a choice of 2 in each case,"},{"Start":"02:54.250 ","End":"02:56.520","Text":"by x or by y,"},{"Start":"02:56.520 ","End":"02:59.730","Text":"with respect to x and with respect to y."},{"Start":"02:59.730 ","End":"03:07.965","Text":"What this is going to equal? Let\u0027s see."},{"Start":"03:07.965 ","End":"03:14.660","Text":"We take 2xy cubed and differentiate it with respect to x."},{"Start":"03:14.660 ","End":"03:22.920","Text":"This is going to equal 2xy cubed differentiated with respect to x,"},{"Start":"03:22.920 ","End":"03:24.165","Text":"I\u0027ll do it in a moment,"},{"Start":"03:24.165 ","End":"03:26.080","Text":"I\u0027m just writing what I have to do."},{"Start":"03:26.080 ","End":"03:31.635","Text":"This will be 2xy cubed differentiated with respect to y."},{"Start":"03:31.635 ","End":"03:36.705","Text":"This will be x squared times 3y squared,"},{"Start":"03:36.705 ","End":"03:39.480","Text":"differentiated with respect to x."},{"Start":"03:39.480 ","End":"03:44.010","Text":"This will be the same x squared 3y squared differentiated with"},{"Start":"03:44.010 ","End":"03:49.065","Text":"respect to y. I think we can get rid of this."},{"Start":"03:49.065 ","End":"03:51.525","Text":"Let\u0027s see, you just put an equals here."},{"Start":"03:51.525 ","End":"04:01.030","Text":"Let me also do a little bit of highlighting to show you that this x is because of this x,"},{"Start":"04:01.030 ","End":"04:06.160","Text":"this y is because of this y, and so on."},{"Start":"04:06.160 ","End":"04:09.485","Text":"Let\u0027s do it. With respect to x,"},{"Start":"04:09.485 ","End":"04:12.680","Text":"differentiate the 2x and the y cubed,"},{"Start":"04:12.680 ","End":"04:14.705","Text":"stays as is, it\u0027s a constant."},{"Start":"04:14.705 ","End":"04:17.480","Text":"With respect to y, the 2x is a constant,"},{"Start":"04:17.480 ","End":"04:22.965","Text":"and I differentiate the y cubed to get 3y squared."},{"Start":"04:22.965 ","End":"04:26.920","Text":"Now here, I differentiate with respect to x,"},{"Start":"04:26.920 ","End":"04:32.850","Text":"so 2x times 3y squared,"},{"Start":"04:32.850 ","End":"04:35.275","Text":"3 and y squared are all constants."},{"Start":"04:35.275 ","End":"04:38.595","Text":"Here, with respect to y,"},{"Start":"04:38.595 ","End":"04:40.380","Text":"the x squared is as is,"},{"Start":"04:40.380 ","End":"04:44.400","Text":"and the 3y squared becomes 6y."},{"Start":"04:44.400 ","End":"04:48.795","Text":"Ultimately, these are equal to 2y cubed,"},{"Start":"04:48.795 ","End":"04:54.570","Text":"6xy squared, 6xy squared,"},{"Start":"04:54.570 ","End":"04:59.190","Text":"and 6x squared y."},{"Start":"04:59.190 ","End":"05:02.130","Text":"Do you notice what I see?"},{"Start":"05:02.130 ","End":"05:06.485","Text":"I\u0027ll highlight them."},{"Start":"05:06.485 ","End":"05:09.115","Text":"These seem to be equal."},{"Start":"05:09.115 ","End":"05:14.030","Text":"I wonder if that\u0027s a coincidence or there\u0027s some reason behind it."},{"Start":"05:14.030 ","End":"05:17.410","Text":"These 2 are called the mixed second derivatives,"},{"Start":"05:17.410 ","End":"05:20.080","Text":"because we have with respect to just x all the time,"},{"Start":"05:20.080 ","End":"05:21.715","Text":"x and x, y and y."},{"Start":"05:21.715 ","End":"05:25.740","Text":"These 2, you do with respect to x then y,"},{"Start":"05:25.740 ","End":"05:26.760","Text":"and here, the other way around."},{"Start":"05:26.760 ","End":"05:28.730","Text":"These 2 are called the mixed second"},{"Start":"05:28.730 ","End":"05:32.405","Text":"derivatives or partial derivatives of the second order."},{"Start":"05:32.405 ","End":"05:34.700","Text":"I can\u0027t exactly say yes or no."},{"Start":"05:34.700 ","End":"05:36.245","Text":"The answer is yes,"},{"Start":"05:36.245 ","End":"05:39.290","Text":"but the function has to satisfy certain conditions."},{"Start":"05:39.290 ","End":"05:41.915","Text":"In fact, as far as you will encounter,"},{"Start":"05:41.915 ","End":"05:44.080","Text":"it will always be equal."},{"Start":"05:44.080 ","End":"05:51.560","Text":"Yes, this theorem does hold that f with respect to x and then with respect to y,"},{"Start":"05:51.560 ","End":"05:53.765","Text":"second order partial derivative,"},{"Start":"05:53.765 ","End":"05:59.340","Text":"is equal to f with respect to y and then with respect to x."},{"Start":"05:59.600 ","End":"06:04.760","Text":"Although, I should add that this is under certain conditions,"},{"Start":"06:04.760 ","End":"06:08.720","Text":"which will almost always hold as far as you will encounter."},{"Start":"06:08.720 ","End":"06:11.335","Text":"That\u0027s useful to know."},{"Start":"06:11.335 ","End":"06:17.100","Text":"I\u0027ve done with this introduction to partial derivatives of the second order."}],"ID":20872},{"Watched":false,"Name":"Exercise 1 part a","Duration":"2m 31s","ChapterTopicVideoID":8578,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"In this exercise, we have to compute the 1st order"},{"Start":"00:03.030 ","End":"00:08.820","Text":"partial derivatives of this function of x and y given here."},{"Start":"00:08.820 ","End":"00:13.875","Text":"Partial derivatives is going to be 2 of them because there\u0027s 2 variables."},{"Start":"00:13.875 ","End":"00:16.740","Text":"We need a partial derivative with respect to x,"},{"Start":"00:16.740 ","End":"00:18.900","Text":"which can be written in several ways,"},{"Start":"00:18.900 ","End":"00:20.610","Text":"but we write it like this."},{"Start":"00:20.610 ","End":"00:25.710","Text":"Then we\u0027re going to compute the partial derivative of f with respect to y."},{"Start":"00:25.710 ","End":"00:29.925","Text":"The idea is that when we differentiate partially with respect to x,"},{"Start":"00:29.925 ","End":"00:34.815","Text":"we treat y like a constant or parameter and vice versa."},{"Start":"00:34.815 ","End":"00:37.040","Text":"Let\u0027s do the first 1,"},{"Start":"00:37.040 ","End":"00:38.910","Text":"f with respect to x."},{"Start":"00:38.910 ","End":"00:46.475","Text":"The variable is x, so we differentiate 4x cubed and that\u0027s 4 times 3 is 12x squared."},{"Start":"00:46.475 ","End":"00:51.000","Text":"Now here it\u0027s minus 3 x squared times this constant."},{"Start":"00:51.000 ","End":"00:56.025","Text":"The constant just sticks and we get 2 times 3 is 6."},{"Start":"00:56.025 ","End":"01:03.270","Text":"We get 6x and y squared being a multiplicative constant just stays."},{"Start":"01:03.270 ","End":"01:08.720","Text":"2x gives us 2 and 3y is just a constant without any x in it,"},{"Start":"01:08.720 ","End":"01:12.035","Text":"so that\u0027s nothing. That\u0027s 1 of them."},{"Start":"01:12.035 ","End":"01:14.735","Text":"Now the other, with respect to y,"},{"Start":"01:14.735 ","End":"01:16.355","Text":"we look at x as a constant."},{"Start":"01:16.355 ","End":"01:18.775","Text":"This gives us nothing."},{"Start":"01:18.775 ","End":"01:21.930","Text":"You know what, maybe I\u0027ll write here plus 0,"},{"Start":"01:21.930 ","End":"01:26.365","Text":"just so you don\u0027t think I\u0027ve forgotten something and here I\u0027ll start out with 0,"},{"Start":"01:26.365 ","End":"01:30.055","Text":"because x is a constant and so is 4x cubed."},{"Start":"01:30.055 ","End":"01:37.075","Text":"Then minus, now derivative of y squared is 2y and the x squared sticks."},{"Start":"01:37.075 ","End":"01:44.115","Text":"It\u0027s minus 2 times 3 is 6x squared y."},{"Start":"01:44.115 ","End":"01:47.505","Text":"Then 2x is a constant."},{"Start":"01:47.505 ","End":"01:51.630","Text":"Again, we get plus 0 and 3y,"},{"Start":"01:51.630 ","End":"01:54.810","Text":"its derivative is 3."},{"Start":"01:54.810 ","End":"01:59.380","Text":"Just to clear out the 0s,"},{"Start":"01:59.380 ","End":"02:01.680","Text":"let me just write this again."},{"Start":"02:01.680 ","End":"02:05.800","Text":"This is 12x squared minus 6x,"},{"Start":"02:05.800 ","End":"02:08.895","Text":"y squared plus 2."},{"Start":"02:08.895 ","End":"02:12.515","Text":"That\u0027s derivative with respect to x and with respect to y,"},{"Start":"02:12.515 ","End":"02:18.455","Text":"we get minus 6x squared y plus 3."},{"Start":"02:18.455 ","End":"02:22.925","Text":"In the future clips, I\u0027m not going to be writing 0s all the time."},{"Start":"02:22.925 ","End":"02:25.720","Text":"Sometimes I like to highlight the answer,"},{"Start":"02:25.720 ","End":"02:31.370","Text":"1 partial derivative and the other partial derivative. We\u0027re done."}],"ID":20873},{"Watched":false,"Name":"Exercise 1 part b","Duration":"56s","ChapterTopicVideoID":8579,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"Here we want to find the 1st order partial derivatives of the function of x and y,"},{"Start":"00:05.280 ","End":"00:07.725","Text":"x^5 natural log y."},{"Start":"00:07.725 ","End":"00:09.929","Text":"There\u0027s going to be 2 partial derivatives,"},{"Start":"00:09.929 ","End":"00:11.600","Text":"because we have 2 variables."},{"Start":"00:11.600 ","End":"00:14.670","Text":"There\u0027s going to be a partial derivative with respect to x,"},{"Start":"00:14.670 ","End":"00:17.100","Text":"and the partial derivative with respect to y."},{"Start":"00:17.100 ","End":"00:19.620","Text":"Let\u0027s take the 1 with respect to x."},{"Start":"00:19.620 ","End":"00:23.190","Text":"In this case, x is the variable and y is a constant."},{"Start":"00:23.190 ","End":"00:25.170","Text":"Natural log of y is also a constant,"},{"Start":"00:25.170 ","End":"00:27.450","Text":"some constant times x^5."},{"Start":"00:27.450 ","End":"00:33.765","Text":"It\u0027s 5x^4 and the constant sticks. That\u0027s it."},{"Start":"00:33.765 ","End":"00:36.770","Text":"Then as far as y goes,"},{"Start":"00:36.770 ","End":"00:38.210","Text":"now y is the variable,"},{"Start":"00:38.210 ","End":"00:40.775","Text":"x is some constant as is x^5."},{"Start":"00:40.775 ","End":"00:47.090","Text":"The constant stays and then the derivative of the natural logarithm is 1 over y."},{"Start":"00:47.090 ","End":"00:49.160","Text":"You can write it times 1 over y,"},{"Start":"00:49.160 ","End":"00:53.225","Text":"or you could put a dividing sign and put y on the denominator,"},{"Start":"00:53.225 ","End":"00:56.460","Text":"whatever. Okay, that\u0027s it."}],"ID":20874},{"Watched":false,"Name":"Exercise 1 part c","Duration":"1m 16s","ChapterTopicVideoID":8580,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:07.409","Text":"In this exercise, we have to compute the first-order partial derivative of this function,"},{"Start":"00:07.409 ","End":"00:10.440","Text":"which looks awful, but only with respect to x."},{"Start":"00:10.440 ","End":"00:13.710","Text":"We don\u0027t need to do the derivative with respect to y."},{"Start":"00:13.710 ","End":"00:16.590","Text":"You\u0027ll see why this is a great relief."},{"Start":"00:16.590 ","End":"00:20.565","Text":"Now, derivative of f with respect to x means that"},{"Start":"00:20.565 ","End":"00:24.374","Text":"x is a variable and y is like a constant or parameter."},{"Start":"00:24.374 ","End":"00:26.910","Text":"But if you look closely at this,"},{"Start":"00:26.910 ","End":"00:31.460","Text":"we see that the only place that x squared appears is here,"},{"Start":"00:31.460 ","End":"00:35.600","Text":"and all this, what I\u0027ve just circled,"},{"Start":"00:35.600 ","End":"00:38.555","Text":"is all a function of y."},{"Start":"00:38.555 ","End":"00:45.610","Text":"This whole thing is like a constant as far as x is concerned."},{"Start":"00:45.610 ","End":"00:48.800","Text":"The answer is just to differentiate the x squared is"},{"Start":"00:48.800 ","End":"00:53.450","Text":"2x and multiplied by this constant so we have to repeat all that."},{"Start":"00:53.450 ","End":"00:56.225","Text":"The main difficulty is just copying it out."},{"Start":"00:56.225 ","End":"01:02.195","Text":"There\u0027s no actual computation to do 5 natural log of y"},{"Start":"01:02.195 ","End":"01:09.095","Text":"over y squared plus 5y plus y to the power of y."},{"Start":"01:09.095 ","End":"01:11.960","Text":"If they had asked with respect to y also,"},{"Start":"01:11.960 ","End":"01:13.835","Text":"then it would have been more difficult."},{"Start":"01:13.835 ","End":"01:16.500","Text":"As it is, we are done."}],"ID":20875},{"Watched":false,"Name":"Exercise 1 part d","Duration":"2m 25s","ChapterTopicVideoID":8581,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this Exercise, we have to compute"},{"Start":"00:02.220 ","End":"00:07.575","Text":"the first order partial derivatives of this function of x and y."},{"Start":"00:07.575 ","End":"00:10.680","Text":"There\u0027s 2 partial derivatives,"},{"Start":"00:10.680 ","End":"00:13.230","Text":"1 with respect to x, 1 with respect to y."},{"Start":"00:13.230 ","End":"00:15.735","Text":"Let\u0027s start with respect to x,"},{"Start":"00:15.735 ","End":"00:18.585","Text":"and then we treat y like a constant."},{"Start":"00:18.585 ","End":"00:21.090","Text":"There\u0027s actually more than 1 way of doing this,"},{"Start":"00:21.090 ","End":"00:26.280","Text":"1 way here would actually be to multiply out and do some algebra and expand."},{"Start":"00:26.280 ","End":"00:29.009","Text":"The other possibility, more general,"},{"Start":"00:29.009 ","End":"00:31.065","Text":"would be to use the product rule."},{"Start":"00:31.065 ","End":"00:34.320","Text":"Just to remind you what the product rule is in general,"},{"Start":"00:34.320 ","End":"00:36.159","Text":"although this is for 1 variable,"},{"Start":"00:36.159 ","End":"00:39.050","Text":"but it also works with partial derivatives."},{"Start":"00:39.050 ","End":"00:41.765","Text":"The derivative of a product,"},{"Start":"00:41.765 ","End":"00:46.175","Text":"we take the derivative of the first multiply by the second,"},{"Start":"00:46.175 ","End":"00:48.110","Text":"and then the first as is,"},{"Start":"00:48.110 ","End":"00:50.075","Text":"times the derivative of the second."},{"Start":"00:50.075 ","End":"00:51.275","Text":"That\u0027s what we\u0027ll do here,"},{"Start":"00:51.275 ","End":"00:53.390","Text":"remembering that x is the variable."},{"Start":"00:53.390 ","End":"00:59.090","Text":"Derivative of the first is 2x because y cubed is a constant,"},{"Start":"00:59.090 ","End":"01:02.675","Text":"times the second as is,"},{"Start":"01:02.675 ","End":"01:06.095","Text":"which is 2x plus 3y."},{"Start":"01:06.095 ","End":"01:10.010","Text":"Then vice versa, the other way round,"},{"Start":"01:10.010 ","End":"01:12.380","Text":"this 1, I keep as is,"},{"Start":"01:12.380 ","End":"01:13.400","Text":"so I just copy it,"},{"Start":"01:13.400 ","End":"01:16.705","Text":"x squared plus y cubed,"},{"Start":"01:16.705 ","End":"01:19.555","Text":"and then the derivative of the second."},{"Start":"01:19.555 ","End":"01:22.300","Text":"Now in the second also, y is a constant,"},{"Start":"01:22.300 ","End":"01:23.830","Text":"so 3y is a constant,"},{"Start":"01:23.830 ","End":"01:27.790","Text":"so we just got the derivative of 2x, which is 2."},{"Start":"01:27.790 ","End":"01:30.610","Text":"Of course we can simplify and multiply out,"},{"Start":"01:30.610 ","End":"01:31.945","Text":"but that\u0027s not our purpose here,"},{"Start":"01:31.945 ","End":"01:34.780","Text":"so I\u0027m going to leave it like this."},{"Start":"01:34.780 ","End":"01:39.005","Text":"Now let\u0027s go to the derivative with respect to y."},{"Start":"01:39.005 ","End":"01:42.235","Text":"Similar, this time y is the variable,"},{"Start":"01:42.235 ","End":"01:43.660","Text":"x is the constant,"},{"Start":"01:43.660 ","End":"01:45.670","Text":"but again the product rule."},{"Start":"01:45.670 ","End":"01:47.995","Text":"We get the derivative of the first,"},{"Start":"01:47.995 ","End":"01:49.570","Text":"this is the constant,"},{"Start":"01:49.570 ","End":"01:56.130","Text":"so this is 3y squared only times the second as is,"},{"Start":"01:56.130 ","End":"02:00.990","Text":"2x plus 3y plus, there\u0027s this plus,"},{"Start":"02:00.990 ","End":"02:08.390","Text":"the first 1 as is x squared plus y cubed times derivative of the second,"},{"Start":"02:08.390 ","End":"02:12.020","Text":"derivative of this, but this time it\u0027s with respect to y,"},{"Start":"02:12.020 ","End":"02:13.774","Text":"so x is the constant,"},{"Start":"02:13.774 ","End":"02:17.260","Text":"and so we just get times 3."},{"Start":"02:17.260 ","End":"02:19.940","Text":"Of course we would tidy it up normally,"},{"Start":"02:19.940 ","End":"02:21.965","Text":"but not in this case."},{"Start":"02:21.965 ","End":"02:25.110","Text":"I\u0027m just declaring that we\u0027re done."}],"ID":20876},{"Watched":false,"Name":"Exercise 1 part e","Duration":"6m 24s","ChapterTopicVideoID":8582,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.165","Text":"In this exercise, we have to compute the first-order partial derivatives of f of x, y."},{"Start":"00:06.165 ","End":"00:10.830","Text":"That means is going to be 2 of them with respect to x and with respect to y."},{"Start":"00:10.830 ","End":"00:16.215","Text":"Let\u0027s start with the partial derivative with respect to x."},{"Start":"00:16.215 ","End":"00:22.605","Text":"Now we notice that this is a quotient and x appears both on top and on the bottom."},{"Start":"00:22.605 ","End":"00:25.650","Text":"Let me remind you of the quotient rule first."},{"Start":"00:25.650 ","End":"00:29.910","Text":"In general, the quotient rule says that if we take"},{"Start":"00:29.910 ","End":"00:36.045","Text":"the derivative and usually it\u0027s a function of 1 variable which we shall soon see."},{"Start":"00:36.045 ","End":"00:39.920","Text":"The derivative of a quotient is a fraction on"},{"Start":"00:39.920 ","End":"00:43.580","Text":"the bottom we have the denominator squared, on the numerator,"},{"Start":"00:43.580 ","End":"00:48.710","Text":"we have the derivative of u times v as"},{"Start":"00:48.710 ","End":"00:53.900","Text":"is minus u times v prime."},{"Start":"00:53.900 ","End":"00:56.870","Text":"In other words, we differentiate u once and leave v alone,"},{"Start":"00:56.870 ","End":"00:58.550","Text":"in the second 1 we leave u alone,"},{"Start":"00:58.550 ","End":"01:04.325","Text":"differentiate v. Now this time I want to really spell things out."},{"Start":"01:04.325 ","End":"01:07.190","Text":"If for example, I wanted to know what is"},{"Start":"01:07.190 ","End":"01:11.345","Text":"a quotient u over v and suppose the functions of x and y,"},{"Start":"01:11.345 ","End":"01:13.250","Text":"and I want to with respect to x,"},{"Start":"01:13.250 ","End":"01:15.320","Text":"the same thing just instead of the prime,"},{"Start":"01:15.320 ","End":"01:17.465","Text":"I\u0027ll write derivative with respect to x,"},{"Start":"01:17.465 ","End":"01:22.820","Text":"v minus u times v with respect to x over,"},{"Start":"01:22.820 ","End":"01:27.080","Text":"and it\u0027s still v squared and later on while I\u0027m at it already,"},{"Start":"01:27.080 ","End":"01:30.710","Text":"I might as well show you how this works with respect to y."},{"Start":"01:30.710 ","End":"01:35.900","Text":"Same thing derivative of 1 with respect to y this time,"},{"Start":"01:35.900 ","End":"01:40.115","Text":"times the other untouched minus the first 1 untouched"},{"Start":"01:40.115 ","End":"01:49.935","Text":"and the second 1 with respect to y and it\u0027s still over the denominator squared."},{"Start":"01:49.935 ","End":"01:54.910","Text":"Armed with all of this and let\u0027s get to the first 1 again."},{"Start":"01:54.910 ","End":"01:57.830","Text":"Like I said, I\u0027m really going to spell things out this time."},{"Start":"01:57.830 ","End":"02:00.800","Text":"That would be u and this would be v,"},{"Start":"02:00.800 ","End":"02:03.365","Text":"that\u0027s the numerator, that\u0027s the denominator."},{"Start":"02:03.365 ","End":"02:06.905","Text":"Using this formula with respect to x,"},{"Start":"02:06.905 ","End":"02:09.365","Text":"we get the first 1,"},{"Start":"02:09.365 ","End":"02:14.000","Text":"x squared minus 3y with respect to x,"},{"Start":"02:14.000 ","End":"02:17.140","Text":"I\u0027ll just write it then I\u0027m going to do it"},{"Start":"02:17.140 ","End":"02:22.385","Text":"times v which is the denominator x plus y squared."},{"Start":"02:22.385 ","End":"02:26.750","Text":"There\u0027s a minus, unlike the product rule where there is a plus."},{"Start":"02:26.990 ","End":"02:30.380","Text":"This time we take the first 1 as is,"},{"Start":"02:30.380 ","End":"02:34.250","Text":"just x squared minus 3y and the second 1,"},{"Start":"02:34.250 ","End":"02:35.915","Text":"x plus y squared."},{"Start":"02:35.915 ","End":"02:38.990","Text":"This 1 differentiated with respect to x because we\u0027re"},{"Start":"02:38.990 ","End":"02:42.080","Text":"doing with respect to x and it\u0027s still"},{"Start":"02:42.080 ","End":"02:50.150","Text":"over denominator squared and this is x plus y squared, squared."},{"Start":"02:50.150 ","End":"02:52.810","Text":"Now we\u0027ll expand it."},{"Start":"02:52.810 ","End":"02:54.530","Text":"We\u0027ll do the derivatives,"},{"Start":"02:54.530 ","End":"02:57.635","Text":"the derivative of this with respect to x and remember,"},{"Start":"02:57.635 ","End":"03:01.325","Text":"y is a constant or parameter and x is the variable."},{"Start":"03:01.325 ","End":"03:06.590","Text":"So all we get from here is 2x because the minus 3y is still"},{"Start":"03:06.590 ","End":"03:14.190","Text":"a constant times this 1 x plus y squared minus and here I\u0027m just copying,"},{"Start":"03:14.190 ","End":"03:18.075","Text":"x squared minus 3y and this with respect to x,"},{"Start":"03:18.075 ","End":"03:19.815","Text":"y squared is a constant."},{"Start":"03:19.815 ","End":"03:26.025","Text":"This is just 1 and I can omit the times 1 if I want but I\u0027ll leave it in,"},{"Start":"03:26.025 ","End":"03:32.385","Text":"over x plus y squared squared."},{"Start":"03:32.385 ","End":"03:36.500","Text":"Normally after this I might simplify but this time I\u0027m not,"},{"Start":"03:36.500 ","End":"03:38.570","Text":"I\u0027m just going to write the word simplify,"},{"Start":"03:38.570 ","End":"03:43.070","Text":"it\u0027s optional because this ready is the partial derivative."},{"Start":"03:43.070 ","End":"03:46.670","Text":"Of course I\u0027d say 2x squared plus 2xy squared and so on and"},{"Start":"03:46.670 ","End":"03:50.700","Text":"so on but I\u0027ll omit that part."},{"Start":"03:50.700 ","End":"03:56.495","Text":"Let\u0027s go on to the second partial derivative because there\u0027s a second variable y so"},{"Start":"03:56.495 ","End":"04:02.345","Text":"derivative of f with respect to y is equal to and using the second formula,"},{"Start":"04:02.345 ","End":"04:07.080","Text":"essentially it\u0027s everything gets here except that little x replaced"},{"Start":"04:07.080 ","End":"04:12.575","Text":"by the little y. I can even just copy x squared minus 3y with respect to y,"},{"Start":"04:12.575 ","End":"04:16.645","Text":"x plus y squared this y is too big."},{"Start":"04:16.645 ","End":"04:19.880","Text":"That\u0027s better, I don\u0027t want to end up multiplying by y,"},{"Start":"04:19.880 ","End":"04:21.575","Text":"it\u0027s the derivative with respect to y."},{"Start":"04:21.575 ","End":"04:26.855","Text":"Actually, some people even leave the prime in but not in this course."},{"Start":"04:26.855 ","End":"04:30.140","Text":"Why bother with the prime if we understand."},{"Start":"04:30.140 ","End":"04:34.950","Text":"Minus continuing from here,"},{"Start":"04:34.950 ","End":"04:43.025","Text":"x squared minus 3y as is and x plus y squared this time with respect to"},{"Start":"04:43.025 ","End":"04:48.180","Text":"y and the same denominator there I just"},{"Start":"04:48.180 ","End":"04:54.090","Text":"copied it from here and now with the actual differentiation partial."},{"Start":"04:54.090 ","End":"04:56.825","Text":"Remember, with respect to y,"},{"Start":"04:56.825 ","End":"04:59.080","Text":"x is like a constant."},{"Start":"04:59.080 ","End":"05:01.430","Text":"The first 1 with respect to y,"},{"Start":"05:01.430 ","End":"05:05.725","Text":"the x squared is a constant so minus 3y becomes minus 3."},{"Start":"05:05.725 ","End":"05:07.570","Text":"This 1 as is,"},{"Start":"05:07.570 ","End":"05:11.570","Text":"x plus y squared minus this 1 just"},{"Start":"05:11.570 ","End":"05:16.279","Text":"copied x squared minus 3y and this 1 with respect to y,"},{"Start":"05:16.279 ","End":"05:18.575","Text":"x is a constant, y squared gives us"},{"Start":"05:18.575 ","End":"05:26.460","Text":"2y and all this and over x plus y squared, squared."},{"Start":"05:26.460 ","End":"05:28.995","Text":"I just copy pasted it from here."},{"Start":"05:28.995 ","End":"05:35.330","Text":"As before, we\u0027re done but I would normally simplify and you know what,"},{"Start":"05:35.330 ","End":"05:39.320","Text":"let me just start simplifying for those who like who like to do it but as I say,"},{"Start":"05:39.320 ","End":"05:41.225","Text":"we are done this is the derivative."},{"Start":"05:41.225 ","End":"05:44.405","Text":"You might say minus 3x,"},{"Start":"05:44.405 ","End":"05:50.120","Text":"minus 3y squared then minus x squared times 2y is"},{"Start":"05:50.120 ","End":"05:56.705","Text":"minus 2x squared y. I usually put numbers in front then x\u0027s and y\u0027s."},{"Start":"05:56.705 ","End":"06:00.770","Text":"The next 1, it\u0027s minus and a minus so it\u0027s"},{"Start":"06:00.770 ","End":"06:07.380","Text":"a plus 3y times 2y is 6y squared so it\u0027s plus 6y squared."},{"Start":"06:07.400 ","End":"06:09.720","Text":"We could still keep going,"},{"Start":"06:09.720 ","End":"06:11.640","Text":"for example, we could collect like terms,"},{"Start":"06:11.640 ","End":"06:15.000","Text":"minus 3y squared plus 6y squared we\u0027d replace"},{"Start":"06:15.000 ","End":"06:19.440","Text":"that by plus 3y squared and so on and that\u0027s just about it."},{"Start":"06:19.440 ","End":"06:24.100","Text":"It\u0027s in more detail than I normally do and we\u0027re done."}],"ID":20877},{"Watched":false,"Name":"Exercise 1 part f","Duration":"1m 58s","ChapterTopicVideoID":8583,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"Here we want to compute the first-order partial derivatives of f of x,"},{"Start":"00:03.960 ","End":"00:07.305","Text":"y, which is actually sine of x, y,"},{"Start":"00:07.305 ","End":"00:15.720","Text":"and 2 partial derivatives with respect to x and with respect to y. I\u0027ll begin with x."},{"Start":"00:15.720 ","End":"00:21.675","Text":"What we have to remember is that y is treated like a constant and that x is the variable."},{"Start":"00:21.675 ","End":"00:24.885","Text":"We want to differentiate sine of x, y."},{"Start":"00:24.885 ","End":"00:28.890","Text":"Now we start off with saying the derivative of sine is"},{"Start":"00:28.890 ","End":"00:33.315","Text":"cosine and you want to say cosine of x, y."},{"Start":"00:33.315 ","End":"00:37.515","Text":"But that\u0027s not all because it\u0027s sine not of x,"},{"Start":"00:37.515 ","End":"00:42.175","Text":"but of x, y so we need the inner derivative from the chain rule."},{"Start":"00:42.175 ","End":"00:45.740","Text":"We need to multiply by the derivative of x, y."},{"Start":"00:45.740 ","End":"00:50.514","Text":"Now x, y, its derivative with respect to x is just y."},{"Start":"00:50.514 ","End":"00:52.080","Text":"If you think about it."},{"Start":"00:52.080 ","End":"00:53.670","Text":"If it was like x, y,"},{"Start":"00:53.670 ","End":"00:55.870","Text":"x times 5 or 5x,"},{"Start":"00:55.870 ","End":"00:59.525","Text":"you would just write 5 and y is like the 5."},{"Start":"00:59.525 ","End":"01:02.800","Text":"That\u0027s the derivative with respect to x."},{"Start":"01:02.800 ","End":"01:06.335","Text":"Similarly, the derivative with respect to y."},{"Start":"01:06.335 ","End":"01:09.600","Text":"Once again, the derivative of sine is cosine."},{"Start":"01:09.600 ","End":"01:11.740","Text":"It\u0027s also cosine of x,"},{"Start":"01:11.740 ","End":"01:14.435","Text":"y only this time what I\u0027ve underlined here,"},{"Start":"01:14.435 ","End":"01:17.290","Text":"we differentiate with respect to y."},{"Start":"01:17.290 ","End":"01:19.830","Text":"It\u0027s like I wrote here 3y."},{"Start":"01:19.830 ","End":"01:22.470","Text":"If I differentiated 3y, that would just be 3,"},{"Start":"01:22.470 ","End":"01:24.155","Text":"only it\u0027s not 3, it\u0027s x,"},{"Start":"01:24.155 ","End":"01:26.495","Text":"which is a constant as far as y goes."},{"Start":"01:26.495 ","End":"01:29.130","Text":"This is our answer."},{"Start":"01:29.210 ","End":"01:34.820","Text":"Usually I would put aesthetically the y in front."},{"Start":"01:34.820 ","End":"01:40.345","Text":"For example, I would write y cosine of x, y."},{"Start":"01:40.345 ","End":"01:43.980","Text":"It\u0027s just certain aesthetics to doing things."},{"Start":"01:43.980 ","End":"01:50.760","Text":"Here also I put x cosine of x, y."},{"Start":"01:50.760 ","End":"01:57.790","Text":"Just looks better. This is fine as long as you have the brackets. We\u0027re done."}],"ID":20878},{"Watched":false,"Name":"Exercise 1 part g","Duration":"4m 6s","ChapterTopicVideoID":8584,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.490","Text":"In this exercise, we have a function of x and y given by this formula,"},{"Start":"00:05.490 ","End":"00:09.945","Text":"and we have to compute the first order partial derivatives."},{"Start":"00:09.945 ","End":"00:11.790","Text":"Derivatives because there\u0027s 2 of them,"},{"Start":"00:11.790 ","End":"00:14.325","Text":"1 with respect to x and 1 with respect to y."},{"Start":"00:14.325 ","End":"00:19.080","Text":"Let\u0027s begin with the derivative with respect to x."},{"Start":"00:19.080 ","End":"00:22.050","Text":"What do we do with the arctangent?"},{"Start":"00:22.050 ","End":"00:26.070","Text":"If you look in your formula sheet in case you\u0027ve forgotten,"},{"Start":"00:26.070 ","End":"00:32.430","Text":"and in the formula sheet you might find something similar to the arctangent of x,"},{"Start":"00:32.430 ","End":"00:38.280","Text":"but I don\u0027t want it to be x I want to just put some placeholder like a square,"},{"Start":"00:38.280 ","End":"00:45.560","Text":"derivative would be 1 over 1 plus this placeholder squared."},{"Start":"00:45.560 ","End":"00:48.634","Text":"Usually it\u0027s with x, but this is more flexible."},{"Start":"00:48.634 ","End":"00:54.110","Text":"Now, I\u0027m going to modify this because suppose it wasn\u0027t x,"},{"Start":"00:54.110 ","End":"00:58.550","Text":"this was some expression involving x or later on y."},{"Start":"00:58.550 ","End":"01:02.060","Text":"What we have to do is take care of the inner derivative,"},{"Start":"01:02.060 ","End":"01:10.530","Text":"so we have to multiply by the derivative of whatever this is."},{"Start":"01:10.690 ","End":"01:14.930","Text":"Just to tidy it up,"},{"Start":"01:14.930 ","End":"01:17.060","Text":"I could put this on the numerator,"},{"Start":"01:17.060 ","End":"01:25.180","Text":"so I would say it\u0027s this thing, prime over 1 plus this thing squared,"},{"Start":"01:25.180 ","End":"01:28.180","Text":"square, squared so to speak."},{"Start":"01:28.390 ","End":"01:32.945","Text":"Now, same thing holds with partial derivatives."},{"Start":"01:32.945 ","End":"01:36.740","Text":"If I was to take the arctangent of"},{"Start":"01:36.740 ","End":"01:41.870","Text":"something and I want to take its partial derivative with respect to x,"},{"Start":"01:41.870 ","End":"01:43.670","Text":"it would just be instead of prime,"},{"Start":"01:43.670 ","End":"01:45.590","Text":"we take derivative with respect to x."},{"Start":"01:45.590 ","End":"01:54.610","Text":"It would be this thing with respect to x over 1 plus this thing squared."},{"Start":"01:54.670 ","End":"02:00.795","Text":"Similarly for y, quick copy paste,"},{"Start":"02:00.795 ","End":"02:07.595","Text":"let\u0027s remove the x and replace it with a little y, there."},{"Start":"02:07.595 ","End":"02:11.510","Text":"Back here, so this is equal to,"},{"Start":"02:11.510 ","End":"02:15.470","Text":"and I\u0027m looking at this formula here because that\u0027s the 1 I want."},{"Start":"02:15.470 ","End":"02:22.565","Text":"I see that my box is just this thing here,"},{"Start":"02:22.565 ","End":"02:29.460","Text":"that\u0027s what I call the box or a square or whatever."},{"Start":"02:30.100 ","End":"02:34.484","Text":"We get dividing line,"},{"Start":"02:34.484 ","End":"02:37.865","Text":"then we want on the numerator,"},{"Start":"02:37.865 ","End":"02:41.545","Text":"the derivative of box with respect to x."},{"Start":"02:41.545 ","End":"02:42.785","Text":"What is that?"},{"Start":"02:42.785 ","End":"02:43.910","Text":"Y is a constant,"},{"Start":"02:43.910 ","End":"02:45.785","Text":"so it\u0027s like 2x plus a constant,"},{"Start":"02:45.785 ","End":"02:48.740","Text":"so this would be just 2."},{"Start":"02:48.740 ","End":"02:52.650","Text":"On the denominator, 1 plus this thing squared,"},{"Start":"02:52.650 ","End":"02:58.005","Text":"which is 2x plus 3y squared."},{"Start":"02:58.005 ","End":"03:05.400","Text":"That\u0027s it as far as this 1 goes and similarly for the next 1."},{"Start":"03:05.400 ","End":"03:08.990","Text":"Those of you who did remember the derivative of arctangent,"},{"Start":"03:08.990 ","End":"03:10.180","Text":"you would have done it quicker,"},{"Start":"03:10.180 ","End":"03:13.235","Text":"you wouldn\u0027t have had to look here, but you can."},{"Start":"03:13.235 ","End":"03:16.175","Text":"You would have just said, we have an arctangent,"},{"Start":"03:16.175 ","End":"03:23.540","Text":"so it\u0027s 1 over the argument of the arctangent in the denominator squared,"},{"Start":"03:23.540 ","End":"03:30.810","Text":"so I put 2x plus 3y squared,"},{"Start":"03:30.810 ","End":"03:32.655","Text":"1 plus of course."},{"Start":"03:32.655 ","End":"03:34.560","Text":"Then we would say, okay,"},{"Start":"03:34.560 ","End":"03:36.335","Text":"instead of putting 1 over,"},{"Start":"03:36.335 ","End":"03:38.690","Text":"I need the derivative of this,"},{"Start":"03:38.690 ","End":"03:45.075","Text":"and the derivative of this with respect to y is just 3 because it\u0027s a constant plus 3y."},{"Start":"03:45.075 ","End":"03:48.435","Text":"I\u0027d put the 3 here only I wouldn\u0027t,"},{"Start":"03:48.435 ","End":"03:53.865","Text":"I would say, 3 at the side so let\u0027s just replace the 1 with a 3."},{"Start":"03:53.865 ","End":"03:56.005","Text":"Of course it\u0027s the same thing as if I did this,"},{"Start":"03:56.005 ","End":"03:58.565","Text":"I have this thing with respect to y,"},{"Start":"03:58.565 ","End":"04:03.095","Text":"which is 3 and 1 plus this thing squared,"},{"Start":"04:03.095 ","End":"04:06.690","Text":"so that\u0027s the answer. We\u0027re done."}],"ID":20879},{"Watched":false,"Name":"Exercise 1 part h","Duration":"1m 57s","ChapterTopicVideoID":8585,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we have to compute"},{"Start":"00:03.060 ","End":"00:06.825","Text":"the 1st order partial derivatives of a function of 2 variables."},{"Start":"00:06.825 ","End":"00:09.390","Text":"Only for a change, instead of x and y,"},{"Start":"00:09.390 ","End":"00:15.750","Text":"we have r and Theta and it might remind you of polar coordinates,"},{"Start":"00:15.750 ","End":"00:18.300","Text":"and this is typically what you might see if you are using"},{"Start":"00:18.300 ","End":"00:23.385","Text":"polar coordinates instead of Cartesian coordinates, the x, y."},{"Start":"00:23.385 ","End":"00:27.660","Text":"No problem, the same idea, just different letters."},{"Start":"00:27.660 ","End":"00:30.135","Text":"We have 2 partial derivatives."},{"Start":"00:30.135 ","End":"00:32.895","Text":"We have derivative of f with respect to"},{"Start":"00:32.895 ","End":"00:37.685","Text":"r and then we\u0027re going to have a derivative of f with respect to Theta."},{"Start":"00:37.685 ","End":"00:39.410","Text":"That\u0027s the Greek letter Theta,"},{"Start":"00:39.410 ","End":"00:42.560","Text":"of course, small Theta."},{"Start":"00:43.040 ","End":"00:46.115","Text":"When we take a derivative with respect to r,"},{"Start":"00:46.115 ","End":"00:49.760","Text":"we treat Theta like a constant so let\u0027s see what we get."},{"Start":"00:49.760 ","End":"00:55.610","Text":"We have r times cosine Theta and the cosine doesn\u0027t affect the fact that it\u0027s a constant."},{"Start":"00:55.610 ","End":"00:59.540","Text":"Cosine of some constant is still a constant so it\u0027s like r times"},{"Start":"00:59.540 ","End":"01:05.950","Text":"5 or 5r, and the derivative of that would be just 5."},{"Start":"01:05.950 ","End":"01:11.630","Text":"Likewise, this 1 would just be the coefficient of r,"},{"Start":"01:11.630 ","End":"01:13.925","Text":"which is cosine Theta,"},{"Start":"01:13.925 ","End":"01:18.410","Text":"the thing that multiplies r. Okay, that\u0027s all there."},{"Start":"01:18.410 ","End":"01:21.874","Text":"Now, as far as with respect to Theta,"},{"Start":"01:21.874 ","End":"01:23.710","Text":"it\u0027s a bit different."},{"Start":"01:23.710 ","End":"01:25.590","Text":"Idea is the same,"},{"Start":"01:25.590 ","End":"01:29.370","Text":"r is a constant this time, Theta\u0027s the variable."},{"Start":"01:29.370 ","End":"01:31.955","Text":"We have a constant times something."},{"Start":"01:31.955 ","End":"01:38.790","Text":"The constant just stays as if it was 19 cosine Theta,"},{"Start":"01:38.790 ","End":"01:42.680","Text":"so you\u0027d put the 19 and then you differentiate cosine Theta."},{"Start":"01:42.680 ","End":"01:46.430","Text":"Derivative of cosine Theta is minus sine Theta so allow me to"},{"Start":"01:46.430 ","End":"01:50.200","Text":"write the sine Theta here and the minus I\u0027ll put upfront."},{"Start":"01:50.200 ","End":"01:53.310","Text":"It\u0027s really r times minus Theta and just put the minus"},{"Start":"01:53.310 ","End":"01:57.250","Text":"here, and that\u0027s all there is to it."}],"ID":20880},{"Watched":false,"Name":"Exercise 1 part i","Duration":"2m 47s","ChapterTopicVideoID":8586,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"In this exercise, we have to compute"},{"Start":"00:02.160 ","End":"00:05.895","Text":"the first-order partial derivatives of this function."},{"Start":"00:05.895 ","End":"00:09.525","Text":"Notice that it\u0027s a function of 3 variables, x, y, and z."},{"Start":"00:09.525 ","End":"00:11.910","Text":"Most of the exercises have been in 2 variables,"},{"Start":"00:11.910 ","End":"00:13.190","Text":"but it\u0027s the same idea."},{"Start":"00:13.190 ","End":"00:15.285","Text":"Just extend it to 3 variables."},{"Start":"00:15.285 ","End":"00:19.575","Text":"Going to be a bit of a longer exercise because we\u0027re going to have 3 partial derivatives."},{"Start":"00:19.575 ","End":"00:21.930","Text":"We\u0027re going to have f with respect to x,"},{"Start":"00:21.930 ","End":"00:23.700","Text":"and we\u0027ll see in a moment what that is,"},{"Start":"00:23.700 ","End":"00:26.160","Text":"we\u0027ll have f with respect to y,"},{"Start":"00:26.160 ","End":"00:28.935","Text":"and we\u0027ll have f with respect to z."},{"Start":"00:28.935 ","End":"00:32.910","Text":"These are the first-order partial derivatives."},{"Start":"00:32.910 ","End":"00:35.880","Text":"You may have seen there\u0027s also second-order and so on,"},{"Start":"00:35.880 ","End":"00:39.570","Text":"but anyway, just take it as partial derivative."},{"Start":"00:39.570 ","End":"00:41.640","Text":"If it\u0027d not show a first-order,"},{"Start":"00:41.640 ","End":"00:44.650","Text":"we can ignore that."},{"Start":"00:45.290 ","End":"00:50.150","Text":"Let\u0027s take them in order of x, y, and then z."},{"Start":"00:50.150 ","End":"00:51.935","Text":"F with respect to x,"},{"Start":"00:51.935 ","End":"00:55.010","Text":"the idea here is that x is the variable,"},{"Start":"00:55.010 ","End":"00:58.950","Text":"and anything else is a constant, or parameters,"},{"Start":"00:58.950 ","End":"01:02.315","Text":"so both y and z will be constants."},{"Start":"01:02.315 ","End":"01:05.300","Text":"This is the expression we\u0027re differentiating with respect to x."},{"Start":"01:05.300 ","End":"01:07.280","Text":"If y and z are constants,"},{"Start":"01:07.280 ","End":"01:09.440","Text":"all this might have been like a squared,"},{"Start":"01:09.440 ","End":"01:11.780","Text":"b cubed or even actual numbers."},{"Start":"01:11.780 ","End":"01:15.500","Text":"Whatever it is, this bit is a constant times x."},{"Start":"01:15.500 ","End":"01:19.250","Text":"Derivative of a constant times x is just that constant,"},{"Start":"01:19.250 ","End":"01:24.090","Text":"so the answer is going to be y squared, z cubed,"},{"Start":"01:24.090 ","End":"01:27.200","Text":"as if it was like x times 4 or something,"},{"Start":"01:27.200 ","End":"01:29.360","Text":"then the answer would be 4."},{"Start":"01:29.360 ","End":"01:32.240","Text":"Now with respect to y, of course,"},{"Start":"01:32.240 ","End":"01:34.085","Text":"if we\u0027re differentiating with respect to y,"},{"Start":"01:34.085 ","End":"01:38.175","Text":"the other 2, x and z are constants."},{"Start":"01:38.175 ","End":"01:45.004","Text":"We have the derivative of y squared, which is 2 y,"},{"Start":"01:45.004 ","End":"01:49.750","Text":"and we\u0027ll put the 2 upfront,"},{"Start":"01:49.750 ","End":"01:55.370","Text":"I need a space here because I want to just keep these constants in,"},{"Start":"01:55.370 ","End":"01:56.765","Text":"but in the same order,"},{"Start":"01:56.765 ","End":"02:01.870","Text":"so I\u0027m going to put the x in here and the z cubed here."},{"Start":"02:01.870 ","End":"02:06.105","Text":"I just didn\u0027t want to break up and put a 2 in the middle."},{"Start":"02:06.105 ","End":"02:09.380","Text":"That\u0027s the partial derivative with respect to y."},{"Start":"02:09.380 ","End":"02:11.645","Text":"But we have 3 variables,"},{"Start":"02:11.645 ","End":"02:14.375","Text":"and we have a third 1 with the z."},{"Start":"02:14.375 ","End":"02:18.900","Text":"This time we look at the z cubed and everything else here is a constant."},{"Start":"02:18.900 ","End":"02:21.200","Text":"We have some constant times z cubed."},{"Start":"02:21.200 ","End":"02:24.665","Text":"This time it\u0027s going to be 3 z squared."},{"Start":"02:24.665 ","End":"02:27.935","Text":"As before, I\u0027ll write little space in the middle,"},{"Start":"02:27.935 ","End":"02:29.780","Text":"the 3 and the z-squared,"},{"Start":"02:29.780 ","End":"02:33.290","Text":"because I want to multiply now by that constant xy squared"},{"Start":"02:33.290 ","End":"02:36.955","Text":"so I put this here, the xy squared."},{"Start":"02:36.955 ","End":"02:40.130","Text":"Instead of having the 3 here and then bringing it to the front,"},{"Start":"02:40.130 ","End":"02:42.335","Text":"I immediately put it in front."},{"Start":"02:42.335 ","End":"02:47.340","Text":"3 variables, 3 partial derivatives, and we\u0027re done."}],"ID":20881},{"Watched":false,"Name":"Exercise 1 part j","Duration":"7m 16s","ChapterTopicVideoID":8587,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"In this exercise, we have the function f of 3 variables,"},{"Start":"00:04.620 ","End":"00:06.750","Text":"note x, y, z,"},{"Start":"00:06.750 ","End":"00:10.020","Text":"u, v, and t given as follows."},{"Start":"00:10.020 ","End":"00:12.765","Text":"We want the first-order partial derivatives,"},{"Start":"00:12.765 ","End":"00:15.240","Text":"it\u0027s going to be 3 of them with respect to u,"},{"Start":"00:15.240 ","End":"00:17.685","Text":"which we\u0027re going to call fu."},{"Start":"00:17.685 ","End":"00:21.135","Text":"Then with respect to v,"},{"Start":"00:21.135 ","End":"00:26.824","Text":"that will be f with respect to v and then f with respect to t."},{"Start":"00:26.824 ","End":"00:34.130","Text":"Remember the idea is that when we differentiate with respect to 1 variable,"},{"Start":"00:34.130 ","End":"00:36.590","Text":"the others, in this case there will be 2 others,"},{"Start":"00:36.590 ","End":"00:39.660","Text":"will be treated like constants or parameters."},{"Start":"00:39.660 ","End":"00:43.490","Text":"Let\u0027s start with the first one, with respect to u."},{"Start":"00:43.490 ","End":"00:46.355","Text":"Actually this is the hardest of the 3,"},{"Start":"00:46.355 ","End":"00:51.230","Text":"they\u0027re all easy but this one because we have u both here and here,"},{"Start":"00:51.230 ","End":"00:54.055","Text":"I\u0027m going to look at it as a product."},{"Start":"00:54.055 ","End":"00:56.280","Text":"Quick reminder of product rule,"},{"Start":"00:56.280 ","End":"01:00.710","Text":"if I have a product and take its derivative the first times the derivative of the"},{"Start":"01:00.710 ","End":"01:06.230","Text":"second plus the derivative of the first times the second."},{"Start":"01:06.230 ","End":"01:09.200","Text":"Only in our case it\u0027s going to be a partial derivative with"},{"Start":"01:09.200 ","End":"01:14.957","Text":"respect to u. I chose a variable u."},{"Start":"01:14.957 ","End":"01:23.974","Text":"I won\u0027t spell it out because actually to avoid a name clash,"},{"Start":"01:23.974 ","End":"01:28.650","Text":"I\u0027ll use f and g. Now I\u0027ve got an f here as well."},{"Start":"01:28.750 ","End":"01:34.670","Text":"I usually write the product rule with u and v but we already have u and v here."},{"Start":"01:34.670 ","End":"01:40.325","Text":"We\u0027re not going to use f and g. Now I have an g and h fine."},{"Start":"01:40.325 ","End":"01:44.365","Text":"G times h derivative."},{"Start":"01:44.365 ","End":"01:46.070","Text":"I just remembered the scheme,"},{"Start":"01:46.070 ","End":"01:49.040","Text":"derivative of the first times the second plus"},{"Start":"01:49.040 ","End":"01:52.790","Text":"the first times the derivative of the second and doesn\u0027t matter what letters."},{"Start":"01:52.790 ","End":"01:55.550","Text":"In our case, we\u0027re going to use the partial derivative"},{"Start":"01:55.550 ","End":"01:58.510","Text":"the same thing works if I take a product,"},{"Start":"01:58.510 ","End":"02:01.520","Text":"and take its partial derivative with respect to u,"},{"Start":"02:01.520 ","End":"02:04.785","Text":"it\u0027s going to be the first with respect to u,"},{"Start":"02:04.785 ","End":"02:06.495","Text":"the other one as is,"},{"Start":"02:06.495 ","End":"02:08.445","Text":"and then the first one as is,"},{"Start":"02:08.445 ","End":"02:15.750","Text":"the second with respect to u. I\u0027m just translating the prime as appropriate."},{"Start":"02:16.790 ","End":"02:21.120","Text":"In our particular case of course this will be the g,"},{"Start":"02:21.120 ","End":"02:24.660","Text":"and this will be the h,"},{"Start":"02:24.660 ","End":"02:28.985","Text":"and I\u0027m looking here and then I say,"},{"Start":"02:28.985 ","End":"02:32.240","Text":"g with respect to u."},{"Start":"02:32.240 ","End":"02:38.510","Text":"That\u0027s going to be e to the power of something derivative,"},{"Start":"02:38.510 ","End":"02:42.770","Text":"it would be just e to the same thing,"},{"Start":"02:42.770 ","End":"02:48.040","Text":"e^uv, but I need to multiply it by the inner derivative,"},{"Start":"02:48.040 ","End":"02:52.110","Text":"which is going to be derivative of"},{"Start":"02:52.110 ","End":"02:58.410","Text":"uv is going to be v. Actually I don\u0027t need the brackets."},{"Start":"02:58.410 ","End":"03:02.594","Text":"For example, suppose it was u times 6,"},{"Start":"03:02.594 ","End":"03:07.060","Text":"or 6u, then the derivative would be just the 6."},{"Start":"03:07.160 ","End":"03:11.885","Text":"So far we\u0027ve just done the g prime part."},{"Start":"03:11.885 ","End":"03:15.380","Text":"Now we need the h which is the other function,"},{"Start":"03:15.380 ","End":"03:22.890","Text":"sine of ut, maybe it\u0027s best to put it in brackets."},{"Start":"03:22.890 ","End":"03:27.200","Text":"Here you can tell its got a font that\u0027s still I would"},{"Start":"03:27.200 ","End":"03:32.405","Text":"have done it again with brackets just to make it absolutely clear."},{"Start":"03:32.405 ","End":"03:36.310","Text":"That\u0027s the g prime h then a plus,"},{"Start":"03:36.310 ","End":"03:40.645","Text":"and then g as is which is e^uv,"},{"Start":"03:40.645 ","End":"03:42.155","Text":"and now I need the derivative."},{"Start":"03:42.155 ","End":"03:46.360","Text":"Actually I\u0027m looking here, derivative of h with respect to u."},{"Start":"03:46.360 ","End":"03:50.265","Text":"Now, I first of all see a sine of something,"},{"Start":"03:50.265 ","End":"03:55.310","Text":"I start off with cosine of that same thing, cosine of ut."},{"Start":"03:55.310 ","End":"03:58.100","Text":"I start off because again we have an inner derivative,"},{"Start":"03:58.100 ","End":"04:03.390","Text":"I need to differentiate ut with respect to u."},{"Start":"04:03.390 ","End":"04:06.750","Text":"Once again, it\u0027s u with a constant,"},{"Start":"04:06.750 ","End":"04:08.880","Text":"this time it\u0027s t,"},{"Start":"04:08.880 ","End":"04:13.865","Text":"I need to multiply that by t which is the derivative of ut."},{"Start":"04:13.865 ","End":"04:16.685","Text":"Other than tidying up,"},{"Start":"04:16.685 ","End":"04:19.645","Text":"this would be the answer."},{"Start":"04:19.645 ","End":"04:23.570","Text":"I could leave this partial derivative just like"},{"Start":"04:23.570 ","End":"04:26.840","Text":"that but I\u0027d like to just tidy it up a bit."},{"Start":"04:26.840 ","End":"04:29.575","Text":"Optional. Bare with me."},{"Start":"04:29.575 ","End":"04:34.170","Text":"The e^uv part appears in both,"},{"Start":"04:34.170 ","End":"04:40.025","Text":"so I\u0027d like to take that out as a common factor in front of the brackets."},{"Start":"04:40.025 ","End":"04:42.785","Text":"Let\u0027s start with e^uv."},{"Start":"04:42.785 ","End":"04:44.525","Text":"Let\u0027s see what we\u0027re left with."},{"Start":"04:44.525 ","End":"04:51.770","Text":"Here we have v times sine of ut."},{"Start":"04:51.770 ","End":"04:55.545","Text":"In the second one I have cosine ut times t,"},{"Start":"04:55.545 ","End":"04:59.630","Text":"but it\u0027s preferable to put the t in front for"},{"Start":"04:59.630 ","End":"05:03.965","Text":"some reason this looks nicer to mathematicians at any rate."},{"Start":"05:03.965 ","End":"05:07.150","Text":"We\u0027ll have t cosine of ut,"},{"Start":"05:07.150 ","End":"05:09.380","Text":"of course you could leave the t at the end,"},{"Start":"05:09.380 ","End":"05:11.610","Text":"no problem with that."},{"Start":"05:12.570 ","End":"05:16.790","Text":"Maybe I\u0027ll change the square brackets."},{"Start":"05:16.980 ","End":"05:21.940","Text":"Once again, it\u0027s just an aesthetic thing if I have already round brackets,"},{"Start":"05:21.940 ","End":"05:25.490","Text":"not to confuse the brackets I\u0027ll use square ones."},{"Start":"05:25.710 ","End":"05:28.180","Text":"That does the first one,"},{"Start":"05:28.180 ","End":"05:30.685","Text":"2 more to go but they\u0027re easier."},{"Start":"05:30.685 ","End":"05:34.480","Text":"If I take the partial with respect to v,"},{"Start":"05:34.480 ","End":"05:40.085","Text":"v only appears in one place and all this second bit is a constant."},{"Start":"05:40.085 ","End":"05:44.145","Text":"In fact, you can ignore the split g, h here."},{"Start":"05:44.145 ","End":"05:49.420","Text":"What I do is, I\u0027m going to differentiate this with respect to"},{"Start":"05:49.420 ","End":"05:55.210","Text":"v and then just stick this thing along because it doesn\u0027t contain v. Each of the uv,"},{"Start":"05:55.210 ","End":"05:58.110","Text":"like before, e to the power of,"},{"Start":"05:58.110 ","End":"05:59.675","Text":"it\u0027s e to the power of,"},{"Start":"05:59.675 ","End":"06:01.160","Text":"then as an inner derivative,"},{"Start":"06:01.160 ","End":"06:08.985","Text":"only this time v is the variable and u is the constant and we get times u, not as before."},{"Start":"06:08.985 ","End":"06:12.030","Text":"Then this thing is a constant,"},{"Start":"06:12.030 ","End":"06:17.640","Text":"so at sine of ut,"},{"Start":"06:17.640 ","End":"06:19.530","Text":"and there\u0027s no product here,"},{"Start":"06:19.530 ","End":"06:21.825","Text":"this is the answer."},{"Start":"06:21.825 ","End":"06:26.970","Text":"I\u0027ll leave it as is although I would put the u in front possibly."},{"Start":"06:26.970 ","End":"06:30.440","Text":"Next one, with respect to t. Once again,"},{"Start":"06:30.440 ","End":"06:34.420","Text":"t only appears in the second path."},{"Start":"06:34.420 ","End":"06:35.910","Text":"There\u0027s no t here,"},{"Start":"06:35.910 ","End":"06:37.425","Text":"so this thing is a constant,"},{"Start":"06:37.425 ","End":"06:41.385","Text":"so I start off with this constant e^v."},{"Start":"06:41.385 ","End":"06:49.535","Text":"Now I just have to differentiate this with respect to t. Sine of something,"},{"Start":"06:49.535 ","End":"06:52.760","Text":"we get cosine of the same thing,"},{"Start":"06:52.760 ","End":"07:00.135","Text":"but it wasn\u0027t just a simple t it\u0027s ut I don\u0027t know 2t,"},{"Start":"07:00.135 ","End":"07:04.980","Text":"then I would put the inner derivative of 2 or in this case u."},{"Start":"07:04.980 ","End":"07:09.855","Text":"Once again, I would put the u upfront,"},{"Start":"07:09.855 ","End":"07:12.220","Text":"but this is fine."},{"Start":"07:12.740 ","End":"07:16.810","Text":"That\u0027s 3 out of 3, we\u0027re done."}],"ID":20882},{"Watched":false,"Name":"Exercise 2 part a","Duration":"4m 53s","ChapterTopicVideoID":8588,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.550","Text":"In this exercise, we\u0027re given a function of 2 variables, x,"},{"Start":"00:05.550 ","End":"00:07.365","Text":"and y as here,"},{"Start":"00:07.365 ","End":"00:11.445","Text":"and we have to compute the second-order partial derivatives."},{"Start":"00:11.445 ","End":"00:15.690","Text":"Now, what are the second-order partial derivatives?"},{"Start":"00:15.690 ","End":"00:18.210","Text":"Turns out there\u0027s 4 of them,"},{"Start":"00:18.210 ","End":"00:22.965","Text":"they\u0027re just the partial derivatives of the first-order partial derivatives."},{"Start":"00:22.965 ","End":"00:29.639","Text":"Let me at the side say that we have first-order partial derivatives."},{"Start":"00:29.639 ","End":"00:32.520","Text":"We have f with respect to x,"},{"Start":"00:32.520 ","End":"00:34.605","Text":"and we\u0027ll compute that in a moment."},{"Start":"00:34.605 ","End":"00:38.705","Text":"Then we also have f with respect to y."},{"Start":"00:38.705 ","End":"00:42.750","Text":"Now if I take this one after I\u0027ve computed it,"},{"Start":"00:42.750 ","End":"00:47.190","Text":"I can take 2 partial derivatives of this with respect to x and y."},{"Start":"00:47.190 ","End":"00:50.810","Text":"What I get is with respect to x and then with respect to"},{"Start":"00:50.810 ","End":"00:54.470","Text":"x will be one of the second-order partial derivatives."},{"Start":"00:54.470 ","End":"01:01.345","Text":"Then we also have with respect to x and then with respect to y."},{"Start":"01:01.345 ","End":"01:03.635","Text":"Then taking the other one,"},{"Start":"01:03.635 ","End":"01:05.900","Text":"it has 2 partial derivatives,"},{"Start":"01:05.900 ","End":"01:10.850","Text":"which will be derivative of y with respect to"},{"Start":"01:10.850 ","End":"01:17.390","Text":"x and the derivative with respect to y and then with respect to y."},{"Start":"01:17.390 ","End":"01:21.140","Text":"Actually, there are 6 exercises in one."},{"Start":"01:21.140 ","End":"01:25.190","Text":"I mean, we have to do the first-order ones in order to get to the second-order ones."},{"Start":"01:25.190 ","End":"01:28.080","Text":"But these are the ones that we\u0027re looking for."},{"Start":"01:28.490 ","End":"01:31.370","Text":"Let\u0027s start. First of all,"},{"Start":"01:31.370 ","End":"01:34.435","Text":"let\u0027s do these first-order ones."},{"Start":"01:34.435 ","End":"01:40.745","Text":"You should be adapted to these by now derivative with respect to x. Y is a constant."},{"Start":"01:40.745 ","End":"01:46.255","Text":"We get from 4x squared, I get 8x."},{"Start":"01:46.255 ","End":"01:48.920","Text":"From here, y is a constant,"},{"Start":"01:48.920 ","End":"01:51.635","Text":"so I get minus 2x."},{"Start":"01:51.635 ","End":"01:59.000","Text":"But the y squared sticks derivative of this is just 4 and 10y is a constant,"},{"Start":"01:59.000 ","End":"02:01.985","Text":"so nothing I\u0027m not going to write plus 0."},{"Start":"02:01.985 ","End":"02:05.195","Text":"Now with respect to y,"},{"Start":"02:05.195 ","End":"02:09.230","Text":"first ones are constants as far as y goes that\u0027s nothing."},{"Start":"02:09.230 ","End":"02:13.220","Text":"We start off with minus x squared is a constant,"},{"Start":"02:13.220 ","End":"02:16.350","Text":"so we have 2y."},{"Start":"02:16.880 ","End":"02:22.430","Text":"The constant stays, so it\u0027s x squared, but it\u0027s 2y."},{"Start":"02:22.430 ","End":"02:27.020","Text":"You know what, I\u0027d like to put the 2 here and the y here."},{"Start":"02:27.020 ","End":"02:30.995","Text":"I don\u0027t like to put a constant in the middle of an expression."},{"Start":"02:30.995 ","End":"02:35.225","Text":"That\u0027s the derivative with respect to y."},{"Start":"02:35.225 ","End":"02:38.375","Text":"This part\u0027s like a warm-up exercise."},{"Start":"02:38.375 ","End":"02:41.390","Text":"Then each of these, we\u0027re going to take"},{"Start":"02:41.390 ","End":"02:45.880","Text":"2 partial derivatives with respect to x and with respect to y."},{"Start":"02:45.880 ","End":"02:50.495","Text":"Here also with respect to x and with respect to y."},{"Start":"02:50.495 ","End":"02:53.195","Text":"We\u0027ll get the 4 here."},{"Start":"02:53.195 ","End":"02:57.980","Text":"This one, I\u0027m just using as a notation,"},{"Start":"02:57.980 ","End":"03:03.720","Text":"mean derivative with respect to x, I get 8."},{"Start":"03:03.720 ","End":"03:05.990","Text":"Then y is a constant, remember,"},{"Start":"03:05.990 ","End":"03:10.460","Text":"so it\u0027s minus 2x is just minus 2."},{"Start":"03:10.460 ","End":"03:14.130","Text":"But the y squared stays."},{"Start":"03:14.200 ","End":"03:16.580","Text":"This is a constant,"},{"Start":"03:16.580 ","End":"03:18.885","Text":"so that\u0027s all we get."},{"Start":"03:18.885 ","End":"03:21.464","Text":"Now this one with respect to y,"},{"Start":"03:21.464 ","End":"03:25.375","Text":"the 8x disappears with respect to y,"},{"Start":"03:25.375 ","End":"03:29.225","Text":"and if differentiate the y squared, it\u0027s 2y."},{"Start":"03:29.225 ","End":"03:35.015","Text":"If it\u0027s 2y, I can combine the 2 with the 2 and get 4,"},{"Start":"03:35.015 ","End":"03:41.150","Text":"2 times 2 is 4, x is a constant stays and the y."},{"Start":"03:42.710 ","End":"03:45.765","Text":"Well, I combine the 2 times 2,"},{"Start":"03:45.765 ","End":"03:47.985","Text":"and write that down."},{"Start":"03:47.985 ","End":"03:50.700","Text":"The 4 of course gives me nothing also,"},{"Start":"03:50.700 ","End":"03:52.560","Text":"so that\u0027s all I get."},{"Start":"03:52.560 ","End":"03:56.685","Text":"Now, let\u0027s go to this one and"},{"Start":"03:56.685 ","End":"04:01.260","Text":"differentiate it partially with respect to x so y is a constant."},{"Start":"04:01.260 ","End":"04:03.360","Text":"The minus 2y stays."},{"Start":"04:03.360 ","End":"04:05.080","Text":"This gives me 2x."},{"Start":"04:05.080 ","End":"04:09.740","Text":"As before, I\u0027ll do the 2 times 2 without the calculator."},{"Start":"04:09.740 ","End":"04:14.630","Text":"I know that the minus was the 2 times 2 is 4."},{"Start":"04:14.630 ","End":"04:19.255","Text":"Then we get the x from the 2x and then the y."},{"Start":"04:19.255 ","End":"04:22.125","Text":"Now this one with respect to y,"},{"Start":"04:22.125 ","End":"04:26.480","Text":"so all this minus 2x squared is a constant and the derivative of y is just 1,"},{"Start":"04:26.480 ","End":"04:30.365","Text":"so we end up with just minus 2x squared."},{"Start":"04:30.365 ","End":"04:32.180","Text":"Well, I just like you to note,"},{"Start":"04:32.180 ","End":"04:33.770","Text":"I\u0027m not going to do anything with it,"},{"Start":"04:33.770 ","End":"04:36.095","Text":"but that these 2 are equal."},{"Start":"04:36.095 ","End":"04:39.000","Text":"Just observe nothing else."},{"Start":"04:39.940 ","End":"04:42.170","Text":"The answer, of course,"},{"Start":"04:42.170 ","End":"04:45.860","Text":"is all this part here."},{"Start":"04:45.860 ","End":"04:48.710","Text":"These are the 4 second-order partial derivatives."},{"Start":"04:48.710 ","End":"04:52.620","Text":"I won\u0027t bother with highlighting them. We are done."}],"ID":20883},{"Watched":false,"Name":"Exercise 2 part b","Duration":"4m 35s","ChapterTopicVideoID":8589,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this exercise, we have to find"},{"Start":"00:02.850 ","End":"00:11.470","Text":"the second-order partial derivatives of the function of 2 variables as follows."},{"Start":"00:11.770 ","End":"00:15.930","Text":"To start off, we should find"},{"Start":"00:15.930 ","End":"00:20.670","Text":"the first-order partial derivatives because we need those to find the second-order."},{"Start":"00:20.670 ","End":"00:23.595","Text":"Let me do those at the side here."},{"Start":"00:23.595 ","End":"00:32.670","Text":"I need first-order with respect to x and the first-order with respect to y."},{"Start":"00:32.670 ","End":"00:36.089","Text":"Let\u0027s see, with respect to x,"},{"Start":"00:36.089 ","End":"00:38.565","Text":"y is a constant."},{"Start":"00:38.565 ","End":"00:41.910","Text":"I just get 4_x cubed,"},{"Start":"00:41.910 ","End":"00:45.330","Text":"and this constant just sticks."},{"Start":"00:45.330 ","End":"00:48.840","Text":"As for the other 1, with respect to y,"},{"Start":"00:48.840 ","End":"00:50.610","Text":"x is a constant."},{"Start":"00:50.610 ","End":"00:59.060","Text":"I have x to the 4th times derivative of this is 1 over y."},{"Start":"00:59.060 ","End":"01:05.270","Text":"So why don\u0027t I just write it as x to the 4th over y instead of times 1 over y."},{"Start":"01:05.270 ","End":"01:10.010","Text":"Now, the way we get the second-order ones is each 1 of these"},{"Start":"01:10.010 ","End":"01:16.515","Text":"I differentiate partially ones with respect to x,"},{"Start":"01:16.515 ","End":"01:20.885","Text":"and ones with respect to y. I\u0027m just writing this symbolically."},{"Start":"01:20.885 ","End":"01:25.710","Text":"This 1 also has 2 partial derivatives with respect to x,"},{"Start":"01:25.710 ","End":"01:27.970","Text":"and with respect to y."},{"Start":"01:29.720 ","End":"01:35.000","Text":"The first 1 will give us what we call fxx,"},{"Start":"01:35.000 ","End":"01:38.525","Text":"second order with respect to x and with respect to x,"},{"Start":"01:38.525 ","End":"01:40.805","Text":"or twice with respect to x."},{"Start":"01:40.805 ","End":"01:44.360","Text":"Then we\u0027ll have an fxy and all the combinations."},{"Start":"01:44.360 ","End":"01:45.995","Text":"Anyway, let\u0027s get started."},{"Start":"01:45.995 ","End":"01:50.075","Text":"The first 1, I need to take this with respect to x,"},{"Start":"01:50.075 ","End":"01:52.650","Text":"y is a constant."},{"Start":"01:52.970 ","End":"01:56.490","Text":"I differentiate 4_x cubed with respect to x,"},{"Start":"01:56.490 ","End":"01:58.080","Text":"that gives me 12_x squared,"},{"Start":"01:58.080 ","End":"01:59.910","Text":"3 times 4 is 12."},{"Start":"01:59.910 ","End":"02:03.735","Text":"12_x squared, and this constant sticks."},{"Start":"02:03.735 ","End":"02:06.300","Text":"That\u0027s 1 of them, easy."},{"Start":"02:06.300 ","End":"02:10.155","Text":"Now, this 1 with respect to y."},{"Start":"02:10.155 ","End":"02:14.045","Text":"We get f with respect to x,"},{"Start":"02:14.045 ","End":"02:17.345","Text":"with respect to y is equal to,"},{"Start":"02:17.345 ","End":"02:20.580","Text":"the 4_x cubed is the constant."},{"Start":"02:21.230 ","End":"02:24.540","Text":"X is a constant so this is all a constant."},{"Start":"02:24.540 ","End":"02:28.880","Text":"Derivative of natural log of y is 1 over y with respect to y."},{"Start":"02:28.880 ","End":"02:33.050","Text":"Let\u0027s put the 1 over y just like we did here in the denominator."},{"Start":"02:33.050 ","End":"02:37.545","Text":"We get 4_x cubed times 1 over y,"},{"Start":"02:37.545 ","End":"02:39.645","Text":"so it looks like this."},{"Start":"02:39.645 ","End":"02:43.290","Text":"That\u0027s the second. Now, third,"},{"Start":"02:43.290 ","End":"02:44.710","Text":"they don\u0027t have a particular order,"},{"Start":"02:44.710 ","End":"02:45.770","Text":"I\u0027m just saying third,"},{"Start":"02:45.770 ","End":"02:51.185","Text":"to be the derivative with respect to y and then with respect to x."},{"Start":"02:51.185 ","End":"02:55.415","Text":"We take this and differentiate it with respect to x."},{"Start":"02:55.415 ","End":"03:00.245","Text":"Now, this denominator is a constant,"},{"Start":"03:00.245 ","End":"03:04.460","Text":"might as well been over 8 or something,"},{"Start":"03:04.460 ","End":"03:07.325","Text":"or over 3, so it would just stay."},{"Start":"03:07.325 ","End":"03:11.440","Text":"This part just stays."},{"Start":"03:11.440 ","End":"03:16.130","Text":"What I need to do is derivative of the numerator with respect to x,"},{"Start":"03:16.130 ","End":"03:19.475","Text":"and that\u0027s 4_x cubed."},{"Start":"03:19.475 ","End":"03:23.640","Text":"Then finally with respect to y twice,"},{"Start":"03:25.460 ","End":"03:29.800","Text":"here we just differentiate this with respect to y. I just"},{"Start":"03:29.800 ","End":"03:35.050","Text":"need to remind you that the derivative of 1 over y,"},{"Start":"03:35.050 ","End":"03:44.260","Text":"regular derivative of 1 variable function is minus 1 over y squared,"},{"Start":"03:44.260 ","End":"03:46.255","Text":"because this is to the minus 1."},{"Start":"03:46.255 ","End":"03:48.255","Text":"Anyway, you\u0027ve seen this before."},{"Start":"03:48.255 ","End":"03:52.200","Text":"All I do now is the x to the 4th is a constant."},{"Start":"03:52.200 ","End":"03:56.190","Text":"It\u0027s x to the 4th times minus 1 over y squared."},{"Start":"03:56.190 ","End":"04:01.485","Text":"I\u0027ll put the y squared here and the minus here, I can do that."},{"Start":"04:01.485 ","End":"04:04.530","Text":"That\u0027s all 4 of them."},{"Start":"04:04.530 ","End":"04:12.620","Text":"I\u0027d just like you to note that these 2 are equal and it may or may not be a coincidence."},{"Start":"04:12.620 ","End":"04:16.080","Text":"These are called the mixed partial"},{"Start":"04:16.080 ","End":"04:21.070","Text":"derivatives because each have an x and a y, mix and match."},{"Start":"04:21.070 ","End":"04:25.170","Text":"I\u0027m not sure what these are called as opposed to mixed."},{"Start":"04:25.170 ","End":"04:35.260","Text":"Anyway, these 4 are the 4 partial derivatives of second-order, and we\u0027re done."}],"ID":20884},{"Watched":false,"Name":"Exercise 2 part c","Duration":"6m 19s","ChapterTopicVideoID":8590,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.805","Text":"Here we have a function of x and y."},{"Start":"00:02.805 ","End":"00:07.800","Text":"This is it. We need the second-order partial derivatives."},{"Start":"00:07.800 ","End":"00:09.449","Text":"To get to the second-order,"},{"Start":"00:09.449 ","End":"00:13.100","Text":"we first have to do the first-order partial derivatives."},{"Start":"00:13.100 ","End":"00:14.995","Text":"Let\u0027s do those at the side."},{"Start":"00:14.995 ","End":"00:16.520","Text":"I have 2 of them."},{"Start":"00:16.520 ","End":"00:22.540","Text":"I have with respect to x and then with respect to y,"},{"Start":"00:22.540 ","End":"00:25.720","Text":"let\u0027s see with respect to x."},{"Start":"00:25.720 ","End":"00:27.350","Text":"Well, in both cases,"},{"Start":"00:27.350 ","End":"00:34.235","Text":"the derivative of sine is going to be cosine 10x plus 4y."},{"Start":"00:34.235 ","End":"00:37.940","Text":"That\u0027s how this 1\u0027s going to start out also,"},{"Start":"00:37.940 ","End":"00:39.815","Text":"the difference will be,"},{"Start":"00:39.815 ","End":"00:42.800","Text":"is that the inner derivative,"},{"Start":"00:42.800 ","End":"00:44.045","Text":"and this is a chain rule."},{"Start":"00:44.045 ","End":"00:49.205","Text":"I have to take the internal or inner derivative of the 10x plus 4y."},{"Start":"00:49.205 ","End":"00:53.645","Text":"But I\u0027m doing with respect to x so y is a constant."},{"Start":"00:53.645 ","End":"01:01.935","Text":"The derivative of 10x plus 4y is 10x plus 7 or something."},{"Start":"01:01.935 ","End":"01:06.880","Text":"The answer would be 10 so I multiply this by 10."},{"Start":"01:06.880 ","End":"01:11.675","Text":"On the other hand, when I\u0027m differentiating with respect to y partially,"},{"Start":"01:11.675 ","End":"01:15.515","Text":"I also get the derivative of sine is a cosine."},{"Start":"01:15.515 ","End":"01:20.150","Text":"But the inner derivative is different this time x is a constant,"},{"Start":"01:20.150 ","End":"01:23.510","Text":"so it\u0027s like I had 4y plus 7 or something,"},{"Start":"01:23.510 ","End":"01:28.000","Text":"and then the derivative would be 4."},{"Start":"01:29.660 ","End":"01:33.675","Text":"I\u0027d like to put the constant in front."},{"Start":"01:33.675 ","End":"01:41.175","Text":"Let me just rewrite this 1 as 10 cosine of 10x plus 4y."},{"Start":"01:41.175 ","End":"01:48.310","Text":"This 1 I\u0027ll write as 4 cosine of 10x plus 4y."},{"Start":"01:49.040 ","End":"01:56.810","Text":"I\u0027ve got these, now each of these will give me 2 second-order partial derivatives."},{"Start":"01:56.810 ","End":"01:59.730","Text":"I can take it with respect to x."},{"Start":"02:00.970 ","End":"02:03.500","Text":"Just thought this looks a bit messy,"},{"Start":"02:03.500 ","End":"02:09.610","Text":"why don\u0027t I just bring the 10 in front here."},{"Start":"02:09.610 ","End":"02:13.070","Text":"Here and here, this looks a bit better."},{"Start":"02:13.070 ","End":"02:15.500","Text":"The numbers look much better in front."},{"Start":"02:15.500 ","End":"02:18.920","Text":"As I was saying, we have 2 partial derivatives of"},{"Start":"02:18.920 ","End":"02:24.635","Text":"the first-order and this with respect to x and y and so on."},{"Start":"02:24.635 ","End":"02:27.845","Text":"That will give us 4 second-order."},{"Start":"02:27.845 ","End":"02:33.590","Text":"Start the first 1 is called fxx means we"},{"Start":"02:33.590 ","End":"02:36.170","Text":"do with respect to x first and then again with respect to"},{"Start":"02:36.170 ","End":"02:39.740","Text":"x. I need to take this 1 with respect to x."},{"Start":"02:39.740 ","End":"02:42.605","Text":"The 10 is a constant, it stays."},{"Start":"02:42.605 ","End":"02:46.640","Text":"The derivative of cosine is"},{"Start":"02:46.640 ","End":"02:56.640","Text":"minus sine of the same thing, 10x plus 4y."},{"Start":"02:58.250 ","End":"03:05.265","Text":"Then I again need an inner derivative of the 10x plus 4y."},{"Start":"03:05.265 ","End":"03:07.430","Text":"Just as before, this,"},{"Start":"03:07.430 ","End":"03:09.950","Text":"with respect to x, this is going to be 10."},{"Start":"03:09.950 ","End":"03:14.555","Text":"It\u0027s times 10. Let\u0027s just rearrange things."},{"Start":"03:14.555 ","End":"03:16.835","Text":"Bring the numbers in front and the minus."},{"Start":"03:16.835 ","End":"03:22.160","Text":"I\u0027ve got minus 100 sine"},{"Start":"03:22.160 ","End":"03:27.910","Text":"of 10x plus 4y."},{"Start":"03:27.910 ","End":"03:30.470","Text":"Now this 1 with respect to y,"},{"Start":"03:30.470 ","End":"03:32.630","Text":"so I\u0027m doing this with respect to x and with respect"},{"Start":"03:32.630 ","End":"03:35.780","Text":"to y and with respect to x and with respect to y."},{"Start":"03:35.780 ","End":"03:44.765","Text":"We get fxy is equal to this with respect to y."},{"Start":"03:44.765 ","End":"03:52.490","Text":"Now, again I have 10 derivative of cosine is minus sine,"},{"Start":"03:52.490 ","End":"03:58.790","Text":"just like before, minus sine of 10x plus 4y."},{"Start":"03:58.790 ","End":"04:05.980","Text":"Only this time, the inner derivative is 4."},{"Start":"04:06.680 ","End":"04:12.255","Text":"Tidying up, we get minus 40"},{"Start":"04:12.255 ","End":"04:19.450","Text":"sine of 10x plus 4y."},{"Start":"04:20.270 ","End":"04:25.745","Text":"That\u0027s 2 down. Now this 1 will give us 2 this with respect to x and with respect to y."},{"Start":"04:25.745 ","End":"04:29.330","Text":"We get fyx."},{"Start":"04:29.330 ","End":"04:31.880","Text":"We\u0027re going to do this a bit quicker."},{"Start":"04:31.880 ","End":"04:36.310","Text":"I mean, I\u0027m going to start arranging the numbers as I go."},{"Start":"04:36.310 ","End":"04:39.530","Text":"With respect to x,"},{"Start":"04:39.530 ","End":"04:42.725","Text":"once again, cosine gives me minus sine."},{"Start":"04:42.725 ","End":"04:48.000","Text":"I know I start off with a minus and that\u0027s going to be 4."},{"Start":"04:48.430 ","End":"04:51.320","Text":"I\u0027ll leave the number here for the moment."},{"Start":"04:51.320 ","End":"04:57.675","Text":"We know it\u0027s going to be minus the sine of 10x plus 4y."},{"Start":"04:57.675 ","End":"05:00.995","Text":"Now, we already had a 4 and we get another"},{"Start":"05:00.995 ","End":"05:05.735","Text":"constant from the inner derivative of the result."},{"Start":"05:05.735 ","End":"05:10.460","Text":"A derivative of this with respect to x we already saw is 10."},{"Start":"05:10.460 ","End":"05:12.845","Text":"This 10 combines with the 4,"},{"Start":"05:12.845 ","End":"05:15.240","Text":"so we get minus."},{"Start":"05:16.610 ","End":"05:20.760","Text":"Then with respect to y, with respect to y,"},{"Start":"05:20.760 ","End":"05:23.210","Text":"the only difference is that with respect to y,"},{"Start":"05:23.210 ","End":"05:25.160","Text":"I\u0027m going to get 4 instead of 10 here."},{"Start":"05:25.160 ","End":"05:27.110","Text":"4 times 4 is 16."},{"Start":"05:27.110 ","End":"05:29.710","Text":"I get minus 16."},{"Start":"05:29.710 ","End":"05:33.980","Text":"In all cases it\u0027s got sine of 10x plus 4y."},{"Start":"05:33.980 ","End":"05:36.860","Text":"Just the number in front comes out different."},{"Start":"05:36.860 ","End":"05:42.820","Text":"Notice, just observe I\u0027m not going do anything with it that these 2"},{"Start":"05:42.820 ","End":"05:48.365","Text":"happened to be equal to 2 mixed second-order partial derivatives are equal."},{"Start":"05:48.365 ","End":"05:51.110","Text":"I\u0027m going to say if it has always happens or sometimes,"},{"Start":"05:51.110 ","End":"05:55.065","Text":"or just a fluke. Just observe."},{"Start":"05:55.065 ","End":"06:00.200","Text":"We are done, but I think I should highlight the results."},{"Start":"06:00.200 ","End":"06:02.920","Text":"This is 1 of them."},{"Start":"06:02.920 ","End":"06:05.640","Text":"This is the second 1."},{"Start":"06:05.640 ","End":"06:08.000","Text":"This is the third 1, they don\u0027t have a particular order."},{"Start":"06:08.000 ","End":"06:10.700","Text":"I\u0027m just saying from the fourth 1 we got anyway,"},{"Start":"06:10.700 ","End":"06:12.965","Text":"there\u0027ll be 4 of them in 2 variables,"},{"Start":"06:12.965 ","End":"06:15.095","Text":"2 times 2."},{"Start":"06:15.095 ","End":"06:18.180","Text":"Okay, that\u0027s it."}],"ID":20885},{"Watched":false,"Name":"Exercise 2 part d","Duration":"4m 46s","ChapterTopicVideoID":8591,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.920","Text":"In this exercise, we have to compute"},{"Start":"00:02.920 ","End":"00:09.180","Text":"the second-order partial derivatives of following function."},{"Start":"00:09.180 ","End":"00:15.860","Text":"The thing that\u0027s a bit different is that it has 3 variables; x, y, and z."},{"Start":"00:15.860 ","End":"00:18.765","Text":"Mostly, we\u0027ve done 2 variables."},{"Start":"00:18.765 ","End":"00:23.010","Text":"Now, this will have 3 first-order partial derivatives."},{"Start":"00:23.010 ","End":"00:27.525","Text":"Each of those will have 3 partial derivatives."},{"Start":"00:27.525 ","End":"00:34.460","Text":"We\u0027re going to end up with 3 times 3 is 9 second-order ones and 3 first-order ones."},{"Start":"00:34.460 ","End":"00:38.680","Text":"We\u0027re really going to have 12 in 1 exercises here."},{"Start":"00:38.680 ","End":"00:42.990","Text":"Let me start with the first-order partial derivatives."},{"Start":"00:42.990 ","End":"00:48.580","Text":"Let\u0027s see what f with respect to x is equal to,"},{"Start":"00:48.580 ","End":"00:53.555","Text":"and then we\u0027ll do f with respect to y,"},{"Start":"00:53.555 ","End":"00:58.500","Text":"and then f with respect to z."},{"Start":"00:59.710 ","End":"01:05.240","Text":"As usual, when we do a partial derivative with respect to 1 variable,"},{"Start":"01:05.240 ","End":"01:08.195","Text":"the others are treated like constants or parameters."},{"Start":"01:08.195 ","End":"01:10.190","Text":"In this case with respect to x,"},{"Start":"01:10.190 ","End":"01:12.305","Text":"y and z are constants."},{"Start":"01:12.305 ","End":"01:14.145","Text":"It\u0027s a constant times x."},{"Start":"01:14.145 ","End":"01:17.885","Text":"The answer is just yz."},{"Start":"01:17.885 ","End":"01:21.469","Text":"Similarly here, y is the variable,"},{"Start":"01:21.469 ","End":"01:23.600","Text":"x and z are the constants."},{"Start":"01:23.600 ","End":"01:26.030","Text":"We get xz."},{"Start":"01:26.030 ","End":"01:29.645","Text":"Here clearly we get xy."},{"Start":"01:29.645 ","End":"01:35.825","Text":"Xy is 7, 7z,"},{"Start":"01:35.825 ","End":"01:37.570","Text":"so we just write the 7."},{"Start":"01:37.570 ","End":"01:44.570","Text":"Now, each of these is going to give us 3 second-order partial derivatives."},{"Start":"01:44.570 ","End":"01:48.770","Text":"This 1 is going to give us fxx,"},{"Start":"01:48.770 ","End":"01:57.290","Text":"fxy, and fxz."},{"Start":"01:57.290 ","End":"02:04.205","Text":"Likewise, this one\u0027s going to give us f with respect to y, with respect to x."},{"Start":"02:04.205 ","End":"02:06.120","Text":"Just do it methodically."},{"Start":"02:06.120 ","End":"02:09.805","Text":"F with respect to y, then with respect to y,"},{"Start":"02:09.805 ","End":"02:13.480","Text":"and with respect to y and z."},{"Start":"02:13.480 ","End":"02:16.990","Text":"This 1 will give us the f with respect to z."},{"Start":"02:16.990 ","End":"02:23.980","Text":"If I take 3 partial derivatives; fz, fz, fz,"},{"Start":"02:23.980 ","End":"02:27.270","Text":"1 is with respect to x, 1 is with respect to y,"},{"Start":"02:27.270 ","End":"02:29.935","Text":"1 is with respect to z, equals,"},{"Start":"02:29.935 ","End":"02:33.640","Text":"equals, equals, equals, equals, equals."},{"Start":"02:33.640 ","End":"02:37.995","Text":"Okay. With respect to x,"},{"Start":"02:37.995 ","End":"02:42.030","Text":"there is no x here, it\u0027s a constant 0."},{"Start":"02:42.030 ","End":"02:45.660","Text":"With respect to y, z is a constant,"},{"Start":"02:45.660 ","End":"02:49.390","Text":"constant times y, so just the constant."},{"Start":"02:49.790 ","End":"02:55.030","Text":"With respect to z, z is the variable,"},{"Start":"02:55.030 ","End":"02:56.290","Text":"y is the constant,"},{"Start":"02:56.290 ","End":"02:57.460","Text":"the constant times z,"},{"Start":"02:57.460 ","End":"03:00.920","Text":"so it\u0027s just the constant y."},{"Start":"03:01.160 ","End":"03:08.920","Text":"Now, this with respect to x, it\u0027s just the z."},{"Start":"03:09.140 ","End":"03:11.430","Text":"With respect to y,"},{"Start":"03:11.430 ","End":"03:13.930","Text":"there is no y, so it\u0027s 0."},{"Start":"03:13.930 ","End":"03:17.119","Text":"With respect to z,"},{"Start":"03:17.119 ","End":"03:21.145","Text":"constant times z, so it\u0027s just the constant."},{"Start":"03:21.145 ","End":"03:25.345","Text":"Like if it was 4z, we would put 4, just the x."},{"Start":"03:25.345 ","End":"03:28.915","Text":"Similarly here, with respect to x,"},{"Start":"03:28.915 ","End":"03:30.660","Text":"we\u0027re left with the y."},{"Start":"03:30.660 ","End":"03:33.970","Text":"With respect to y,"},{"Start":"03:33.970 ","End":"03:38.210","Text":"we\u0027re left with the x."},{"Start":"03:39.950 ","End":"03:42.575","Text":"With respect to z,"},{"Start":"03:42.575 ","End":"03:45.845","Text":"there is no z here, so it\u0027s 0."},{"Start":"03:45.845 ","End":"03:49.895","Text":"These are all the 9 second-order partial derivatives."},{"Start":"03:49.895 ","End":"03:56.089","Text":"I\u0027d just like you to note that f with respect to x with respect to y,"},{"Start":"03:56.089 ","End":"03:58.220","Text":"although in principle, it might be"},{"Start":"03:58.220 ","End":"04:01.300","Text":"different than with respect to y and with respect to x,"},{"Start":"04:01.300 ","End":"04:03.940","Text":"it happens to give the same answer."},{"Start":"04:03.940 ","End":"04:07.505","Text":"This will be discussed more another time,"},{"Start":"04:07.505 ","End":"04:09.845","Text":"whether it\u0027s a coincidence or not."},{"Start":"04:09.845 ","End":"04:15.320","Text":"Notice also that if I do partial with respect to x and then z,"},{"Start":"04:15.320 ","End":"04:19.850","Text":"which is y, it comes out the same as with"},{"Start":"04:19.850 ","End":"04:25.800","Text":"respect to z and with respect to x, another coincidence."},{"Start":"04:26.350 ","End":"04:32.510","Text":"Also, notice that yz and"},{"Start":"04:32.510 ","End":"04:39.820","Text":"zy partial derivatives also come out the same."},{"Start":"04:40.010 ","End":"04:42.995","Text":"Just observe, I\u0027m not doing anything with that."},{"Start":"04:42.995 ","End":"04:45.900","Text":"Anyway, we are done."}],"ID":20886},{"Watched":false,"Name":"Exercise 3","Duration":"8m 47s","ChapterTopicVideoID":8592,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"Parts, but it all relates to this function f of x,"},{"Start":"00:03.990 ","End":"00:06.570","Text":"y, which is defined one way for x,"},{"Start":"00:06.570 ","End":"00:07.830","Text":"y not equal to 0,"},{"Start":"00:07.830 ","End":"00:09.660","Text":"0, but at 0, 0,"},{"Start":"00:09.660 ","End":"00:11.610","Text":"it makes no sense because it\u0027s 0 over 0,"},{"Start":"00:11.610 ","End":"00:13.335","Text":"so we define it differently,"},{"Start":"00:13.335 ","End":"00:15.510","Text":"we define it to be 0 at the origin."},{"Start":"00:15.510 ","End":"00:17.175","Text":"Then the 3 questions,"},{"Start":"00:17.175 ","End":"00:20.040","Text":"the first one is to find the partial derivatives,"},{"Start":"00:20.040 ","End":"00:24.885","Text":"and then, we have to say whether the function\u0027s continuous at that point."},{"Start":"00:24.885 ","End":"00:28.540","Text":"Well, we\u0027ll get to the third question we\u0027ll read it."},{"Start":"00:28.760 ","End":"00:32.190","Text":"Let\u0027s first of all take the first part"},{"Start":"00:32.190 ","End":"00:34.585","Text":"and find the partial derivatives, there\u0027s 2 of them."},{"Start":"00:34.585 ","End":"00:39.740","Text":"We\u0027ll start off with the derivative with respect to x at the point 0,"},{"Start":"00:39.740 ","End":"00:42.985","Text":"0 and because it\u0027s defined this way,"},{"Start":"00:42.985 ","End":"00:46.530","Text":"piece-wise, we\u0027re going to work straight from the definition."},{"Start":"00:46.530 ","End":"00:50.945","Text":"I think I\u0027ll start in general not at the point 0, 0."},{"Start":"00:50.945 ","End":"00:52.640","Text":"Just remember what it is."},{"Start":"00:52.640 ","End":"00:54.470","Text":"We have a general point, x naught,"},{"Start":"00:54.470 ","End":"00:57.830","Text":"y naught that would be the limit as"},{"Start":"00:57.830 ","End":"01:05.795","Text":"some h small quantity goes to 0 of f of x naught plus h,"},{"Start":"01:05.795 ","End":"01:10.530","Text":"y naught, minus f of x naught,"},{"Start":"01:10.530 ","End":"01:16.650","Text":"y naught over h. In our case,"},{"Start":"01:16.650 ","End":"01:19.500","Text":"x naught, y naught is 0,"},{"Start":"01:19.500 ","End":"01:28.040","Text":"0 here, and then so we get limit as h goes to 0,"},{"Start":"01:28.040 ","End":"01:33.285","Text":"f of 0 plus h is just h,"},{"Start":"01:33.285 ","End":"01:36.600","Text":"0 minus f of 0,"},{"Start":"01:36.600 ","End":"01:43.245","Text":"0 over h, and this is equal to."},{"Start":"01:43.245 ","End":"01:45.470","Text":"Now, for the first part,"},{"Start":"01:45.470 ","End":"01:49.595","Text":"we take this definition because h is not 0, it tends to 0."},{"Start":"01:49.595 ","End":"01:55.580","Text":"What we get is this is h times 0,"},{"Start":"01:55.580 ","End":"01:58.165","Text":"which is already 0."},{"Start":"01:58.165 ","End":"02:02.240","Text":"This part, let me just write limit h goes to 0,"},{"Start":"02:02.240 ","End":"02:07.050","Text":"this part is 0, f of 0, 0 is 0,"},{"Start":"02:07.050 ","End":"02:11.915","Text":"minus 0 over h. Since the numerator is 0,"},{"Start":"02:11.915 ","End":"02:16.640","Text":"this is just the limit of 0 so clearly, this is 0."},{"Start":"02:16.640 ","End":"02:19.320","Text":"That\u0027s the first part."},{"Start":"02:19.540 ","End":"02:21.630","Text":"I don\u0027t have much space here,"},{"Start":"02:21.630 ","End":"02:22.740","Text":"let me do it at the side."},{"Start":"02:22.740 ","End":"02:26.180","Text":"The partial derivative with respect to y,"},{"Start":"02:26.180 ","End":"02:27.665","Text":"I\u0027ll do it in a different color."},{"Start":"02:27.665 ","End":"02:29.885","Text":"In general, at the point x naught,"},{"Start":"02:29.885 ","End":"02:32.465","Text":"y naught, this is equal to the limit."},{"Start":"02:32.465 ","End":"02:33.860","Text":"It\u0027s very similar to this,"},{"Start":"02:33.860 ","End":"02:35.585","Text":"h goes to 0,"},{"Start":"02:35.585 ","End":"02:37.190","Text":"f of, but this time,"},{"Start":"02:37.190 ","End":"02:40.970","Text":"x naught stays fixed and we take y naught plus h,"},{"Start":"02:40.970 ","End":"02:45.140","Text":"minus f at the point itself, x naught,"},{"Start":"02:45.140 ","End":"02:50.840","Text":"y naught over h. Let\u0027s see."},{"Start":"02:50.840 ","End":"02:56.395","Text":"This is equal to, I should have mentioned here."},{"Start":"02:56.395 ","End":"03:02.375","Text":"I should have mentioned that this is fx at the 0,0 in our case and similarly here,"},{"Start":"03:02.375 ","End":"03:06.505","Text":"we have fy of 0,0,"},{"Start":"03:06.505 ","End":"03:09.740","Text":"and this is equal to the limit h goes to 0."},{"Start":"03:09.740 ","End":"03:14.940","Text":"Now, this will be f of 0,"},{"Start":"03:14.940 ","End":"03:18.000","Text":"and then 0 plus h is h,"},{"Start":"03:18.000 ","End":"03:24.515","Text":"minus f of 0,0 over h,"},{"Start":"03:24.515 ","End":"03:28.820","Text":"and this is equal to the limit h goes to 0."},{"Start":"03:28.820 ","End":"03:31.310","Text":"Now, f of 0, h,"},{"Start":"03:31.310 ","End":"03:34.954","Text":"we read it from here, but 0 times h is 0,"},{"Start":"03:34.954 ","End":"03:37.315","Text":"so this is just 0."},{"Start":"03:37.315 ","End":"03:42.060","Text":"This is 0 times h over something,"},{"Start":"03:42.060 ","End":"03:47.670","Text":"doesn\u0027t matter, over 0 squared plus h squared, the same thing here,"},{"Start":"03:47.670 ","End":"03:49.650","Text":"because it looks messy I\u0027ll just erase it,"},{"Start":"03:49.650 ","End":"03:54.975","Text":"but this comes out to be 0 minus,"},{"Start":"03:54.975 ","End":"03:56.580","Text":"and now this one, I\u0027m reading off here,"},{"Start":"03:56.580 ","End":"04:01.110","Text":"it\u0027s this 0 and over h. Same thing as here."},{"Start":"04:01.110 ","End":"04:03.150","Text":"Again, we get 0."},{"Start":"04:03.150 ","End":"04:12.395","Text":"The partial derivatives of the function are both 0,"},{"Start":"04:12.395 ","End":"04:15.180","Text":"and here they are."},{"Start":"04:15.260 ","End":"04:18.840","Text":"That was the first part. Now, the second part."},{"Start":"04:18.840 ","End":"04:23.225","Text":"Is the function continuous at 0, 0 and remember,"},{"Start":"04:23.225 ","End":"04:32.180","Text":"continuous basically means that the limit at the point equals the value at the point."},{"Start":"04:32.180 ","End":"04:33.740","Text":"This is just a mnemonic for me,"},{"Start":"04:33.740 ","End":"04:34.880","Text":"the limit equals value."},{"Start":"04:34.880 ","End":"04:37.760","Text":"In other words, we\u0027ll take the limit as x, y goes to 0,"},{"Start":"04:37.760 ","End":"04:42.915","Text":"0 and ask is it equal to the value at 0, 0."},{"Start":"04:42.915 ","End":"04:47.399","Text":"Let\u0027s do the value 1 first that\u0027s easier."},{"Start":"04:47.399 ","End":"04:51.455","Text":"F of 0, 0, which is given to us, it\u0027s equal to 0."},{"Start":"04:51.455 ","End":"04:54.740","Text":"Now we have to check, does this equal the limit as x,"},{"Start":"04:54.740 ","End":"04:56.600","Text":"y goes to 0, 0 of f of x,"},{"Start":"04:56.600 ","End":"04:59.105","Text":"y. I\u0027m going to scroll."},{"Start":"04:59.105 ","End":"05:03.560","Text":"But the other one is the limit as x,"},{"Start":"05:03.560 ","End":"05:05.930","Text":"y goes to 0,"},{"Start":"05:05.930 ","End":"05:07.940","Text":"0 of f of x,"},{"Start":"05:07.940 ","End":"05:10.220","Text":"y, which is x,"},{"Start":"05:10.220 ","End":"05:14.970","Text":"y over x squared plus y squared."},{"Start":"05:14.970 ","End":"05:18.925","Text":"We going to see what this equals and check if it\u0027s equal to 0."},{"Start":"05:18.925 ","End":"05:21.350","Text":"Now actually, I\u0027m not going to compute this limit."},{"Start":"05:21.350 ","End":"05:24.035","Text":"The reason, very good reason that there is no limit."},{"Start":"05:24.035 ","End":"05:26.270","Text":"Instead, I\u0027m going to show you that there\u0027s no limit."},{"Start":"05:26.270 ","End":"05:28.910","Text":"There are several, more than one way of doing this,"},{"Start":"05:28.910 ","End":"05:34.745","Text":"but one way of showing there\u0027s no limit is computing the limit along 2 different paths."},{"Start":"05:34.745 ","End":"05:38.240","Text":"What I\u0027m going to do is, I\u0027m going to write that down."},{"Start":"05:38.240 ","End":"05:41.970","Text":"I\u0027m going to do limit along"},{"Start":"05:42.020 ","End":"05:48.260","Text":"paths and show that 2 different paths give me 2 different limits."},{"Start":"05:48.260 ","End":"05:52.270","Text":"The paths that I\u0027m going to try are going to be,"},{"Start":"05:52.270 ","End":"05:58.655","Text":"I\u0027m going to want y to be something that will give me all the same powers of x."},{"Start":"05:58.655 ","End":"06:03.455","Text":"For example, if I let y equals x as being one path,"},{"Start":"06:03.455 ","End":"06:07.400","Text":"then I\u0027ll get everything in terms of x squared."},{"Start":"06:07.400 ","End":"06:10.160","Text":"Actually, if I let y equal some other number,"},{"Start":"06:10.160 ","End":"06:13.280","Text":"say 2x, I\u0027ll still get x squared plus 2x all squared,"},{"Start":"06:13.280 ","End":"06:15.140","Text":"I still get everything in terms of x squared."},{"Start":"06:15.140 ","End":"06:18.570","Text":"These will be the 2 paths that I choose."},{"Start":"06:18.620 ","End":"06:24.380","Text":"This one b, a and this one b and I\u0027ll show that the 2 paths give different limits."},{"Start":"06:24.380 ","End":"06:30.485","Text":"For a, we get the limit as x goes to 0."},{"Start":"06:30.485 ","End":"06:36.080","Text":"When x goes to 0, y also goes to 0 so we\u0027re fine of x times y,"},{"Start":"06:36.080 ","End":"06:39.680","Text":"which is x times x over x squared,"},{"Start":"06:39.680 ","End":"06:48.570","Text":"and y squared is also x squared as x goes to 0 and this equals the limit."},{"Start":"06:48.570 ","End":"06:54.615","Text":"That\u0027s the x squared over 2x squared is just 1/2."},{"Start":"06:54.615 ","End":"06:56.190","Text":"I mean the numerator is x squared,"},{"Start":"06:56.190 ","End":"07:00.360","Text":"denominator\u0027s 2x squared, the x squared cancels,"},{"Start":"07:00.360 ","End":"07:03.820","Text":"and so I just get a limit of 1/2,"},{"Start":"07:04.490 ","End":"07:10.295","Text":"x goes to 0 of 1/2, the limit of a constant, it\u0027s just 1/2."},{"Start":"07:10.295 ","End":"07:11.840","Text":"On the other hand,"},{"Start":"07:11.840 ","End":"07:13.700","Text":"if I do option b,"},{"Start":"07:13.700 ","End":"07:15.500","Text":"where y equals to 2x,"},{"Start":"07:15.500 ","End":"07:18.949","Text":"I get the limit as x goes to 0,"},{"Start":"07:18.949 ","End":"07:24.390","Text":"x times 2x over x squared plus"},{"Start":"07:24.390 ","End":"07:29.945","Text":"2x squared and this is 2x squared."},{"Start":"07:29.945 ","End":"07:33.590","Text":"This is x squared plus 4x squared is 5x squared,"},{"Start":"07:33.590 ","End":"07:36.115","Text":"and the x squareds cancel."},{"Start":"07:36.115 ","End":"07:43.420","Text":"I get the limit as x goes to 0 of 2/5, which is 2/5."},{"Start":"07:44.390 ","End":"07:51.460","Text":"Now, this and this are not equal and so there is no limit."},{"Start":"07:51.530 ","End":"07:57.110","Text":"Certainly, 0 is not the same as no limit but I could have dispensed with this,"},{"Start":"07:57.110 ","End":"08:00.230","Text":"if there\u0027s no limit, already it\u0027s not continuous."},{"Start":"08:00.230 ","End":"08:04.950","Text":"The answer to Part 2 is not continuous."},{"Start":"08:06.290 ","End":"08:08.490","Text":"That\u0027s the second part."},{"Start":"08:08.490 ","End":"08:11.445","Text":"Now, let\u0027s do the third part,"},{"Start":"08:11.445 ","End":"08:14.719","Text":"and let\u0027s remember what the third part was."},{"Start":"08:14.719 ","End":"08:20.300","Text":"It asks if a function is partially differentiable at a point,"},{"Start":"08:20.300 ","End":"08:23.855","Text":"must it also be continuous at that point?"},{"Start":"08:23.855 ","End":"08:27.500","Text":"The answer is clearly no."},{"Start":"08:27.500 ","End":"08:29.435","Text":"Because we have an example,"},{"Start":"08:29.435 ","End":"08:33.080","Text":"our function f, it is partially differentiable."},{"Start":"08:33.080 ","End":"08:37.565","Text":"We found the derivative with respect to x and with respect to y."},{"Start":"08:37.565 ","End":"08:39.230","Text":"No problem there."},{"Start":"08:39.230 ","End":"08:44.090","Text":"But further down, we also showed that it\u0027s not continuous at that point."},{"Start":"08:44.090 ","End":"08:45.410","Text":"The answer is no,"},{"Start":"08:45.410 ","End":"08:47.490","Text":"and we are done."}],"ID":20887},{"Watched":false,"Name":"Exercise 4","Duration":"1m 8s","ChapterTopicVideoID":8593,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In this exercise, we\u0027re asking if the following function"},{"Start":"00:03.540 ","End":"00:06.780","Text":"is differentiable at 0,0 at the origin."},{"Start":"00:06.780 ","End":"00:09.900","Text":"Now, I\u0027m presuming that you\u0027ve already done"},{"Start":"00:09.900 ","End":"00:13.500","Text":"the previous exercise in which this function also"},{"Start":"00:13.500 ","End":"00:20.175","Text":"appears and where we show that f is not continuous."},{"Start":"00:20.175 ","End":"00:22.785","Text":"This is important that we showed this."},{"Start":"00:22.785 ","End":"00:26.025","Text":"You should go back and look for the exercise if you don\u0027t remember,"},{"Start":"00:26.025 ","End":"00:28.290","Text":"we showed it by taking the limit"},{"Start":"00:28.290 ","End":"00:31.110","Text":"along 2 different paths and showing that they were different."},{"Start":"00:31.110 ","End":"00:34.335","Text":"In any event, this function is not continuous."},{"Start":"00:34.335 ","End":"00:42.775","Text":"Now, there\u0027s a theorem that if a function is differentiable at a point,"},{"Start":"00:42.775 ","End":"00:47.970","Text":"then it\u0027s also continuous at that point."},{"Start":"00:47.970 ","End":"00:51.030","Text":"Both of these are at a point."},{"Start":"00:51.030 ","End":"00:54.705","Text":"In our case, it\u0027s the point at 0,0."},{"Start":"00:54.705 ","End":"00:57.425","Text":"We could argue if it were differentiable,"},{"Start":"00:57.425 ","End":"01:01.015","Text":"then it would also be continuous at 0,0,"},{"Start":"01:01.015 ","End":"01:03.645","Text":"but it isn\u0027t and so it\u0027s not."},{"Start":"01:03.645 ","End":"01:08.320","Text":"The answer is no. We\u0027re done."}],"ID":20888},{"Watched":false,"Name":"Exercise 5 part a","Duration":"12m 50s","ChapterTopicVideoID":8610,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.750","Text":"In this exercise, we have to figure out if the following function is differentiable 0,"},{"Start":"00:06.750 ","End":"00:08.205","Text":"0, which is the origin."},{"Start":"00:08.205 ","End":"00:10.230","Text":"The function is given piece-wise,"},{"Start":"00:10.230 ","End":"00:13.170","Text":"given 1 way for every x,"},{"Start":"00:13.170 ","End":"00:16.635","Text":"y naught at the origin and 0 at the origin."},{"Start":"00:16.635 ","End":"00:21.555","Text":"We\u0027re going to work off the definition of differentiability. Let me bring it."},{"Start":"00:21.555 ","End":"00:25.335","Text":"Definition of differentiability at a general point,"},{"Start":"00:25.335 ","End":"00:28.080","Text":"x naught, y naught, in our case 0,"},{"Start":"00:28.080 ","End":"00:33.720","Text":"0 depends on the following limit of 2 variables,"},{"Start":"00:33.720 ","End":"00:38.190","Text":"Delta x and Delta y going to 0 of this thing and this,"},{"Start":"00:38.190 ","End":"00:41.760","Text":"I\u0027ll put a question mark here because that\u0027s what we\u0027re going to figure out."},{"Start":"00:41.760 ","End":"00:43.600","Text":"If it\u0027s equal to 0,"},{"Start":"00:43.600 ","End":"00:47.480","Text":"it\u0027s differentiable and otherwise not."},{"Start":"00:47.480 ","End":"00:50.000","Text":"Now in our case, as I say,"},{"Start":"00:50.000 ","End":"00:51.860","Text":"we have that x naught,"},{"Start":"00:51.860 ","End":"00:55.710","Text":"y naught is equal to 0, 0."},{"Start":"00:55.710 ","End":"01:02.155","Text":"Let\u0027s just replace everywhere and what we\u0027ll get and I\u0027ll bring in a different picture,"},{"Start":"01:02.155 ","End":"01:06.140","Text":"is the following, replacing x naught,"},{"Start":"01:06.140 ","End":"01:07.805","Text":"y naught by 0, 0."},{"Start":"01:07.805 ","End":"01:09.680","Text":"Again, we want to check,"},{"Start":"01:09.680 ","End":"01:12.740","Text":"we\u0027re going to work on the left-hand side and figure out the limit."},{"Start":"01:12.740 ","End":"01:15.025","Text":"If it exists and equals 0,"},{"Start":"01:15.025 ","End":"01:18.220","Text":"we\u0027re differentiable, otherwise, not."},{"Start":"01:18.470 ","End":"01:21.620","Text":"We need to compute several things."},{"Start":"01:21.620 ","End":"01:25.295","Text":"This 1 is easy, it\u0027s just f of 0, 0."},{"Start":"01:25.295 ","End":"01:28.700","Text":"We can immediately write that as equal to 0."},{"Start":"01:28.700 ","End":"01:30.065","Text":"I\u0027m taking it up from here."},{"Start":"01:30.065 ","End":"01:33.230","Text":"Then we have to compute the 2 partial derivatives of 0,"},{"Start":"01:33.230 ","End":"01:36.245","Text":"0 and we\u0027ll work straight off the definition."},{"Start":"01:36.245 ","End":"01:38.300","Text":"Fx at the point 0,"},{"Start":"01:38.300 ","End":"01:43.385","Text":"0 is equal to, well, in general,"},{"Start":"01:43.385 ","End":"01:47.315","Text":"we take f of this plus h,"},{"Start":"01:47.315 ","End":"01:50.460","Text":"which is just h and the other 1,"},{"Start":"01:50.460 ","End":"01:51.840","Text":"the y naught as is,"},{"Start":"01:51.840 ","End":"01:54.030","Text":"minus f at the point 0,"},{"Start":"01:54.030 ","End":"02:02.675","Text":"0 over h. I forgot to write limit as h goes to 0, of course."},{"Start":"02:02.675 ","End":"02:04.880","Text":"This is equal to,"},{"Start":"02:04.880 ","End":"02:06.835","Text":"let\u0027s see the limit,"},{"Start":"02:06.835 ","End":"02:08.760","Text":"again, h goes to 0."},{"Start":"02:08.760 ","End":"02:12.885","Text":"F of h 0, h is not 0, so we work off here."},{"Start":"02:12.885 ","End":"02:17.750","Text":"It\u0027s going to be h cubed plus 0 cubed over 2h squared,"},{"Start":"02:17.750 ","End":"02:23.925","Text":"and so on, we\u0027ll basically get h cubed over 2h squared,"},{"Start":"02:23.925 ","End":"02:27.045","Text":"because the y is 0 over,"},{"Start":"02:27.045 ","End":"02:34.545","Text":"the h from here up and minus f of naught, naught, 0, 0 is 0."},{"Start":"02:34.545 ","End":"02:37.850","Text":"Now, this expression can easily be simplified."},{"Start":"02:37.850 ","End":"02:39.320","Text":"It\u0027s just h cubed."},{"Start":"02:39.320 ","End":"02:41.075","Text":"This goes to the denominator."},{"Start":"02:41.075 ","End":"02:45.920","Text":"We get h cubed over 2h squared h is h cubed,"},{"Start":"02:45.920 ","End":"02:56.270","Text":"and that\u0027s just equal to 1/2 and the limit of 1/2 at the limit of a constant is just 1/2."},{"Start":"02:56.270 ","End":"02:58.850","Text":"Next, we want to compute this 1."},{"Start":"02:58.850 ","End":"03:01.310","Text":"We\u0027ve got already 3 of the quantities,"},{"Start":"03:01.310 ","End":"03:05.835","Text":"we\u0027ve got this quantity is here,"},{"Start":"03:05.835 ","End":"03:10.740","Text":"this quantity is here and now, we need this 1,"},{"Start":"03:10.740 ","End":"03:16.440","Text":"f with respect to y. F with respect to y at 0,"},{"Start":"03:16.440 ","End":"03:18.815","Text":"0 is a similar limit,"},{"Start":"03:18.815 ","End":"03:21.575","Text":"limit h goes to 0."},{"Start":"03:21.575 ","End":"03:27.809","Text":"This time, the first coordinate stays as is and then we add 0 plus h,"},{"Start":"03:27.809 ","End":"03:29.520","Text":"also minus f of 0,"},{"Start":"03:29.520 ","End":"03:33.735","Text":"0, which of course is 0, it was 0 here."},{"Start":"03:33.735 ","End":"03:40.380","Text":"This time if we substitute,"},{"Start":"03:40.380 ","End":"03:42.270","Text":"we\u0027re going to get x is 0,"},{"Start":"03:42.270 ","End":"03:46.275","Text":"y is h, we\u0027ll just get h cubed over h squared."},{"Start":"03:46.275 ","End":"03:49.110","Text":"I just remember that when I scroll."},{"Start":"03:49.110 ","End":"03:54.840","Text":"We get, yeah, limit h goes to 0,"},{"Start":"03:54.840 ","End":"03:58.190","Text":"h cubed over h squared again minus 0,"},{"Start":"03:58.190 ","End":"04:01.850","Text":"which I need not to bother writing anyway."},{"Start":"04:01.850 ","End":"04:06.455","Text":"This time, it\u0027s simplifies to h cubed over h cubed,"},{"Start":"04:06.455 ","End":"04:12.615","Text":"which is 1 and the limit of 1 is just 1."},{"Start":"04:12.615 ","End":"04:16.145","Text":"We have another quantity that\u0027s that 1 here."},{"Start":"04:16.145 ","End":"04:19.560","Text":"Now, we have enough to plug-in to the limit."},{"Start":"04:20.180 ","End":"04:25.710","Text":"Now, what we need to compute is the limit."},{"Start":"04:25.710 ","End":"04:35.260","Text":"Delta x goes to 0 and Delta y goes to 0,"},{"Start":"04:36.740 ","End":"04:40.140","Text":"f of Delta x, Delta y,"},{"Start":"04:40.140 ","End":"04:43.940","Text":"we\u0027re going to be working off the definition."},{"Start":"04:43.940 ","End":"04:46.130","Text":"I\u0027ll have to go look again."},{"Start":"04:46.130 ","End":"04:48.860","Text":"We\u0027re going to need just like this,"},{"Start":"04:48.860 ","End":"04:51.845","Text":"but with Delta x cubed plus Delta y cubed,"},{"Start":"04:51.845 ","End":"04:54.460","Text":"I\u0027ll just write it."},{"Start":"04:54.460 ","End":"05:00.185","Text":"We\u0027re going to get Delta x cubed"},{"Start":"05:00.185 ","End":"05:06.680","Text":"plus Delta y cubed over 2,"},{"Start":"05:06.680 ","End":"05:13.025","Text":"Delta x squared plus Delta y squared."},{"Start":"05:13.025 ","End":"05:15.010","Text":"That\u0027s just this bit."},{"Start":"05:15.010 ","End":"05:20.700","Text":"Minus f of 0, 0, which is this,"},{"Start":"05:20.700 ","End":"05:29.000","Text":"is 0, minus partial derivative with respect to x is 1/2."},{"Start":"05:29.000 ","End":"05:35.540","Text":"Then Delta x and then minus f with respect to y is 1."},{"Start":"05:35.540 ","End":"05:38.930","Text":"I\u0027ll write the 1 in just to remind us,"},{"Start":"05:38.930 ","End":"05:47.370","Text":"times Delta y and all this over the square root of Delta x squared plus Delta y squared"},{"Start":"05:47.680 ","End":"05:59.430","Text":"over the square root of Delta x squared plus Delta y squared."},{"Start":"05:59.430 ","End":"06:04.000","Text":"Now, this thing looks messy and we can actually"},{"Start":"06:04.000 ","End":"06:08.500","Text":"simplify it because everything involves Delta x and Delta y."},{"Start":"06:08.500 ","End":"06:10.015","Text":"There\u0027s no other variables."},{"Start":"06:10.015 ","End":"06:16.890","Text":"Just to simplify it, let\u0027s replace Delta x by x and replace Delta y by y."},{"Start":"06:16.890 ","End":"06:18.520","Text":"You could do any 2 variables,"},{"Start":"06:18.520 ","End":"06:23.880","Text":"you could call it h and k if you wanted or a and b but we\u0027ll call them x and y."},{"Start":"06:23.880 ","End":"06:29.425","Text":"Then we\u0027ll just get a simpler expression with the same limit."},{"Start":"06:29.425 ","End":"06:35.600","Text":"The limit as x goes to naught,"},{"Start":"06:35.600 ","End":"06:40.545","Text":"y goes to naught of x cubed"},{"Start":"06:40.545 ","End":"06:46.010","Text":"plus y cubed over 2x squared plus y squared."},{"Start":"06:46.010 ","End":"06:49.130","Text":"Just replacing this everywhere,"},{"Start":"06:49.130 ","End":"06:51.230","Text":"minus 0, I don\u0027t need it,"},{"Start":"06:51.230 ","End":"06:57.110","Text":"minus x over 2 minus y,"},{"Start":"06:57.110 ","End":"07:05.165","Text":"all this over the square root of x squared plus y squared."},{"Start":"07:05.165 ","End":"07:08.855","Text":"Now, we can do a little bit of simplification."},{"Start":"07:08.855 ","End":"07:10.490","Text":"This was equal to this,"},{"Start":"07:10.490 ","End":"07:11.795","Text":"was equal to this,"},{"Start":"07:11.795 ","End":"07:14.335","Text":"was equal to the limit."},{"Start":"07:14.335 ","End":"07:16.230","Text":"Again, x goes to naught,"},{"Start":"07:16.230 ","End":"07:25.425","Text":"y goes to naught multiplying top and bottom by 2x squared plus y squared, top and bottom."},{"Start":"07:25.425 ","End":"07:31.820","Text":"Same thing. Here, we just get x cubed plus y cubed."},{"Start":"07:31.820 ","End":"07:33.890","Text":"The denominator disappeared."},{"Start":"07:33.890 ","End":"07:39.070","Text":"Here, we get minus x/2."},{"Start":"07:39.070 ","End":"07:40.610","Text":"You know what? Let me change my mind."},{"Start":"07:40.610 ","End":"07:45.800","Text":"Let me multiply by twice to x squared plus y squared."},{"Start":"07:45.800 ","End":"07:49.560","Text":"Then I can get rid of this 2 as well."},{"Start":"07:50.090 ","End":"07:53.655","Text":"Now, I need an extra 2 here."},{"Start":"07:53.655 ","End":"07:56.699","Text":"Over here, 2 disappears,"},{"Start":"07:56.699 ","End":"08:05.515","Text":"so I\u0027ve got just 2x squared plus y squared and here,"},{"Start":"08:05.515 ","End":"08:07.295","Text":"I get the whole thing,"},{"Start":"08:07.295 ","End":"08:13.362","Text":"2 times y times 2x squared plus y squared,"},{"Start":"08:13.362 ","End":"08:16.175","Text":"but we still have the denominator,"},{"Start":"08:16.175 ","End":"08:19.835","Text":"which I just copied from here."},{"Start":"08:19.835 ","End":"08:22.580","Text":"Here too, and here,"},{"Start":"08:22.580 ","End":"08:26.555","Text":"2x squared plus y squared."},{"Start":"08:26.555 ","End":"08:29.305","Text":"I want to do a bit of simplification."},{"Start":"08:29.305 ","End":"08:33.720","Text":"Note that multiply out here,"},{"Start":"08:33.720 ","End":"08:35.370","Text":"I get 2x cubed,"},{"Start":"08:35.370 ","End":"08:37.440","Text":"but this also gives to x cubed,"},{"Start":"08:37.440 ","End":"08:39.600","Text":"so it\u0027s like this cancels with this."},{"Start":"08:39.600 ","End":"08:42.710","Text":"Also this with this gives 2y cubed."},{"Start":"08:42.710 ","End":"08:46.195","Text":"On the other hand, this with this also gives minus 2y cubed,"},{"Start":"08:46.195 ","End":"08:48.670","Text":"so it\u0027s like this cancels with this."},{"Start":"08:48.670 ","End":"08:55.305","Text":"That means that all we\u0027re left with is minus xy"},{"Start":"08:55.305 ","End":"09:00.090","Text":"squared from here and"},{"Start":"09:00.090 ","End":"09:07.200","Text":"minus 2 times 2 is 4x squared y from here."},{"Start":"09:07.200 ","End":"09:09.435","Text":"Our big question is,"},{"Start":"09:09.435 ","End":"09:15.335","Text":"does this limit exist and specifically does it equal 0?"},{"Start":"09:15.335 ","End":"09:18.410","Text":"I\u0027m claiming that the answer is no,"},{"Start":"09:18.410 ","End":"09:20.950","Text":"and I\u0027m going to show you why not."},{"Start":"09:20.950 ","End":"09:25.835","Text":"I\u0027m going to use the concept of a limit along the path."},{"Start":"09:25.835 ","End":"09:31.670","Text":"Let\u0027s just suppose for a moment that we did have this limit existing and equaling 0."},{"Start":"09:31.670 ","End":"09:33.455","Text":"If there\u0027s a limit,"},{"Start":"09:33.455 ","End":"09:35.750","Text":"we know from a theorem that this limit is also"},{"Start":"09:35.750 ","End":"09:38.785","Text":"the limit along any path that lead to 0, 0."},{"Start":"09:38.785 ","End":"09:41.480","Text":"Let\u0027s choose the path."},{"Start":"09:41.780 ","End":"09:48.865","Text":"The 45-degree line, y equals x. I mean,"},{"Start":"09:48.865 ","End":"09:53.030","Text":"looks like this is the y-axis and this is the x axis,"},{"Start":"09:53.030 ","End":"09:55.505","Text":"we don\u0027t need a picture, but y equals x."},{"Start":"09:55.505 ","End":"10:01.205","Text":"In fact, let\u0027s even go along the positive in the first quadrant, y equals x,"},{"Start":"10:01.205 ","End":"10:12.960","Text":"and move along to 0, 0 this way, so we would have the limit."},{"Start":"10:12.960 ","End":"10:16.910","Text":"All we need now is x goes to 0 because y"},{"Start":"10:16.910 ","End":"10:21.005","Text":"equals x, then y also goes to 0, but let\u0027s make it 0 plus,"},{"Start":"10:21.005 ","End":"10:23.120","Text":"so we\u0027re going along here."},{"Start":"10:23.120 ","End":"10:26.005","Text":"You could do it from the other direction too."},{"Start":"10:26.005 ","End":"10:29.445","Text":"Now, we substitute y equals x here,"},{"Start":"10:29.445 ","End":"10:36.120","Text":"so we get minus, xy squared is x cubed,"},{"Start":"10:36.120 ","End":"10:41.050","Text":"y equals x here, minus 4x cubed."},{"Start":"10:43.550 ","End":"10:47.275","Text":"Now the square root of x squared plus y squared,"},{"Start":"10:47.275 ","End":"10:52.280","Text":"I\u0027ll just write it as square root of 2x squared."},{"Start":"10:52.280 ","End":"11:00.265","Text":"Here, we have 2x squared plus x squared."},{"Start":"11:00.265 ","End":"11:03.550","Text":"Let\u0027s do another simplification."},{"Start":"11:04.790 ","End":"11:10.660","Text":"Again, we need to scroll a bit."},{"Start":"11:12.020 ","End":"11:19.190","Text":"We get the limit as x goes to 0 from the right."},{"Start":"11:19.190 ","End":"11:23.820","Text":"Here, we have minus 5x cubed."},{"Start":"11:24.140 ","End":"11:27.080","Text":"What do we have in the denominator?"},{"Start":"11:27.080 ","End":"11:30.980","Text":"We have 2, then we have a square root of 2,"},{"Start":"11:30.980 ","End":"11:33.594","Text":"then the square root of x squared."},{"Start":"11:33.594 ","End":"11:37.340","Text":"Normally, it\u0027s absolute value of x but in our case,"},{"Start":"11:37.340 ","End":"11:41.285","Text":"if x going to 0 from the positive, it\u0027s just x."},{"Start":"11:41.285 ","End":"11:47.790","Text":"Here, we have 3x squared."},{"Start":"11:49.880 ","End":"11:56.090","Text":"We can actually cancel x and x squared canceled with x cubed."},{"Start":"11:56.090 ","End":"12:01.970","Text":"This expression here is just a number minus 5 over 2 root 2 times"},{"Start":"12:01.970 ","End":"12:08.060","Text":"3 and so this limit is equal to minus 5 over,"},{"Start":"12:08.060 ","End":"12:10.430","Text":"doesn\u0027t really matter what the number is, it\u0027s not 0,"},{"Start":"12:10.430 ","End":"12:13.715","Text":"2 times 3 is 6 root 2."},{"Start":"12:13.715 ","End":"12:17.195","Text":"In any event, it\u0027s not equal to 0."},{"Start":"12:17.195 ","End":"12:21.890","Text":"This is a contradiction because if we did have a limit equaling 0,"},{"Start":"12:21.890 ","End":"12:26.210","Text":"and it would be 0 along any path but along this path it\u0027s not 0,"},{"Start":"12:26.210 ","End":"12:31.235","Text":"so either the limit\u0027s not 0 and it\u0027s equal to minus 5 over 6 root 2,"},{"Start":"12:31.235 ","End":"12:35.300","Text":"or it doesn\u0027t exist, but either way, we\u0027re not differentiable."},{"Start":"12:35.300 ","End":"12:42.550","Text":"As a conclusion, we just write the original function is not differentiable."},{"Start":"12:42.550 ","End":"12:47.870","Text":"If you want to really spell it out at 0,"},{"Start":"12:47.870 ","End":"12:51.690","Text":"0, we\u0027re done, the answer is no."}],"ID":20889},{"Watched":false,"Name":"Exercise 5 part b","Duration":"6m 29s","ChapterTopicVideoID":8611,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In this exercise, we\u0027re given this function f,"},{"Start":"00:03.540 ","End":"00:09.329","Text":"it\u0027s defined piece-wise, and we have to decide if it\u0027s differentiable at the origin."},{"Start":"00:09.329 ","End":"00:17.340","Text":"Notice that it\u0027s defined 1 way at the origin where it\u0027s 0 and this expression elsewhere."},{"Start":"00:17.340 ","End":"00:20.550","Text":"Let me remind you what it means for a function to be"},{"Start":"00:20.550 ","End":"00:25.215","Text":"differentiable in general at a point x_0, y_0."},{"Start":"00:25.215 ","End":"00:29.460","Text":"I just copied the formula, the definition."},{"Start":"00:29.460 ","End":"00:35.945","Text":"Basically, what it says is that if this limit exists and is equal to 0,"},{"Start":"00:35.945 ","End":"00:40.025","Text":"which we check, then the answer would be yes,"},{"Start":"00:40.025 ","End":"00:43.760","Text":"it\u0027s differentiable at x_0, y_0."},{"Start":"00:43.760 ","End":"00:45.770","Text":"Otherwise, not. Now in our case,"},{"Start":"00:45.770 ","End":"00:50.850","Text":"we replace x_0, y_0 by 0,"},{"Start":"00:50.850 ","End":"00:52.940","Text":"0, so it\u0027s slightly simpler."},{"Start":"00:52.940 ","End":"00:59.205","Text":"Let me just replace it, and here\u0027s what it is, with a bit of color."},{"Start":"00:59.205 ","End":"01:03.830","Text":"We have to decide again, whether this limit exists and equal 0,"},{"Start":"01:03.830 ","End":"01:05.150","Text":"in which case, the answer is yes,"},{"Start":"01:05.150 ","End":"01:07.820","Text":"it\u0027s differentiable, otherwise not."},{"Start":"01:07.820 ","End":"01:10.850","Text":"We need to, first of all,"},{"Start":"01:10.850 ","End":"01:16.070","Text":"compute the partial derivatives and we need to know the value of the function."},{"Start":"01:16.070 ","End":"01:17.315","Text":"Well, this one\u0027s the easiest,"},{"Start":"01:17.315 ","End":"01:19.160","Text":"let\u0027s get that out the way."},{"Start":"01:19.160 ","End":"01:22.050","Text":"F of 0, 0, it just is written here,"},{"Start":"01:22.050 ","End":"01:23.745","Text":"at 0, 0, it\u0027s 0."},{"Start":"01:23.745 ","End":"01:26.270","Text":"Let\u0027s do these 2 partial derivatives and afterwards,"},{"Start":"01:26.270 ","End":"01:27.950","Text":"we\u0027ll plug in and get the limit."},{"Start":"01:27.950 ","End":"01:32.150","Text":"The derivative of f with respect to x at 0,"},{"Start":"01:32.150 ","End":"01:37.385","Text":"0, working off the formula is the formula with the h,"},{"Start":"01:37.385 ","End":"01:43.575","Text":"where we have the function at x_0 plus h,"},{"Start":"01:43.575 ","End":"01:46.605","Text":"in this case, just h, 0 plus h,"},{"Start":"01:46.605 ","End":"01:50.340","Text":"and then y_0 minus f of x _0,"},{"Start":"01:50.340 ","End":"01:53.355","Text":"y_0, in our case, 0, 0,"},{"Start":"01:53.355 ","End":"02:00.575","Text":"over h. Now, if I plug in x equals h and y equals 0,"},{"Start":"02:00.575 ","End":"02:01.970","Text":"and working off this line,"},{"Start":"02:01.970 ","End":"02:04.760","Text":"because h is non-zero, it only tends to 0."},{"Start":"02:04.760 ","End":"02:10.040","Text":"We get the limit as h goes to"},{"Start":"02:10.040 ","End":"02:16.520","Text":"0 of h squared plus 0 squared is just h squared."},{"Start":"02:16.520 ","End":"02:24.590","Text":"Here, we have sine"},{"Start":"02:24.590 ","End":"02:30.480","Text":"of 1 over the square root of h squared,"},{"Start":"02:30.480 ","End":"02:39.785","Text":"and all this over h. Well,"},{"Start":"02:39.785 ","End":"02:46.065","Text":"first of all, we can simplify by canceling h with 1 of the h\u0027s here,"},{"Start":"02:46.065 ","End":"02:52.360","Text":"so I\u0027ll just put a line between the two and now, we have the limit of a product."},{"Start":"02:52.360 ","End":"02:55.505","Text":"Now, h goes to 0,"},{"Start":"02:55.505 ","End":"02:59.035","Text":"and this bit, the sine of something,"},{"Start":"02:59.035 ","End":"03:06.600","Text":"all this bit, is bounded, because the sine is between minus 1 and 1,"},{"Start":"03:06.600 ","End":"03:09.720","Text":"minus 1 is less than sine of anything."},{"Start":"03:09.720 ","End":"03:16.345","Text":"Once that\u0027s bounded, and we know that something goes to 0 times something bounded,"},{"Start":"03:16.345 ","End":"03:20.255","Text":"the limit is just 0."},{"Start":"03:20.255 ","End":"03:22.770","Text":"We\u0027ve used this theorem before."},{"Start":"03:22.810 ","End":"03:26.330","Text":"That\u0027s one of the quantities,"},{"Start":"03:26.330 ","End":"03:28.685","Text":"now, we need this one,"},{"Start":"03:28.685 ","End":"03:34.800","Text":"very similar, f with respect to y at 0,"},{"Start":"03:34.800 ","End":"03:36.515","Text":"0. You know what?"},{"Start":"03:36.515 ","End":"03:38.460","Text":"It\u0027s exactly the same thing, x and y,"},{"Start":"03:38.460 ","End":"03:42.140","Text":"it\u0027s totally symmetrical, so equals also 0."},{"Start":"03:42.140 ","End":"03:43.790","Text":"I\u0027ll just write, similarly,"},{"Start":"03:43.790 ","End":"03:49.065","Text":"there\u0027s absolutely no difference, so let\u0027s just take a shortcut."},{"Start":"03:49.065 ","End":"03:51.470","Text":"Now, we have 3 quantities."},{"Start":"03:51.470 ","End":"03:55.550","Text":"We have that this bit is 0,"},{"Start":"03:55.550 ","End":"03:57.755","Text":"this bit is 0,"},{"Start":"03:57.755 ","End":"04:00.810","Text":"and this bit is 0."},{"Start":"04:00.890 ","End":"04:06.420","Text":"If we plug in, what we will get here is,"},{"Start":"04:06.420 ","End":"04:12.285","Text":"we need to check the limit as Delta x goes to 0,"},{"Start":"04:12.285 ","End":"04:16.149","Text":"as well as Delta y going to 0,"},{"Start":"04:17.060 ","End":"04:20.030","Text":"just put a dividing line here,"},{"Start":"04:20.030 ","End":"04:22.940","Text":"f of Delta x, Delta y, in a moment."},{"Start":"04:22.940 ","End":"04:25.055","Text":"I\u0027ll Just scroll backup to see."},{"Start":"04:25.055 ","End":"04:31.590","Text":"Over the square root of Delta x squared plus Delta y squared."},{"Start":"04:31.590 ","End":"04:33.374","Text":"Now, I need this,"},{"Start":"04:33.374 ","End":"04:37.165","Text":"Delta x and Delta y are not both 0."},{"Start":"04:37.165 ","End":"04:41.430","Text":"I\u0027ll just copy this expression here,"},{"Start":"04:41.430 ","End":"04:44.295","Text":"which is, let\u0027s see,"},{"Start":"04:44.295 ","End":"04:45.915","Text":"but with Delta x, Delta y."},{"Start":"04:45.915 ","End":"04:51.540","Text":"We have Delta x squared plus Delta y squared"},{"Start":"04:51.540 ","End":"05:00.370","Text":"sine of 1 over Delta x squared plus Delta y squared."},{"Start":"05:02.330 ","End":"05:06.139","Text":"Yeah, this is the limit we need to compute."},{"Start":"05:06.139 ","End":"05:08.090","Text":"Now, I\u0027m going to simplify this."},{"Start":"05:08.090 ","End":"05:10.850","Text":"Notice that this and this,"},{"Start":"05:10.850 ","End":"05:17.640","Text":"it\u0027s like we have some kind of a over the square root of a, and from algebra,"},{"Start":"05:17.640 ","End":"05:20.240","Text":"that\u0027s just equal to square root of a."},{"Start":"05:20.240 ","End":"05:24.695","Text":"We can rewrite this as follows,"},{"Start":"05:24.695 ","End":"05:27.675","Text":"the limit, same thing,"},{"Start":"05:27.675 ","End":"05:37.665","Text":"of just the square root of Delta x squared plus Delta y squared times the sine of,"},{"Start":"05:37.665 ","End":"05:41.810","Text":"I\u0027ll just copy paste it from here and the limit."},{"Start":"05:41.810 ","End":"05:47.680","Text":"Now, once again, we have a situation of some things that goes to 0, because of course,"},{"Start":"05:47.680 ","End":"05:50.770","Text":"if Delta x and Delta y both go to 0, so does this."},{"Start":"05:50.770 ","End":"05:57.154","Text":"This, once again is bounded, because it\u0027s the sine of something."},{"Start":"05:57.154 ","End":"06:01.030","Text":"Something tends to 0 times something bounded,"},{"Start":"06:01.030 ","End":"06:04.455","Text":"so this limit is equal to 0."},{"Start":"06:04.455 ","End":"06:07.820","Text":"Now remember, originally, we said that if"},{"Start":"06:07.820 ","End":"06:11.915","Text":"this limit exists and equals 0, then we\u0027re differentiable."},{"Start":"06:11.915 ","End":"06:18.400","Text":"We can finally write that the answer is yes,"},{"Start":"06:18.400 ","End":"06:27.585","Text":"I\u0027m going to spell it out, f is differentiable at 0, 0."},{"Start":"06:27.585 ","End":"06:29.980","Text":"Yes, and we\u0027re done."}],"ID":20890},{"Watched":false,"Name":"Exercise 6","Duration":"12m 55s","ChapterTopicVideoID":8612,"CourseChapterTopicPlaylistID":4972,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.839","Text":"In this exercise we\u0027re given a function,"},{"Start":"00:03.839 ","End":"00:06.540","Text":"defined piece-wise and we have to check it\u0027s"},{"Start":"00:06.540 ","End":"00:14.500","Text":"differentiability but everywhere in its domain of definition."},{"Start":"00:15.140 ","End":"00:22.950","Text":"Now the domain of this function is actually all x, y."},{"Start":"00:22.950 ","End":"00:29.850","Text":"In other words, the domain is"},{"Start":"00:29.850 ","End":"00:37.050","Text":"the whole 2D-plane or"},{"Start":"00:37.050 ","End":"00:40.260","Text":"you could say in other words,"},{"Start":"00:40.260 ","End":"00:45.325","Text":"for all x, y."},{"Start":"00:45.325 ","End":"00:49.700","Text":"But I want to separate it into 2 cases."},{"Start":"00:49.700 ","End":"00:52.910","Text":"I want to check differentiability when we\u0027re"},{"Start":"00:52.910 ","End":"00:58.195","Text":"not 0,0 and differentiability separately at 0,0."},{"Start":"00:58.195 ","End":"01:02.875","Text":"If x, y is not equal to 0,0,"},{"Start":"01:02.875 ","End":"01:07.175","Text":"I\u0027m going to use the theorem that if we have"},{"Start":"01:07.175 ","End":"01:16.115","Text":"continuous partial derivatives then that implies differentiability."},{"Start":"01:16.115 ","End":"01:24.290","Text":"I\u0027m still working on the case where x,"},{"Start":"01:24.290 ","End":"01:26.150","Text":"y is not 0,0."},{"Start":"01:26.150 ","End":"01:30.070","Text":"Let\u0027s check if the partial derivatives are continuous."},{"Start":"01:30.070 ","End":"01:37.775","Text":"Now the partial derivative with respect to x is equal to."},{"Start":"01:37.775 ","End":"01:42.375","Text":"We can use the chain rule;"},{"Start":"01:42.375 ","End":"01:44.675","Text":"we have e to the power of something."},{"Start":"01:44.675 ","End":"01:51.095","Text":"It\u0027s e to the power of minus 1 over x squared plus y squared"},{"Start":"01:51.095 ","End":"01:58.040","Text":"times the inner derivative which is the derivative of this."},{"Start":"01:58.040 ","End":"02:03.105","Text":"But remember x is the variable and y is the constant,"},{"Start":"02:03.105 ","End":"02:06.365","Text":"so we have to take the derivative of this."},{"Start":"02:06.365 ","End":"02:14.065","Text":"Now the derivative of 1 over something is minus 1 over that something squared."},{"Start":"02:14.065 ","End":"02:18.810","Text":"I\u0027ll just write that and then I\u0027ll explain that"},{"Start":"02:18.810 ","End":"02:23.510","Text":"at the end in a moment how we get from here to here."},{"Start":"02:23.510 ","End":"02:30.810","Text":"Then we still have an inner derivative of the x squared plus y squared which is 2x."},{"Start":"02:30.810 ","End":"02:34.270","Text":"Just again, from here to here,"},{"Start":"02:37.220 ","End":"02:40.425","Text":"let\u0027s say if it was 1 over x,"},{"Start":"02:40.425 ","End":"02:42.360","Text":"well, that\u0027s 1 over something,"},{"Start":"02:42.360 ","End":"02:48.310","Text":"the derivative is minus 1 over that something squared times the derivative of that."},{"Start":"02:48.310 ","End":"02:51.500","Text":"But if I have a minus in here,"},{"Start":"02:51.500 ","End":"02:56.055","Text":"so this was a minus then this would become a plus,"},{"Start":"02:56.055 ","End":"02:57.960","Text":"which is what we have here."},{"Start":"02:57.960 ","End":"03:01.660","Text":"Then we had the inner derivative."},{"Start":"03:03.410 ","End":"03:08.650","Text":"In a completely similar way because x and y are really very symmetrical,"},{"Start":"03:08.650 ","End":"03:11.420","Text":"it\u0027s just going to be the same thing."},{"Start":"03:11.790 ","End":"03:15.880","Text":"The only difference is at the end, let\u0027s see,"},{"Start":"03:15.880 ","End":"03:17.935","Text":"x squared plus y squared,"},{"Start":"03:17.935 ","End":"03:20.875","Text":"well, we\u0027ll get to y."},{"Start":"03:20.875 ","End":"03:27.830","Text":"Now, if x and y are not equal to 0,0 then again,"},{"Start":"03:27.830 ","End":"03:32.060","Text":"these are elementary functions, exponents, and polynomials."},{"Start":"03:32.060 ","End":"03:40.730","Text":"These are continuous, both f with respect to x and f with respect to y are continuous."},{"Start":"03:40.730 ","End":"03:43.730","Text":"Appealing to the theorem that if we have"},{"Start":"03:43.730 ","End":"03:48.675","Text":"continuous partial derivatives then the function is differentiable."},{"Start":"03:48.675 ","End":"03:54.830","Text":"We\u0027ve got out of the way the case where x and y are not 0,0."},{"Start":"03:54.830 ","End":"04:00.350","Text":"Now we have to take the case where x and y are 0,0."},{"Start":"04:00.350 ","End":"04:04.100","Text":"What happens if x,"},{"Start":"04:04.100 ","End":"04:07.475","Text":"y equals 0,0?"},{"Start":"04:07.475 ","End":"04:10.225","Text":"The second case."},{"Start":"04:10.225 ","End":"04:17.780","Text":"For sure I\u0027m going to need the partial derivative at 0,0."},{"Start":"04:17.780 ","End":"04:19.505","Text":"We\u0027re going to do this from the definition."},{"Start":"04:19.505 ","End":"04:24.815","Text":"This is going to be equal the limit as h goes to"},{"Start":"04:24.815 ","End":"04:33.020","Text":"0 of f of this plus h and the other one as is,"},{"Start":"04:33.020 ","End":"04:42.300","Text":"minus f at the point over h. Now"},{"Start":"04:42.300 ","End":"04:47.190","Text":"this was equal to e to the power"},{"Start":"04:47.190 ","End":"04:53.010","Text":"of minus 1 over x squared plus y squared,"},{"Start":"04:53.010 ","End":"04:56.460","Text":"is just h squared."},{"Start":"04:56.460 ","End":"05:05.400","Text":"F of 0,0, we defined it as 0 over h. The limit as h goes to 0,"},{"Start":"05:05.400 ","End":"05:07.585","Text":"that, of course, I almost forgot."},{"Start":"05:07.585 ","End":"05:11.630","Text":"Now the question is how to compute this limit?"},{"Start":"05:11.630 ","End":"05:15.830","Text":"I recommend doing this by using a substitution."},{"Start":"05:15.830 ","End":"05:24.115","Text":"Let\u0027s let t equals 1 over h. Perhaps we"},{"Start":"05:24.115 ","End":"05:28.530","Text":"better separate the limit 0 from"},{"Start":"05:28.530 ","End":"05:33.830","Text":"the right or 0 from the left because 1 over 0 from the right is infinity,"},{"Start":"05:33.830 ","End":"05:35.600","Text":"in other words, it\u0027s minus infinity."},{"Start":"05:35.600 ","End":"05:43.610","Text":"Let\u0027s say we\u0027re taking first of all the limit as h goes to 0 from the right,"},{"Start":"05:43.610 ","End":"05:50.670","Text":"and that will correspond to the limit 1 over h is t as t goes to plus infinity."},{"Start":"05:50.670 ","End":"05:58.110","Text":"Similarly later we\u0027ll do the 0 from the left and they\u0027ll both come out the same,"},{"Start":"05:58.110 ","End":"05:59.445","Text":"so it will mean there\u0027s a limit."},{"Start":"05:59.445 ","End":"06:02.920","Text":"If I do the substitution I\u0027ll get that this is equal to"},{"Start":"06:02.920 ","End":"06:12.775","Text":"the limit as t goes to infinity and later I have to also take care of minus infinity."},{"Start":"06:12.775 ","End":"06:15.890","Text":"You know what? We can do them both at the same time."},{"Start":"06:15.890 ","End":"06:20.640","Text":"Let me replace this with plus or minus infinity."},{"Start":"06:20.640 ","End":"06:22.950","Text":"I\u0027m meaning we do separately in parallel,"},{"Start":"06:22.950 ","End":"06:27.060","Text":"one\u0027s infinity and one\u0027s minus infinity of;"},{"Start":"06:27.060 ","End":"06:34.499","Text":"this will be e to the minus t squared,"},{"Start":"06:34.499 ","End":"06:39.160","Text":"because 1 over h is t. I don\u0027t need"},{"Start":"06:39.160 ","End":"06:45.230","Text":"a dividing line because 1 over h is t and I can stick the t in front."},{"Start":"06:45.230 ","End":"06:51.290","Text":"We\u0027ve got this limit separately at infinity and minus infinity."},{"Start":"06:52.010 ","End":"06:56.890","Text":"When we have an infinity times 0 situations,"},{"Start":"06:56.890 ","End":"07:02.095","Text":"which is what we have here because t squared goes to infinity either way,"},{"Start":"07:02.095 ","End":"07:04.525","Text":"minus t squared goes to minus infinity."},{"Start":"07:04.525 ","End":"07:08.875","Text":"This is a 0 times infinity situation,"},{"Start":"07:08.875 ","End":"07:17.390","Text":"which means that we can use L\u0027Hopital,"},{"Start":"07:17.390 ","End":"07:21.020","Text":"but to be precise, it\u0027s not this."},{"Start":"07:21.020 ","End":"07:24.830","Text":"It\u0027s plus or minus infinity times 0."},{"Start":"07:24.830 ","End":"07:27.200","Text":"But anyway, we\u0027re going to use L\u0027Hopital,"},{"Start":"07:27.200 ","End":"07:29.965","Text":"but before that we have to rewrite it;"},{"Start":"07:29.965 ","End":"07:33.375","Text":"limit t goes to plus or minus infinity."},{"Start":"07:33.375 ","End":"07:37.145","Text":"In a more convenient form that at 0 over 0 or infinity over infinity."},{"Start":"07:37.145 ","End":"07:42.350","Text":"What I\u0027ll do is I\u0027ll write this as t over e to the power of t squared."},{"Start":"07:42.350 ","End":"07:47.480","Text":"Then we have a plus or minus infinity over infinity situation."},{"Start":"07:47.480 ","End":"07:55.655","Text":"Then we can use L\u0027Hopital\u0027s rule and differentiate numerator and denominator."},{"Start":"07:55.655 ","End":"08:00.215","Text":"We get the limit t goes to plus or minus infinity."},{"Start":"08:00.215 ","End":"08:02.285","Text":"Derivative of this is 1."},{"Start":"08:02.285 ","End":"08:06.005","Text":"The derivative of this is first of all,"},{"Start":"08:06.005 ","End":"08:10.200","Text":"e to the t squared times internal derivative which is 2t."},{"Start":"08:10.210 ","End":"08:12.980","Text":"Now what I\u0027m saying is that either way,"},{"Start":"08:12.980 ","End":"08:16.400","Text":"whether we take it to infinity or to minus infinity we\u0027re going"},{"Start":"08:16.400 ","End":"08:20.855","Text":"to get 0 because this goes to infinity,"},{"Start":"08:20.855 ","End":"08:24.725","Text":"this goes to plus or minus infinity."},{"Start":"08:24.725 ","End":"08:31.110","Text":"But either way we get 1 over plus or minus infinity this is equal to 0."},{"Start":"08:31.520 ","End":"08:34.445","Text":"I\u0027ll just copy it up here."},{"Start":"08:34.445 ","End":"08:39.500","Text":"Now that we know that this now we can write this equals 0."},{"Start":"08:39.500 ","End":"08:45.800","Text":"Next we want the partial derivative according"},{"Start":"08:45.800 ","End":"08:51.470","Text":"to y at 0-0 but it\u0027s completely analogous."},{"Start":"08:51.470 ","End":"08:53.990","Text":"I mean x and y it\u0027s very symmetrical."},{"Start":"08:53.990 ","End":"08:57.940","Text":"It\u0027s just repeating the same thing almost word for word."},{"Start":"08:57.940 ","End":"09:01.860","Text":"This is also equal to 0."},{"Start":"09:01.860 ","End":"09:07.070","Text":"Now let\u0027s get to the matter of differentiability at 0,0."},{"Start":"09:07.070 ","End":"09:13.130","Text":"I\u0027m going to bring in something that comes straight out of the definition."},{"Start":"09:13.130 ","End":"09:16.310","Text":"This here is what I mean."},{"Start":"09:16.310 ","End":"09:18.230","Text":"I\u0027ve used this before."},{"Start":"09:18.230 ","End":"09:23.405","Text":"There is a more general version where we have x naught, y naught,"},{"Start":"09:23.405 ","End":"09:28.590","Text":"but this is the general definition that was adapted to 0,0,"},{"Start":"09:28.590 ","End":"09:30.455","Text":"which says that basically,"},{"Start":"09:30.455 ","End":"09:34.460","Text":"if this limit exists and equals 0 then we have differentiability."},{"Start":"09:34.460 ","End":"09:38.305","Text":"At the moment we don\u0027t know this and this is what we\u0027re going to check."},{"Start":"09:38.305 ","End":"09:43.260","Text":"Now the good thing is that we all ready"},{"Start":"09:43.260 ","End":"09:49.640","Text":"computed the derivative with respect to x at 0,0 is 0."},{"Start":"09:49.640 ","End":"09:51.649","Text":"This bit is 0,"},{"Start":"09:51.649 ","End":"09:54.755","Text":"this bit here is 0."},{"Start":"09:54.755 ","End":"09:57.620","Text":"The function f at 0,0,"},{"Start":"09:57.620 ","End":"09:59.615","Text":"let\u0027s just go back up and look,"},{"Start":"09:59.615 ","End":"10:03.814","Text":"this was given here as 0,"},{"Start":"10:03.814 ","End":"10:08.370","Text":"so we even have that bit is 0."},{"Start":"10:08.370 ","End":"10:10.440","Text":"That makes work easier."},{"Start":"10:10.440 ","End":"10:13.890","Text":"All I have to do now is plug in Delta x,"},{"Start":"10:13.890 ","End":"10:17.520","Text":"Delta y here and compute the limit."},{"Start":"10:17.520 ","End":"10:28.220","Text":"We get the limit as Delta x goes to 0 and so does Delta y of this function."},{"Start":"10:28.220 ","End":"10:30.370","Text":"I\u0027m not going to scroll back.."},{"Start":"10:30.370 ","End":"10:35.280","Text":"It was e to the minus 1 over x squared plus y squared or in this case,"},{"Start":"10:35.280 ","End":"10:41.400","Text":"Delta x squared plus Delta y squared."},{"Start":"10:41.400 ","End":"10:44.270","Text":"That\u0027s when they\u0027re not both 0,"},{"Start":"10:44.270 ","End":"10:46.865","Text":"which they\u0027re not because they\u0027re tending to 0."},{"Start":"10:46.865 ","End":"10:50.030","Text":"That\u0027s this part here. All the rest of this is 0."},{"Start":"10:50.030 ","End":"10:51.800","Text":"I need a dividing line."},{"Start":"10:51.800 ","End":"10:58.230","Text":"I need here the square root of Delta x squared plus Delta y squared."},{"Start":"10:58.230 ","End":"11:07.055","Text":"We have to show that this is equal to 0 and then we have differentiability or not."},{"Start":"11:07.055 ","End":"11:09.490","Text":"Let\u0027s tackle this limit."},{"Start":"11:09.490 ","End":"11:12.815","Text":"Now we actually have done this in a form."},{"Start":"11:12.815 ","End":"11:17.225","Text":"I don\u0027t want to scroll down because I want to somehow use this result because look,"},{"Start":"11:17.225 ","End":"11:22.325","Text":"if I substitute h to equal"},{"Start":"11:22.325 ","End":"11:28.990","Text":"the square root of Delta x squared plus Delta y squared,"},{"Start":"11:28.990 ","End":"11:32.450","Text":"then what we get here is"},{"Start":"11:32.450 ","End":"11:40.140","Text":"the limit as h goes to 0, I claim."},{"Start":"11:40.140 ","End":"11:44.390","Text":"In fact actually h goes to 0 plus because when Delta x and"},{"Start":"11:44.390 ","End":"11:48.785","Text":"Delta y tends to 0 this thing is always positive but it goes to 0."},{"Start":"11:48.785 ","End":"11:50.680","Text":"This is what we have."},{"Start":"11:50.680 ","End":"11:59.930","Text":"This we get e to the minus 1 over."},{"Start":"11:59.930 ","End":"12:04.295","Text":"Now this thing is just h squared because it\u0027s the square root squared."},{"Start":"12:04.295 ","End":"12:07.595","Text":"On the denominator we have h itself."},{"Start":"12:07.595 ","End":"12:11.900","Text":"This limit we\u0027ve all ready computed to be equal."},{"Start":"12:11.900 ","End":"12:16.290","Text":"We started here and we ended up with 0."},{"Start":"12:16.430 ","End":"12:21.920","Text":"We pretty much got it for free once we got the right substitution which was this,"},{"Start":"12:21.920 ","End":"12:24.440","Text":"and so 0 is what we wanted."},{"Start":"12:24.440 ","End":"12:25.685","Text":"This is very good,"},{"Start":"12:25.685 ","End":"12:28.700","Text":"which means that f of x,"},{"Start":"12:28.700 ","End":"12:31.870","Text":"y is differentiable at 0."},{"Start":"12:31.870 ","End":"12:38.440","Text":"F is differentiable at 0,0."},{"Start":"12:38.440 ","End":"12:44.405","Text":"We\u0027ve already proven that it\u0027s differentiable everywhere except 0,0,"},{"Start":"12:44.405 ","End":"12:48.885","Text":"so it\u0027s differentiable therefore everywhere."},{"Start":"12:48.885 ","End":"12:55.770","Text":"Well, everywhere in the 2D-plane hence everywhere and we\u0027re done."}],"ID":20891}],"Thumbnail":null,"ID":4972}]

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