Introduction to Chain Rule
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- The Chain Rule 1
- The Chain Rule 2
- The Chain Rule 3
- The Chain Rule 4
- The Chain Rule 5
- The Chain Rule 6
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13 part a
- Exercise 13 part b
- Exercise 13 part c
- Exercise 14 part a
- Exercise 14 part b
- Exercise 15
- Exercise 16

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[{"Name":"Introduction to Chain Rule","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Chain Rule 1","Duration":"19m 6s","ChapterTopicVideoID":8613,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8613.jpeg","UploadDate":"2020-02-26T11:52:18.6400000","DurationForVideoObject":"PT19M6S","Description":null,"MetaTitle":"The Chain Rule 1: Video + Workbook | Proprep","MetaDescription":"Chain Rule - Introduction to Chain Rule. 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/chain-rule/introduction-to-chain-rule/vid8959","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.125","Text":"In this clip, I\u0027ll be talking about the chain rule for functions of 2 or more variables."},{"Start":"00:07.125 ","End":"00:09.570","Text":"Just so you have some idea what it\u0027s all about,"},{"Start":"00:09.570 ","End":"00:11.955","Text":"let\u0027s start straight away with an example."},{"Start":"00:11.955 ","End":"00:14.550","Text":"Let\u0027s take a function of 2 variables,"},{"Start":"00:14.550 ","End":"00:16.650","Text":"f of x, y."},{"Start":"00:16.650 ","End":"00:23.850","Text":"Let it equal x/y times e^x over"},{"Start":"00:23.850 ","End":"00:32.370","Text":"y plus xy times the log of xy."},{"Start":"00:32.370 ","End":"00:38.180","Text":"Suppose that we are required to compute what is the derivative of f with"},{"Start":"00:38.180 ","End":"00:44.870","Text":"respect to x and what is the derivative of f with respect to y."},{"Start":"00:44.870 ","End":"00:49.340","Text":"Now of course, I can differentiate it with the usual techniques,"},{"Start":"00:49.340 ","End":"00:50.690","Text":"but it\u0027s a bit messy,"},{"Start":"00:50.690 ","End":"00:54.350","Text":"here I have a quotient and here I have a quotient and then it\u0027s a product."},{"Start":"00:54.350 ","End":"00:59.195","Text":"Then I have a product of 3 things and it might be a bit messy, you could do it."},{"Start":"00:59.195 ","End":"01:01.505","Text":"But I\u0027d like to show you another approach"},{"Start":"01:01.505 ","End":"01:04.670","Text":"which might or might not be easier, I think it is."},{"Start":"01:04.670 ","End":"01:05.870","Text":"But in any event,"},{"Start":"01:05.870 ","End":"01:07.730","Text":"it will lead us to the chain rule which"},{"Start":"01:07.730 ","End":"01:11.605","Text":"is necessary for us to learn in order to progress."},{"Start":"01:11.605 ","End":"01:14.980","Text":"So let me show you the other technique."},{"Start":"01:14.980 ","End":"01:19.135","Text":"I happened to notice that if you look at it,"},{"Start":"01:19.135 ","End":"01:22.555","Text":"that this is x/y here,"},{"Start":"01:22.555 ","End":"01:25.580","Text":"and this is x/y here."},{"Start":"01:25.580 ","End":"01:27.705","Text":"Here I have xy,"},{"Start":"01:27.705 ","End":"01:29.555","Text":"and here I have xy."},{"Start":"01:29.555 ","End":"01:34.180","Text":"Now, I\u0027m going to take advantage of this that I don\u0027t have any x\u0027s or"},{"Start":"01:34.180 ","End":"01:38.830","Text":"y\u0027s separately other than in the form x/y or xy."},{"Start":"01:38.830 ","End":"01:43.945","Text":"What I\u0027m going to do is to let u"},{"Start":"01:43.945 ","End":"01:51.350","Text":"equals x/y and set v equals xy."},{"Start":"01:51.350 ","End":"01:54.755","Text":"Now if I look at this function,"},{"Start":"01:54.755 ","End":"02:00.220","Text":"this is equal to u times e to the power of"},{"Start":"02:00.220 ","End":"02:07.615","Text":"u plus v times log v. Now,"},{"Start":"02:07.615 ","End":"02:10.870","Text":"this is a function of x and y directly,"},{"Start":"02:10.870 ","End":"02:12.970","Text":"but of u and v, so I\u0027ll write f of u,"},{"Start":"02:12.970 ","End":"02:16.960","Text":"v. This seems much easier if I was asked to"},{"Start":"02:16.960 ","End":"02:21.610","Text":"differentiate this according to u and according to v. How will this help me?"},{"Start":"02:21.610 ","End":"02:23.980","Text":"You will see, it\u0027s not clear that the"},{"Start":"02:23.980 ","End":"02:26.500","Text":"moment that will help make this not answering the original question,"},{"Start":"02:26.500 ","End":"02:28.605","Text":"but play along with me."},{"Start":"02:28.605 ","End":"02:31.900","Text":"Let\u0027s try and differentiate this with respect to u"},{"Start":"02:31.900 ","End":"02:34.925","Text":"and v. Remembering that this, after all,"},{"Start":"02:34.925 ","End":"02:39.470","Text":"does equal the original f because if I put u equals x/y and v equals x,"},{"Start":"02:39.470 ","End":"02:43.725","Text":"y back in, then I\u0027ve got my original function."},{"Start":"02:43.725 ","End":"02:50.629","Text":"Let\u0027s first of all compute what is f with respect to u,"},{"Start":"02:50.629 ","End":"02:53.330","Text":"the derivative of u,"},{"Start":"02:53.330 ","End":"02:59.635","Text":"v, and also what is f by v of u,"},{"Start":"02:59.635 ","End":"03:04.490","Text":"v. Then we\u0027ll see how it helps us to find what we originally wanted."},{"Start":"03:04.490 ","End":"03:08.720","Text":"Now, this is straightforward because if I differentiate with respect to u,"},{"Start":"03:08.720 ","End":"03:11.230","Text":"then it means that v is a constant."},{"Start":"03:11.230 ","End":"03:16.280","Text":"So what I get is just the derivative of this and this becomes 0."},{"Start":"03:16.280 ","End":"03:18.440","Text":"It\u0027s a straightforward product rule."},{"Start":"03:18.440 ","End":"03:19.790","Text":"It\u0027s a derivative of this,"},{"Start":"03:19.790 ","End":"03:25.190","Text":"which is 1 times e^u plus u times the derivative of this,"},{"Start":"03:25.190 ","End":"03:27.320","Text":"which is just e^u."},{"Start":"03:27.320 ","End":"03:31.970","Text":"If I want, I could simplify it as u plus 1 times e^u,"},{"Start":"03:31.970 ","End":"03:33.485","Text":"or just leave it like this."},{"Start":"03:33.485 ","End":"03:39.090","Text":"Similarly, the derivative of f by v is u is a constant."},{"Start":"03:39.090 ","End":"03:40.550","Text":"This thing becomes 0,"},{"Start":"03:40.550 ","End":"03:43.220","Text":"and now I have the product of these 2."},{"Start":"03:43.220 ","End":"03:50.090","Text":"It\u0027s this thing differentiated is 1 times log v as is,"},{"Start":"03:50.090 ","End":"03:53.690","Text":"when say log, I mean natural log just for brevity."},{"Start":"03:53.690 ","End":"03:56.960","Text":"Plus v as is,"},{"Start":"03:56.960 ","End":"04:00.840","Text":"and log v differentiated is 1/v,"},{"Start":"04:00.880 ","End":"04:08.830","Text":"which is equal to natural log v plus 1."},{"Start":"04:09.140 ","End":"04:11.630","Text":"This is pretty straightforward."},{"Start":"04:11.630 ","End":"04:14.120","Text":"I have with respect to u, with respect to v. Now,"},{"Start":"04:14.120 ","End":"04:19.235","Text":"how do I get to with respect to x and with respect to y?"},{"Start":"04:19.235 ","End":"04:20.900","Text":"To help us in this task,"},{"Start":"04:20.900 ","End":"04:25.850","Text":"I\u0027d like to show you a very useful visual aid that really helps."},{"Start":"04:25.850 ","End":"04:28.760","Text":"Just let me get this out of the way."},{"Start":"04:28.760 ","End":"04:33.125","Text":"What I\u0027m going to draw here is what I call a dependency tree."},{"Start":"04:33.125 ","End":"04:38.875","Text":"We start off with the function f. We see that f,"},{"Start":"04:38.875 ","End":"04:42.950","Text":"according to this depends on u and v. It depends on"},{"Start":"04:42.950 ","End":"04:47.450","Text":"u and it depends on v. This is how I draw the dependencies,"},{"Start":"04:47.450 ","End":"04:49.790","Text":"I go down the branches."},{"Start":"04:49.790 ","End":"04:58.694","Text":"Now, u is dependent on x and y,"},{"Start":"04:58.694 ","End":"05:02.880","Text":"so I write in x and y,"},{"Start":"05:02.880 ","End":"05:08.475","Text":"and v also depends on x and y."},{"Start":"05:08.475 ","End":"05:11.985","Text":"Ultimately, f depends on x,"},{"Start":"05:11.985 ","End":"05:14.700","Text":"there\u0027s 1 path and another path."},{"Start":"05:14.700 ","End":"05:18.975","Text":"That also depends on y through here and here."},{"Start":"05:18.975 ","End":"05:22.425","Text":"Let me do this 1 first,"},{"Start":"05:22.425 ","End":"05:25.215","Text":"f with respect to x."},{"Start":"05:25.215 ","End":"05:27.605","Text":"Because it\u0027s with respect to x,"},{"Start":"05:27.605 ","End":"05:31.730","Text":"I take a look at where I get x,"},{"Start":"05:31.730 ","End":"05:37.175","Text":"the branches which gives me x. I can go this way and this way,"},{"Start":"05:37.175 ","End":"05:41.760","Text":"or I can go this way and this way,"},{"Start":"05:41.760 ","End":"05:44.070","Text":"2 ways of getting to x."},{"Start":"05:44.070 ","End":"05:47.190","Text":"The procedure is this."},{"Start":"05:47.190 ","End":"05:51.440","Text":"In order to take f by x,"},{"Start":"05:51.440 ","End":"05:54.320","Text":"what we do is we take each branch,"},{"Start":"05:54.320 ","End":"05:56.345","Text":"let\u0027s first of all take this branch,"},{"Start":"05:56.345 ","End":"05:59.930","Text":"and within the branch, I do a multiplication."},{"Start":"05:59.930 ","End":"06:02.584","Text":"I say it\u0027s f by u,"},{"Start":"06:02.584 ","End":"06:09.050","Text":"u by x. I mean with respect to I just say by for short."},{"Start":"06:09.050 ","End":"06:13.115","Text":"Plus, now take the other branch."},{"Start":"06:13.115 ","End":"06:15.035","Text":"Between branches I have a plus,"},{"Start":"06:15.035 ","End":"06:17.855","Text":"then within the branch I have multiplication."},{"Start":"06:17.855 ","End":"06:23.615","Text":"It\u0027s f by v times v by x,"},{"Start":"06:23.615 ","End":"06:25.190","Text":"but we still have to expand."},{"Start":"06:25.190 ","End":"06:26.690","Text":"We have these 2,"},{"Start":"06:26.690 ","End":"06:30.665","Text":"f by u and f by v here and here."},{"Start":"06:30.665 ","End":"06:35.000","Text":"What we\u0027re missing is u by x and v by x,"},{"Start":"06:35.000 ","End":"06:39.290","Text":"which is easy enough to compute from these 2."},{"Start":"06:39.290 ","End":"06:43.325","Text":"I can differentiate this with respect to x and this with respect to x."},{"Start":"06:43.325 ","End":"06:46.530","Text":"Let\u0027s continue with the computation."},{"Start":"06:46.960 ","End":"06:51.060","Text":"Let me bring these down here closer."},{"Start":"06:51.550 ","End":"06:59.090","Text":"What I get is f by x equals u plus"},{"Start":"06:59.090 ","End":"07:07.630","Text":"1 e^u and u by x is,"},{"Start":"07:07.630 ","End":"07:09.780","Text":"y is a constant,"},{"Start":"07:09.780 ","End":"07:13.045","Text":"so if I differentiate with respect to x,"},{"Start":"07:13.045 ","End":"07:15.070","Text":"the over y is just a constant,"},{"Start":"07:15.070 ","End":"07:18.550","Text":"so I get 1/y,"},{"Start":"07:18.550 ","End":"07:24.600","Text":"sorry, times 1/y here."},{"Start":"07:24.600 ","End":"07:28.095","Text":"Then I\u0027ll take this second bit."},{"Start":"07:28.095 ","End":"07:32.095","Text":"That\u0027s plus f by v is from here,"},{"Start":"07:32.095 ","End":"07:35.950","Text":"natural log of v plus"},{"Start":"07:35.950 ","End":"07:43.470","Text":"1 times v by x."},{"Start":"07:43.470 ","End":"07:45.165","Text":"If I differentiate v according x,"},{"Start":"07:45.165 ","End":"07:47.100","Text":"again y is a constant,"},{"Start":"07:47.100 ","End":"07:50.050","Text":"so I\u0027m just left with the y."},{"Start":"07:51.070 ","End":"07:56.770","Text":"But this is not the end because I still have the presence of u and v"},{"Start":"07:56.770 ","End":"08:02.085","Text":"and I want everything in terms of x and y. No problem."},{"Start":"08:02.085 ","End":"08:05.000","Text":"We have these formulae."},{"Start":"08:05.000 ","End":"08:10.130","Text":"So I can write it finally as f by x is,"},{"Start":"08:10.130 ","End":"08:12.560","Text":"now u is x/y,"},{"Start":"08:12.560 ","End":"08:23.100","Text":"so x/y plus 1 e^x/y times 1/y."},{"Start":"08:23.630 ","End":"08:28.515","Text":"The second 1 is natural log of,"},{"Start":"08:28.515 ","End":"08:37.485","Text":"v is xy plus 1 times y."},{"Start":"08:37.485 ","End":"08:40.425","Text":"This is the answer."},{"Start":"08:40.425 ","End":"08:43.190","Text":"You might say, well, okay,"},{"Start":"08:43.190 ","End":"08:45.770","Text":"I could have done it the old fashioned way."},{"Start":"08:45.770 ","End":"08:47.720","Text":"If I go back here,"},{"Start":"08:47.720 ","End":"08:49.490","Text":"I could say yes, there is low problem,"},{"Start":"08:49.490 ","End":"08:53.765","Text":"I don\u0027t need u and v and you\u0027d be right."},{"Start":"08:53.765 ","End":"08:58.130","Text":"But later on, there\u0027ll be cases where you can\u0027t do it that way,"},{"Start":"08:58.130 ","End":"08:59.690","Text":"you have to do it directly."},{"Start":"08:59.690 ","End":"09:03.655","Text":"Besides, we need to learn the chain rule for other things."},{"Start":"09:03.655 ","End":"09:06.740","Text":"Bear with me. It\u0027s something new,"},{"Start":"09:06.740 ","End":"09:09.800","Text":"yes, but it\u0027s necessary."},{"Start":"09:11.400 ","End":"09:16.060","Text":"Let\u0027s continue now to do the other partial"},{"Start":"09:16.060 ","End":"09:21.025","Text":"derivative because we were asked for the derivative with respect to x."},{"Start":"09:21.025 ","End":"09:25.375","Text":"But we also need to do the other 1,"},{"Start":"09:25.375 ","End":"09:33.985","Text":"the derivative of f according to y. I\u0027ll do it a little bit quicker this time."},{"Start":"09:33.985 ","End":"09:37.990","Text":"Let\u0027s see, we do still need our tree."},{"Start":"09:37.990 ","End":"09:43.725","Text":"But I have to modify it because now I want to take according to y."},{"Start":"09:43.725 ","End":"09:47.145","Text":"Here\u0027s the revised tree."},{"Start":"09:47.145 ","End":"09:51.400","Text":"To get to y, we can go along this branch here and here,"},{"Start":"09:51.400 ","End":"09:53.620","Text":"or along here and along here."},{"Start":"09:53.620 ","End":"09:56.020","Text":"I think I\u0027ll bring this further down,"},{"Start":"09:56.020 ","End":"10:00.065","Text":"there, it\u0027s now closer."},{"Start":"10:00.065 ","End":"10:03.935","Text":"Let\u0027s do the same thing with f by y,"},{"Start":"10:03.935 ","End":"10:07.055","Text":"which is equal f by u,"},{"Start":"10:07.055 ","End":"10:12.000","Text":"u by y product,"},{"Start":"10:12.000 ","End":"10:17.290","Text":"and then plus f by v, v by y."},{"Start":"10:20.430 ","End":"10:27.865","Text":"Now this equals, this we have, see where it is."},{"Start":"10:27.865 ","End":"10:31.450","Text":"It\u0027s still on the board just about."},{"Start":"10:31.450 ","End":"10:34.240","Text":"This is u plus 1 e^u."},{"Start":"10:34.240 ","End":"10:35.905","Text":"I can copy it from here."},{"Start":"10:35.905 ","End":"10:44.450","Text":"So it\u0027s u plus 1 e^u, u_y."},{"Start":"10:45.030 ","End":"10:48.880","Text":"The x is a constant, so it stays."},{"Start":"10:48.880 ","End":"10:53.650","Text":"The derivative of 1 over y is minus 1 over y squared."},{"Start":"10:53.650 ","End":"10:56.200","Text":"You\u0027ve probably seen this enough times already."},{"Start":"10:56.200 ","End":"11:06.270","Text":"It\u0027s minus, I\u0027ll put the minus upfront and x over y squared."},{"Start":"11:06.270 ","End":"11:09.250","Text":"The x is a constant and it stays."},{"Start":"11:09.830 ","End":"11:14.155","Text":"Plus the other branch,"},{"Start":"11:14.155 ","End":"11:16.855","Text":"f_v, which we already have,"},{"Start":"11:16.855 ","End":"11:26.060","Text":"is natural log of v plus 1 times v_y,"},{"Start":"11:26.940 ","End":"11:32.590","Text":"because x is a constant, it\u0027s just x."},{"Start":"11:32.590 ","End":"11:35.425","Text":"As if it was like 4y,"},{"Start":"11:35.425 ","End":"11:37.705","Text":"so the answer would be 4."},{"Start":"11:37.705 ","End":"11:48.760","Text":"Now, the final step is here to replace u and v with x and y."},{"Start":"11:48.760 ","End":"11:51.925","Text":"So u is x over y."},{"Start":"11:51.925 ","End":"12:03.010","Text":"We have minus x over y plus 1 e^x over y,"},{"Start":"12:03.010 ","End":"12:07.240","Text":"times x over y squared,"},{"Start":"12:07.240 ","End":"12:10.150","Text":"plus natural log of v,"},{"Start":"12:10.150 ","End":"12:17.900","Text":"which is xy plus 1 times x."},{"Start":"12:18.030 ","End":"12:21.505","Text":"We have done what we were asked for."},{"Start":"12:21.505 ","End":"12:23.665","Text":"This one is done."},{"Start":"12:23.665 ","End":"12:25.660","Text":"This one is done."},{"Start":"12:25.660 ","End":"12:28.690","Text":"That\u0027s what we got."},{"Start":"12:28.690 ","End":"12:32.980","Text":"We got f according to x over here,"},{"Start":"12:32.980 ","End":"12:38.630","Text":"and f_y over here."},{"Start":"12:39.810 ","End":"12:43.300","Text":"Now to get onto another example."},{"Start":"12:43.300 ","End":"12:52.900","Text":"In this example, f of xy is x cubed plus y^4."},{"Start":"12:52.900 ","End":"12:54.250","Text":"Suppose for whatever reason,"},{"Start":"12:54.250 ","End":"13:02.500","Text":"I have to substitute x equals m squared plus"},{"Start":"13:02.500 ","End":"13:12.025","Text":"4n and y equals 10m minus 20n."},{"Start":"13:12.025 ","End":"13:15.250","Text":"Where m and n are variables,"},{"Start":"13:15.250 ","End":"13:25.060","Text":"I would like to know what is the partial derivative of f with respect to m. I would"},{"Start":"13:25.060 ","End":"13:31.690","Text":"also like to know what is the partial derivative of f with respect to n. But I"},{"Start":"13:31.690 ","End":"13:38.875","Text":"don\u0027t want to do it by substituting x and y into this function."},{"Start":"13:38.875 ","End":"13:44.545","Text":"I want to use the same concept of the dependency tree and the chain rule."},{"Start":"13:44.545 ","End":"13:47.515","Text":"Let\u0027s draw our tree first."},{"Start":"13:47.515 ","End":"13:49.375","Text":"At the top of the tree,"},{"Start":"13:49.375 ","End":"13:54.820","Text":"we have f, and"},{"Start":"13:54.820 ","End":"14:01.630","Text":"f in this case depends on x and y."},{"Start":"14:01.630 ","End":"14:08.755","Text":"But each of x and y is dependent on m and n. This one depends on m"},{"Start":"14:08.755 ","End":"14:11.980","Text":"and n. It\u0027s similar to the previous except that"},{"Start":"14:11.980 ","End":"14:16.195","Text":"there we had x and y at the bottom and u and v at the middle level."},{"Start":"14:16.195 ","End":"14:20.215","Text":"Whereas here we have x and y at the middle level and m and n at the bottom."},{"Start":"14:20.215 ","End":"14:23.530","Text":"The same rules apply when you go a longer tree."},{"Start":"14:23.530 ","End":"14:29.125","Text":"If I want to see what is f according to m, then I highlight."},{"Start":"14:29.125 ","End":"14:34.599","Text":"I have to get to m so I can go here and here,"},{"Start":"14:34.599 ","End":"14:38.740","Text":"or I can go here and here and end up with"},{"Start":"14:38.740 ","End":"14:45.610","Text":"m. What we get is that f_m is equal to,"},{"Start":"14:45.610 ","End":"14:49.030","Text":"now this branch gives me f_x, x_m."},{"Start":"14:49.030 ","End":"14:53.980","Text":"F_x and x_m and a plus for"},{"Start":"14:53.980 ","End":"15:00.355","Text":"the other branch, f_y and y_m."},{"Start":"15:00.355 ","End":"15:04.645","Text":"I should say y with respect to m. Now let\u0027s start expanding this."},{"Start":"15:04.645 ","End":"15:07.765","Text":"F_x is from here,"},{"Start":"15:07.765 ","End":"15:13.840","Text":"I differentiate where x is the variable and y is the constant or parameter."},{"Start":"15:13.840 ","End":"15:18.339","Text":"This is nothing, and this is 3x squared."},{"Start":"15:18.339 ","End":"15:21.760","Text":"So we get 3x squared all together."},{"Start":"15:21.760 ","End":"15:27.234","Text":"Now x_m, I go to here and take m as the variable,"},{"Start":"15:27.234 ","End":"15:28.525","Text":"n is the constant."},{"Start":"15:28.525 ","End":"15:38.290","Text":"So I get 2m plus f_y, similarly,"},{"Start":"15:38.290 ","End":"15:46.285","Text":"x is the constant, it\u0027s 4y cubed and y_m from here,"},{"Start":"15:46.285 ","End":"15:50.110","Text":"m is the variable it\u0027s just 10."},{"Start":"15:50.110 ","End":"15:56.290","Text":"Finally, we just have to substitute x and y in terms"},{"Start":"15:56.290 ","End":"15:59.350","Text":"of m and n. Because when we"},{"Start":"15:59.350 ","End":"16:02.770","Text":"talk about these 2 partial derivatives with respect to m and n,"},{"Start":"16:02.770 ","End":"16:08.349","Text":"we are assuming that the variables are also m and n. We get 3."},{"Start":"16:08.349 ","End":"16:16.975","Text":"Let\u0027s see. x is m squared plus 4n squared,"},{"Start":"16:16.975 ","End":"16:22.345","Text":"times 2m plus 4 something cubed."},{"Start":"16:22.345 ","End":"16:30.700","Text":"There it is, 10m minus 20n cubed, and times 10."},{"Start":"16:30.700 ","End":"16:32.830","Text":"This can be simplified a bit."},{"Start":"16:32.830 ","End":"16:34.270","Text":"I\u0027m not going to do that."},{"Start":"16:34.270 ","End":"16:39.010","Text":"I\u0027m going to go on to f with respect to n. Let\u0027s see,"},{"Start":"16:39.010 ","End":"16:41.810","Text":"we have to modify the tree."},{"Start":"16:42.390 ","End":"16:45.070","Text":"The top part stays the same,"},{"Start":"16:45.070 ","End":"16:48.610","Text":"but I take the paths to end this time."},{"Start":"16:48.610 ","End":"16:53.499","Text":"So along here and along here."},{"Start":"16:53.499 ","End":"17:01.645","Text":"Now we\u0027re going to write a formula so that f with respect to n is equal to,"},{"Start":"17:01.645 ","End":"17:04.070","Text":"according to the tree f_x,"},{"Start":"17:05.160 ","End":"17:10.730","Text":"x_n plus f_y, y_n."},{"Start":"17:10.730 ","End":"17:20.245","Text":"Now f_x is just 3x squared because y is a constant."},{"Start":"17:20.245 ","End":"17:28.960","Text":"So this is equal 3x squared times x_n."},{"Start":"17:28.960 ","End":"17:33.924","Text":"x_n would be m the constant,"},{"Start":"17:33.924 ","End":"17:38.230","Text":"so it\u0027s 4 and f_y,"},{"Start":"17:38.230 ","End":"17:48.640","Text":"we already have that is 4y cubed and y_n is m the constant,"},{"Start":"17:48.640 ","End":"17:52.010","Text":"so it\u0027s minus 2."},{"Start":"17:54.480 ","End":"18:05.245","Text":"Now we just have to substitute the x and y in terms of m and n. So we get 3 times"},{"Start":"18:05.245 ","End":"18:12.430","Text":"m squared plus 4n squared times"},{"Start":"18:12.430 ","End":"18:17.680","Text":"4 plus 4 10m minus"},{"Start":"18:17.680 ","End":"18:26.350","Text":"2n cubed times minus 2."},{"Start":"18:26.350 ","End":"18:28.270","Text":"Now of course we could simplify this,"},{"Start":"18:28.270 ","End":"18:31.675","Text":"the constants 3 times 4 and 4 times minus 2 and so on."},{"Start":"18:31.675 ","End":"18:33.235","Text":"We\u0027re going to leave it as is."},{"Start":"18:33.235 ","End":"18:35.980","Text":"You just need to get the idea of how to do this."},{"Start":"18:35.980 ","End":"18:39.504","Text":"This whole procedure is the chain rule as it was before."},{"Start":"18:39.504 ","End":"18:42.220","Text":"It\u0027s a chain rule, well you can see the chains here."},{"Start":"18:42.220 ","End":"18:44.800","Text":"Anyway, it\u0027s called the chain rule and what we did before"},{"Start":"18:44.800 ","End":"18:47.050","Text":"also is when a function is"},{"Start":"18:47.050 ","End":"18:50.125","Text":"dependent on some variables and these are dependent on the other."},{"Start":"18:50.125 ","End":"18:54.520","Text":"I want to find the partial derivatives with respect to the variables on the bottom."},{"Start":"18:54.520 ","End":"18:57.280","Text":"So it\u0027s a 2 step thing and it\u0027s the chain rule."},{"Start":"18:57.280 ","End":"19:01.090","Text":"We\u0027re done here. In the next clip we\u0027ll go on to"},{"Start":"19:01.090 ","End":"19:07.160","Text":"more complicated examples and we\u0027ll do some more generalization."}],"ID":8959},{"Watched":false,"Name":"The Chain Rule 2","Duration":"7m 15s","ChapterTopicVideoID":8614,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.535","Text":"Continuing with the chain rule,"},{"Start":"00:02.535 ","End":"00:06.060","Text":"let\u0027s take the following example exercise."},{"Start":"00:06.060 ","End":"00:08.670","Text":"What we are is given,"},{"Start":"00:08.670 ","End":"00:14.400","Text":"that z, if you\u0027re in America,"},{"Start":"00:14.400 ","End":"00:20.265","Text":"is equal to f of x squared plus y squared,"},{"Start":"00:20.265 ","End":"00:27.630","Text":"and we have to prove that y times the derivative of z with respect to x"},{"Start":"00:27.630 ","End":"00:37.830","Text":"minus x times the derivative of z with respect to y is equal to 0."},{"Start":"00:37.830 ","End":"00:40.920","Text":"The solution, we\u0027ll write here,"},{"Start":"00:40.920 ","End":"00:45.840","Text":"but first I\u0027d like to explain what does this mean,"},{"Start":"00:46.360 ","End":"00:49.715","Text":"I\u0027ll say z, there\u0027s more Americans than British,"},{"Start":"00:49.715 ","End":"00:54.540","Text":"that z equals f of x squared plus y squared."},{"Start":"00:54.540 ","End":"00:58.100","Text":"Well, we\u0027re not given what the function f is,"},{"Start":"00:58.100 ","End":"01:03.545","Text":"but we know that z is made up only of x-squared plus y-squared."},{"Start":"01:03.545 ","End":"01:05.915","Text":"I\u0027ll give an example that will be easiest."},{"Start":"01:05.915 ","End":"01:10.190","Text":"Suppose I have that z equals,"},{"Start":"01:10.190 ","End":"01:14.970","Text":"and I\u0027ll give a bit of an elaborate example, this monstrosity."},{"Start":"01:14.970 ","End":"01:23.480","Text":"Notice that everywhere in this expression,"},{"Start":"01:23.480 ","End":"01:26.630","Text":"you don\u0027t get x on its own or y on its own,"},{"Start":"01:26.630 ","End":"01:28.400","Text":"or xy, or x cubed,"},{"Start":"01:28.400 ","End":"01:33.639","Text":"you always get x-squared plus y-squared together here,"},{"Start":"01:33.639 ","End":"01:39.840","Text":"here, here, here, here."},{"Start":"01:39.840 ","End":"01:41.520","Text":"You don\u0027t get x or y,"},{"Start":"01:41.520 ","End":"01:43.305","Text":"or any other combination,"},{"Start":"01:43.305 ","End":"01:45.870","Text":"except x-squared plus y-squared."},{"Start":"01:45.870 ","End":"01:48.590","Text":"It\u0027s a function of x squared plus y squared."},{"Start":"01:48.590 ","End":"01:51.155","Text":"Of course, it doesn\u0027t have to be so complicated,"},{"Start":"01:51.155 ","End":"01:53.525","Text":"we could also take a much simpler example,"},{"Start":"01:53.525 ","End":"02:00.649","Text":"such as z equals x squared plus y squared cubed."},{"Start":"02:00.649 ","End":"02:06.260","Text":"The point is that it\u0027s some combination of x-squared plus y-squared taken as a whole."},{"Start":"02:06.260 ","End":"02:09.480","Text":"Could also look at it as z equals"},{"Start":"02:09.480 ","End":"02:13.745","Text":"some function of t and then replace t with x squared plus y squared."},{"Start":"02:13.745 ","End":"02:17.930","Text":"We have to prove that this expression is true,"},{"Start":"02:17.930 ","End":"02:21.320","Text":"that y times the derivative of z with respect to"},{"Start":"02:21.320 ","End":"02:25.805","Text":"x minus x times derivative z with respect to y is 0."},{"Start":"02:25.805 ","End":"02:32.615","Text":"Even though, I can\u0027t actually compute what these 2 derivatives are,"},{"Start":"02:32.615 ","End":"02:35.345","Text":"the partial derivatives with respect to x and y,"},{"Start":"02:35.345 ","End":"02:37.820","Text":"because I don\u0027t have the exact expression."},{"Start":"02:37.820 ","End":"02:41.945","Text":"Nevertheless, it will turn out that for all such functions,"},{"Start":"02:41.945 ","End":"02:44.480","Text":"this will hold and we have to prove it."},{"Start":"02:44.480 ","End":"02:47.029","Text":"Let\u0027s get on to the solution."},{"Start":"02:47.029 ","End":"02:50.060","Text":"I might add that it would be a total nightmare if we"},{"Start":"02:50.060 ","End":"02:53.750","Text":"actually tried to compute it given a function like this."},{"Start":"02:53.750 ","End":"02:56.915","Text":"Now, what do you think is going to help us with the solution?"},{"Start":"02:56.915 ","End":"02:59.920","Text":"You guessed it, the chain rule."},{"Start":"02:59.920 ","End":"03:02.685","Text":"Let me just clear up this junk."},{"Start":"03:02.685 ","End":"03:06.875","Text":"What I\u0027m going to do is something I just mentioned previously,"},{"Start":"03:06.875 ","End":"03:11.074","Text":"is that since x squared plus y squared appears as a whole,"},{"Start":"03:11.074 ","End":"03:13.295","Text":"I\u0027m going to treat it like a variable."},{"Start":"03:13.295 ","End":"03:19.484","Text":"I\u0027m going to set t is equal to x squared plus y squared,"},{"Start":"03:19.484 ","End":"03:27.120","Text":"and then what I\u0027ll get is that z is f of t. Now,"},{"Start":"03:27.120 ","End":"03:29.795","Text":"if we sketch our dependency tree,"},{"Start":"03:29.795 ","End":"03:32.330","Text":"like I showed you before in the previous clip,"},{"Start":"03:32.330 ","End":"03:35.910","Text":"what we\u0027ll get is that the function f,"},{"Start":"03:35.910 ","End":"03:37.910","Text":"or the variable z,"},{"Start":"03:37.910 ","End":"03:39.620","Text":"practically the same thing,"},{"Start":"03:39.620 ","End":"03:43.500","Text":"depends only on t,"},{"Start":"03:43.720 ","End":"03:52.195","Text":"but t depends both on x and on y."},{"Start":"03:52.195 ","End":"03:56.480","Text":"To figure out what is z with respect to x,"},{"Start":"03:56.480 ","End":"04:02.575","Text":"the derivative, we use one branch of the tree, namely,"},{"Start":"04:02.575 ","End":"04:06.525","Text":"this one will give us z with respect to x,"},{"Start":"04:06.525 ","End":"04:09.240","Text":"and what we get is,"},{"Start":"04:09.240 ","End":"04:10.815","Text":"like we learned to do,"},{"Start":"04:10.815 ","End":"04:12.500","Text":"we take all the branches, in this case,"},{"Start":"04:12.500 ","End":"04:14.955","Text":"there\u0027s only 1, all the paths that lead to x,"},{"Start":"04:14.955 ","End":"04:19.070","Text":"and we multiply, so we get z with respect to t,"},{"Start":"04:19.070 ","End":"04:21.160","Text":"t with respect to x."},{"Start":"04:21.160 ","End":"04:26.185","Text":"Z by t, t by x,"},{"Start":"04:26.185 ","End":"04:31.545","Text":"and we know what t by x is,"},{"Start":"04:31.545 ","End":"04:33.690","Text":"because here it is."},{"Start":"04:33.690 ","End":"04:39.215","Text":"T is this, so the derivative of t with respect to x is just 2x,"},{"Start":"04:39.215 ","End":"04:41.610","Text":"y as a constant."},{"Start":"04:41.990 ","End":"04:50.450","Text":"This is equal to 2 x times c with respect to t."},{"Start":"04:50.450 ","End":"04:56.285","Text":"Similarly, if I want z by y,"},{"Start":"04:56.285 ","End":"04:59.915","Text":"let\u0027s highlight this tree differently,"},{"Start":"04:59.915 ","End":"05:04.735","Text":"this time I\u0027ve highlighted the right path from here,"},{"Start":"05:04.735 ","End":"05:08.730","Text":"that\u0027s how I get z with respect to y,"},{"Start":"05:08.730 ","End":"05:12.555","Text":"and make it z with respect to t,"},{"Start":"05:12.555 ","End":"05:17.030","Text":"this bit, times t with respect to y derivative."},{"Start":"05:17.030 ","End":"05:20.885","Text":"This is equal to t with respect to y,"},{"Start":"05:20.885 ","End":"05:24.390","Text":"x is a constant, so this is 2y,"},{"Start":"05:24.800 ","End":"05:28.977","Text":"and also z with respect to t."},{"Start":"05:28.977 ","End":"05:32.950","Text":"I don\u0027t know what z with respect to t is because I don\u0027t have the function f,"},{"Start":"05:32.950 ","End":"05:36.590","Text":"but I don\u0027t need to know because in order to prove this,"},{"Start":"05:36.590 ","End":"05:39.260","Text":"all we have to do is substitute,"},{"Start":"05:39.260 ","End":"05:41.075","Text":"and let\u0027s see what we get."},{"Start":"05:41.075 ","End":"05:43.849","Text":"Let\u0027s take the left-hand side,"},{"Start":"05:43.849 ","End":"05:48.055","Text":"and we get that this is equal to,"},{"Start":"05:48.055 ","End":"05:50.255","Text":"I won\u0027t copy it, or maybe I will."},{"Start":"05:50.255 ","End":"05:52.145","Text":"Yes, let\u0027s copy it again."},{"Start":"05:52.145 ","End":"05:57.530","Text":"Minus x, z with respect to y is equal to y."},{"Start":"05:57.530 ","End":"06:00.780","Text":"This I\u0027ll take from here,"},{"Start":"06:00.780 ","End":"06:07.425","Text":"which is this, so I get 2x z with respect to t,"},{"Start":"06:07.425 ","End":"06:13.125","Text":"then minus x, z with respect to y I take from here,"},{"Start":"06:13.125 ","End":"06:16.430","Text":"2y z with respect to t,"},{"Start":"06:16.430 ","End":"06:18.590","Text":"and if I expand this,"},{"Start":"06:18.590 ","End":"06:25.520","Text":"here I get 2xy z by t,"},{"Start":"06:25.520 ","End":"06:29.830","Text":"minus x 2 y is 2xy,"},{"Start":"06:29.830 ","End":"06:36.980","Text":"also z by t. This term and this term are exactly the same,"},{"Start":"06:36.980 ","End":"06:41.655","Text":"so this minus itself is equal to 0,"},{"Start":"06:41.655 ","End":"06:44.055","Text":"and that\u0027s what we had to prove."},{"Start":"06:44.055 ","End":"06:47.085","Text":"We write the Latin, Q-E-D,"},{"Start":"06:47.085 ","End":"06:50.985","Text":"that which was required to prove quod erat demonstrandum."},{"Start":"06:50.985 ","End":"06:53.174","Text":"Done with this exercise."},{"Start":"06:53.174 ","End":"06:57.950","Text":"Just like to emphasize how the chain rule came to"},{"Start":"06:57.950 ","End":"07:03.680","Text":"our help in proving all sorts of general claims about general functions,"},{"Start":"07:03.680 ","End":"07:05.780","Text":"because f could have been many things,"},{"Start":"07:05.780 ","End":"07:08.720","Text":"any function I take of x-squared plus y-squared,"},{"Start":"07:08.720 ","End":"07:10.900","Text":"and this will hold true."},{"Start":"07:10.900 ","End":"07:15.120","Text":"Okay. Let\u0027s go on to another example similar to this."}],"ID":8960},{"Watched":false,"Name":"The Chain Rule 3","Duration":"7m 29s","ChapterTopicVideoID":8615,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.215","Text":"Here\u0027s another example exercise regarding the chain rule."},{"Start":"00:04.215 ","End":"00:06.570","Text":"It\u0027s a bit similar to the previous 1."},{"Start":"00:06.570 ","End":"00:11.880","Text":"Given z equals a function of this time,"},{"Start":"00:11.880 ","End":"00:19.650","Text":"x plus y over x minus y."},{"Start":"00:19.650 ","End":"00:27.830","Text":"What we have to show is that x times the partial derivative of z with"},{"Start":"00:27.830 ","End":"00:37.040","Text":"respect to x plus y times the derivative of z with respect to y equals 0."},{"Start":"00:37.040 ","End":"00:39.560","Text":"We don\u0027t know what the function f is."},{"Start":"00:39.560 ","End":"00:42.725","Text":"It could be anything, I can now give you an example."},{"Start":"00:42.725 ","End":"00:52.040","Text":"We could have that z equals x plus y over x minus y"},{"Start":"00:52.040 ","End":"01:00.590","Text":"squared times the natural log of x plus y over x minus y"},{"Start":"01:00.590 ","End":"01:10.550","Text":"over the square root of x plus y over x minus y plus 4."},{"Start":"01:10.550 ","End":"01:13.670","Text":"For instance, it could be something much simpler,"},{"Start":"01:13.670 ","End":"01:20.200","Text":"perhaps just z equals natural log of x plus y over x minus y could be,"},{"Start":"01:20.200 ","End":"01:21.765","Text":"we don\u0027t know what it is,"},{"Start":"01:21.765 ","End":"01:24.155","Text":"but regardless of what f is,"},{"Start":"01:24.155 ","End":"01:27.005","Text":"this equality will hold,"},{"Start":"01:27.005 ","End":"01:28.760","Text":"and this is what we have to show."},{"Start":"01:28.760 ","End":"01:34.594","Text":"Now we\u0027ll go to the solution and as before,"},{"Start":"01:34.594 ","End":"01:38.240","Text":"the main tool that we\u0027ll be using will be the chain rule."},{"Start":"01:38.240 ","End":"01:44.255","Text":"I want to stress this concept of being a function of x plus y over x minus y."},{"Start":"01:44.255 ","End":"01:46.340","Text":"Z is a function of x and y,"},{"Start":"01:46.340 ","End":"01:53.675","Text":"but x and y don\u0027t appear just as x or y or even xy or even x squared or x over y,"},{"Start":"01:53.675 ","End":"01:58.850","Text":"it always appears in parcels of x plus y over x minus y."},{"Start":"01:58.850 ","End":"02:04.620","Text":"So the function is of this and here I have it once,"},{"Start":"02:04.620 ","End":"02:06.105","Text":"here I have it again,"},{"Start":"02:06.105 ","End":"02:12.070","Text":"it\u0027s never separate that\u0027s what makes it a function of this, away with this."},{"Start":"02:12.070 ","End":"02:14.900","Text":"So as in the previous exercise,"},{"Start":"02:14.900 ","End":"02:18.665","Text":"we\u0027ll create a new variable, call it t,"},{"Start":"02:18.665 ","End":"02:23.000","Text":"which is equal to x plus y over"},{"Start":"02:23.000 ","End":"02:30.785","Text":"x minus y then we get that f is a function of t. In other words,"},{"Start":"02:30.785 ","End":"02:33.965","Text":"z is just f of t,"},{"Start":"02:33.965 ","End":"02:40.970","Text":"because this is in fact just t. It\u0027s time to do our dependency tree."},{"Start":"02:40.970 ","End":"02:45.725","Text":"At the very top of the tree we\u0027ll have z."},{"Start":"02:45.725 ","End":"02:52.265","Text":"z depends on t directly,"},{"Start":"02:52.265 ","End":"02:57.005","Text":"but t depends on x and y."},{"Start":"02:57.005 ","End":"03:02.000","Text":"So z depends indirectly on x and on y."},{"Start":"03:02.000 ","End":"03:06.620","Text":"What we have to do is we\u0027re going to compute z with respect"},{"Start":"03:06.620 ","End":"03:11.135","Text":"to x derivative and the partial derivative of z with respect to y."},{"Start":"03:11.135 ","End":"03:13.115","Text":"So let\u0027s start with one of them."},{"Start":"03:13.115 ","End":"03:16.354","Text":"Let\u0027s start with z by x."},{"Start":"03:16.354 ","End":"03:21.800","Text":"Remember what we do, we see the path on the tree to get from z to x,"},{"Start":"03:21.800 ","End":"03:27.560","Text":"and we go down here and then along here."},{"Start":"03:27.560 ","End":"03:31.280","Text":"What we do is each path along the tree,"},{"Start":"03:31.280 ","End":"03:37.280","Text":"we multiply the branches that make it up so that this is equal to z by"},{"Start":"03:37.280 ","End":"03:45.410","Text":"t times t by x, the partial derivatives."},{"Start":"03:45.410 ","End":"03:51.305","Text":"We can\u0027t compute z by t because we don\u0027t know what the function f is but we can compute"},{"Start":"03:51.305 ","End":"03:58.010","Text":"t by x because all we have to do is differentiate this with respect to x."},{"Start":"03:58.010 ","End":"04:01.220","Text":"We have x on the numerator and x on the denominator."},{"Start":"04:01.220 ","End":"04:03.440","Text":"We\u0027ll use the quotient rule,"},{"Start":"04:03.440 ","End":"04:06.220","Text":"u over v derivative,"},{"Start":"04:06.220 ","End":"04:08.660","Text":"I like to start with the denominator squared,"},{"Start":"04:08.660 ","End":"04:11.930","Text":"and then it\u0027s the derivative of the numerator times"},{"Start":"04:11.930 ","End":"04:17.095","Text":"denominator minus the numerator times the derivative of the denominator."},{"Start":"04:17.095 ","End":"04:19.590","Text":"In our case, we\u0027ll get,"},{"Start":"04:19.590 ","End":"04:25.115","Text":"we keep the z by t. We just differentiate this and there\u0027s where we get the quotient"},{"Start":"04:25.115 ","End":"04:32.760","Text":"x minus y squared then derivative of numerator is 1,"},{"Start":"04:32.760 ","End":"04:36.034","Text":"remember we\u0027re differentiating with respect to x."},{"Start":"04:36.034 ","End":"04:37.650","Text":"So y is a constant,"},{"Start":"04:37.650 ","End":"04:42.770","Text":"times the denominator x minus y minus"},{"Start":"04:42.770 ","End":"04:49.700","Text":"the numerator x plus y times the derivative of the denominator,"},{"Start":"04:49.700 ","End":"04:58.525","Text":"which is 1, and x minus y less x plus y is just minus 2y."},{"Start":"04:58.525 ","End":"05:08.010","Text":"So ultimately we get minus 2y over x minus y squared."},{"Start":"05:08.010 ","End":"05:12.365","Text":"Now similarly, the partial of z by y."},{"Start":"05:12.365 ","End":"05:14.625","Text":"We have to modify this tree."},{"Start":"05:14.625 ","End":"05:17.614","Text":"This time we go on the other side."},{"Start":"05:17.614 ","End":"05:24.295","Text":"That\u0027s how we get from z to y. z by t,"},{"Start":"05:24.295 ","End":"05:31.550","Text":"t by y and this is equal z by t. This time if I differentiate with respect to y,"},{"Start":"05:31.550 ","End":"05:32.690","Text":"I get something similar,"},{"Start":"05:32.690 ","End":"05:34.100","Text":"but not quite the same."},{"Start":"05:34.100 ","End":"05:42.020","Text":"1 times x minus y minus the numerator as is,"},{"Start":"05:42.020 ","End":"05:51.590","Text":"is x plus y and the derivative of the denominator with respect to y is minus 1."},{"Start":"05:51.590 ","End":"05:55.900","Text":"The only difference between these 2 is the 1 and the minus 1 and of course,"},{"Start":"05:55.900 ","End":"05:58.595","Text":"I need the denominator squared."},{"Start":"05:58.595 ","End":"06:00.965","Text":"So if it\u0027s minus 1,"},{"Start":"06:00.965 ","End":"06:03.725","Text":"then minus with this minus becomes plus,"},{"Start":"06:03.725 ","End":"06:07.295","Text":"so it\u0027s x minus y plus x plus y."},{"Start":"06:07.295 ","End":"06:10.790","Text":"So this gives me plus 2x."},{"Start":"06:10.790 ","End":"06:19.310","Text":"So we get 2x over x minus y squared."},{"Start":"06:19.310 ","End":"06:23.990","Text":"So this is my zx and this is the zy."},{"Start":"06:23.990 ","End":"06:26.020","Text":"Maybe I should highlight them."},{"Start":"06:26.020 ","End":"06:29.745","Text":"Okay, where this was zx, this was zy."},{"Start":"06:29.745 ","End":"06:33.859","Text":"Now finally we can get to show this."},{"Start":"06:33.859 ","End":"06:35.735","Text":"Let\u0027s evaluate it."},{"Start":"06:35.735 ","End":"06:41.240","Text":"So xz with respect to x plus y,"},{"Start":"06:41.240 ","End":"06:47.510","Text":"z by y is equal to x. I multiply by this."},{"Start":"06:47.510 ","End":"06:50.270","Text":"And so I get minus 2."},{"Start":"06:50.270 ","End":"06:52.190","Text":"Let me stick the x in here,"},{"Start":"06:52.190 ","End":"06:57.770","Text":"xy over x minus y squared,"},{"Start":"06:57.770 ","End":"07:02.445","Text":"and then plus y times zy."},{"Start":"07:02.445 ","End":"07:04.320","Text":"So y times this,"},{"Start":"07:04.320 ","End":"07:06.155","Text":"we\u0027ll put the y after the x."},{"Start":"07:06.155 ","End":"07:12.815","Text":"So I get 2xy over x minus y squared and look,"},{"Start":"07:12.815 ","End":"07:16.070","Text":"this is just the same thing with opposite signs."},{"Start":"07:16.070 ","End":"07:18.560","Text":"Here\u0027s a negative and here\u0027s a positive."},{"Start":"07:18.560 ","End":"07:21.260","Text":"So the sum is equal to 0,"},{"Start":"07:21.260 ","End":"07:23.409","Text":"which is what we had to prove."},{"Start":"07:23.409 ","End":"07:25.910","Text":"In mathematics, we write QED."},{"Start":"07:25.910 ","End":"07:30.390","Text":"Okay, we\u0027re done with this exercise and on to the next."}],"ID":8961},{"Watched":false,"Name":"The Chain Rule 4","Duration":"7m 16s","ChapterTopicVideoID":8616,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"Another example exercise. This time,"},{"Start":"00:04.020 ","End":"00:14.999","Text":"we\u0027re given that Z equals a function of x squared minus y squared,"},{"Start":"00:14.999 ","End":"00:19.815","Text":"y squared minus x squared."},{"Start":"00:19.815 ","End":"00:22.380","Text":"As opposed to the previous exercise,"},{"Start":"00:22.380 ","End":"00:24.645","Text":"this is a function in 2 variables,"},{"Start":"00:24.645 ","End":"00:28.170","Text":"the comma shows us the first variable, second variable."},{"Start":"00:28.170 ","End":"00:37.950","Text":"What we have to show or to prove is the following identity x times"},{"Start":"00:37.950 ","End":"00:44.045","Text":"the derivative of Z with respect to x. I mean Z plus"},{"Start":"00:44.045 ","End":"00:52.100","Text":"y times the derivative of Z with respect to y is equal to 0,"},{"Start":"00:52.100 ","End":"00:55.085","Text":"and the next will be solution."},{"Start":"00:55.085 ","End":"00:57.140","Text":"But before the solution,"},{"Start":"00:57.140 ","End":"01:00.290","Text":"I\u0027d like to give you an example of what such a function might look"},{"Start":"01:00.290 ","End":"01:04.805","Text":"like depending on what f is it could be something like,"},{"Start":"01:04.805 ","End":"01:10.040","Text":"as you see, it doesn\u0027t have just x and y in any form it\u0027s always"},{"Start":"01:10.040 ","End":"01:15.260","Text":"either in the form x squared minus y squared or y squared minus x squared."},{"Start":"01:15.260 ","End":"01:24.700","Text":"For example, the x-squared minus y-squared part would appear here, and also here."},{"Start":"01:25.060 ","End":"01:31.465","Text":"The y squared minus x squared do that in a different color would be here,"},{"Start":"01:31.465 ","End":"01:34.840","Text":"would be here, and here."},{"Start":"01:34.840 ","End":"01:38.455","Text":"Now, in the previous exercise,"},{"Start":"01:38.455 ","End":"01:43.750","Text":"if you recall, what we did was to substitute t equals something."},{"Start":"01:43.750 ","End":"01:50.590","Text":"This time we\u0027ll need 2 variables for substituting separately for the yellow,"},{"Start":"01:50.590 ","End":"01:52.390","Text":"well, for the x-squared minus y-squared,"},{"Start":"01:52.390 ","End":"01:53.770","Text":"and separately for this."},{"Start":"01:53.770 ","End":"01:54.880","Text":"I\u0027ll need 2 letters,"},{"Start":"01:54.880 ","End":"02:02.095","Text":"let\u0027s take u and v. We\u0027ll take u is equal to x squared minus y squared."},{"Start":"02:02.095 ","End":"02:05.605","Text":"We\u0027ll substitute v equals the other 1,"},{"Start":"02:05.605 ","End":"02:08.665","Text":"the y squared minus x squared,"},{"Start":"02:08.665 ","End":"02:17.350","Text":"and then what we\u0027ll get is that Z will equal f of u,"},{"Start":"02:17.350 ","End":"02:22.590","Text":"v. It\u0027s time for our dependence tree I\u0027ll erase this,"},{"Start":"02:22.590 ","End":"02:27.195","Text":"so at the top of the tree will have Z."},{"Start":"02:27.195 ","End":"02:32.710","Text":"Z depends on u, and on v,"},{"Start":"02:32.720 ","End":"02:35.420","Text":"but if you look at u and v,"},{"Start":"02:35.420 ","End":"02:40.060","Text":"each of them depends both on x and on y,"},{"Start":"02:40.060 ","End":"02:43.800","Text":"on x and on y."},{"Start":"02:43.800 ","End":"02:45.500","Text":"In order to show this,"},{"Start":"02:45.500 ","End":"02:48.830","Text":"we have to compute both this and this."},{"Start":"02:48.830 ","End":"02:55.190","Text":"Let\u0027s start with the derivative of Z with respect to x,"},{"Start":"02:55.190 ","End":"03:01.470","Text":"and what this will equal if I highlight with respect to x,"},{"Start":"03:01.470 ","End":"03:04.230","Text":"I can go this way,"},{"Start":"03:04.230 ","End":"03:09.135","Text":"and this way, or I can go this way,"},{"Start":"03:09.135 ","End":"03:11.880","Text":"and this way to get to x,"},{"Start":"03:11.880 ","End":"03:14.400","Text":"there\u0027s this 2 paths."},{"Start":"03:14.400 ","End":"03:19.025","Text":"We have this something times something plus something times something,"},{"Start":"03:19.025 ","End":"03:22.170","Text":"and that comes out to be Z by u,"},{"Start":"03:22.170 ","End":"03:27.185","Text":"u by x derivative of Z with respect to u,"},{"Start":"03:27.185 ","End":"03:31.355","Text":"u by x plus for the 2 branches."},{"Start":"03:31.355 ","End":"03:34.920","Text":"It\u0027s Z by v,"},{"Start":"03:34.920 ","End":"03:39.360","Text":"v by x, and this equals."},{"Start":"03:39.360 ","End":"03:45.360","Text":"Now u by x is just 2x because y is like a constant."},{"Start":"03:45.360 ","End":"03:49.620","Text":"It\u0027s Z by u times 2x"},{"Start":"03:49.620 ","End":"03:54.545","Text":"plus Z by v I can\u0027t do that because I don\u0027t know what the function is."},{"Start":"03:54.545 ","End":"03:59.260","Text":"Then v by x is minus 2x"},{"Start":"03:59.260 ","End":"04:05.880","Text":"I could take 2x outside the brackets we\u0027ll see if it\u0027s useful or not."},{"Start":"04:05.880 ","End":"04:08.790","Text":"Then Z by y well,"},{"Start":"04:08.790 ","End":"04:10.770","Text":"just have to modify the paths here,"},{"Start":"04:10.770 ","End":"04:14.165","Text":"this time we take the ones that end in y,"},{"Start":"04:14.165 ","End":"04:16.760","Text":"and so we get Z by u,"},{"Start":"04:16.760 ","End":"04:25.670","Text":"U by y, plus it\u0027s basically the same except wherever we had x, we now have y."},{"Start":"04:25.670 ","End":"04:28.670","Text":"This also equals Z by u nothing to be done,"},{"Start":"04:28.670 ","End":"04:32.760","Text":"u by y is minus 2y,"},{"Start":"04:35.390 ","End":"04:43.470","Text":"and v by y is 2y bit lengthy,"},{"Start":"04:43.470 ","End":"04:47.659","Text":"and then we have to substitute in here and see that we get 0."},{"Start":"04:47.659 ","End":"04:54.680","Text":"I have the feeling it might be useful to take the 2x outside the brackets here."},{"Start":"04:54.680 ","End":"04:57.720","Text":"I\u0027ll get 2x times,"},{"Start":"04:57.720 ","End":"05:02.050","Text":"let\u0027s see, Z_u minus Z_v,"},{"Start":"05:04.820 ","End":"05:10.400","Text":"and if we take 2y outside the brackets here,"},{"Start":"05:10.400 ","End":"05:14.795","Text":"we\u0027re left with minus Z_u plus"},{"Start":"05:14.795 ","End":"05:24.395","Text":"Z_v I just realized that I copied this thing wrongly let me just fix it."},{"Start":"05:24.395 ","End":"05:28.580","Text":"YZ_x minus xZ_y, also I just noticed"},{"Start":"05:28.580 ","End":"05:32.950","Text":"something this thing is the same as this thing just in reverse."},{"Start":"05:32.950 ","End":"05:37.715","Text":"Instead of this, I could put the minus in front and reverse these,"},{"Start":"05:37.715 ","End":"05:41.935","Text":"and rewrite it as minus 2y."},{"Start":"05:41.935 ","End":"05:45.390","Text":"Now, it will also be Z_u minus Z_v."},{"Start":"05:45.390 ","End":"05:54.840","Text":"Finally, we\u0027re substituting, so we start with yZ_x minus xZ_y,"},{"Start":"05:54.840 ","End":"06:02.300","Text":"and this is equal to y times that Z_x in terms of Z_u and Z_v is from here."},{"Start":"06:02.300 ","End":"06:06.160","Text":"It\u0027s y times 2x"},{"Start":"06:06.160 ","End":"06:13.845","Text":"times Z_u minus Z_v and then minus."},{"Start":"06:13.845 ","End":"06:23.440","Text":"The other 1 is minus 2y, Z_u minus Z_v."},{"Start":"06:23.470 ","End":"06:26.210","Text":"Allow me to change this minus,"},{"Start":"06:26.210 ","End":"06:29.015","Text":"minus to a plus,"},{"Start":"06:29.015 ","End":"06:31.625","Text":"and also I forgot this x here."},{"Start":"06:31.625 ","End":"06:38.505","Text":"Now look, this is the same as this y times 2x is 2xy,"},{"Start":"06:38.505 ","End":"06:42.100","Text":"and this thing here is also 2xy."},{"Start":"06:43.280 ","End":"06:48.150","Text":"I must really apologize today sounds like I I still miscopied it,"},{"Start":"06:48.150 ","End":"06:50.719","Text":"that this is supposed to be a plus."},{"Start":"06:50.719 ","End":"06:53.945","Text":"I have to make it go back to be a minus,"},{"Start":"06:53.945 ","End":"06:56.150","Text":"this is exactly the same thing as this,"},{"Start":"06:56.150 ","End":"06:57.980","Text":"but there\u0027s a minus sign,"},{"Start":"06:57.980 ","End":"06:59.900","Text":"and if there\u0027s a minus sign,"},{"Start":"06:59.900 ","End":"07:05.410","Text":"then definitely when we add something and it\u0027s negative, we get 0."},{"Start":"07:05.410 ","End":"07:09.405","Text":"That\u0027s what we have to show the 0 here,"},{"Start":"07:09.405 ","End":"07:12.795","Text":"Q-E-D as we say in Latin."},{"Start":"07:12.795 ","End":"07:16.840","Text":"End of this exercise and on to the next."}],"ID":8962},{"Watched":false,"Name":"The Chain Rule 5","Duration":"7m 26s","ChapterTopicVideoID":8617,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:03.585","Text":"Continuing with the chain rule,"},{"Start":"00:03.585 ","End":"00:06.885","Text":"a slightly more difficult example now."},{"Start":"00:06.885 ","End":"00:14.550","Text":"What we are given is that z equals x"},{"Start":"00:14.550 ","End":"00:23.350","Text":"squared plus y squared times a function of x/y."},{"Start":"00:25.070 ","End":"00:31.410","Text":"What we have to prove is"},{"Start":"00:31.410 ","End":"00:39.750","Text":"that x times z with respect to x,"},{"Start":"00:39.750 ","End":"00:46.389","Text":"plus y times derivative of z with respect to y,"},{"Start":"00:46.389 ","End":"00:49.910","Text":"instead of equaling 0 as in the previous 2 exercises,"},{"Start":"00:49.910 ","End":"00:53.670","Text":"this time it\u0027s got to equal 2z."},{"Start":"00:55.630 ","End":"00:59.435","Text":"Let\u0027s see how we deal with this variation."},{"Start":"00:59.435 ","End":"01:04.310","Text":"But I\u0027d like to just illustrate what z might look like."},{"Start":"01:04.310 ","End":"01:06.800","Text":"After all, we don\u0027t know what f is,"},{"Start":"01:06.800 ","End":"01:10.175","Text":"so I\u0027m just giving you an example of for instance."},{"Start":"01:10.175 ","End":"01:18.200","Text":"It could be that z equals x squared plus y squared."},{"Start":"01:18.810 ","End":"01:21.590","Text":"It could look like this,"},{"Start":"01:21.590 ","End":"01:24.830","Text":"where this thing is the function of x/y."},{"Start":"01:24.830 ","End":"01:28.160","Text":"We have x/y here, and over here,"},{"Start":"01:28.160 ","End":"01:30.110","Text":"we have just x/y,"},{"Start":"01:30.110 ","End":"01:33.575","Text":"no separate x\u0027s or y\u0027s or x squared or anything."},{"Start":"01:33.575 ","End":"01:38.600","Text":"Just x/y, x/y, x/y."},{"Start":"01:38.600 ","End":"01:41.645","Text":"Let\u0027s start with the solution."},{"Start":"01:41.645 ","End":"01:44.510","Text":"As usual, we do some kind of substitution,"},{"Start":"01:44.510 ","End":"01:47.435","Text":"and this time, I\u0027m going to substitute for x/y,"},{"Start":"01:47.435 ","End":"01:52.670","Text":"and we let it equal t. If t is x/y,"},{"Start":"01:52.670 ","End":"01:57.440","Text":"that means that z is equal to"},{"Start":"01:57.440 ","End":"02:03.620","Text":"x squared plus y squared times f of t,"},{"Start":"02:03.620 ","End":"02:09.964","Text":"because x/y is t. Now it\u0027s time to build our tree."},{"Start":"02:09.964 ","End":"02:12.725","Text":"Thing is, and this is very important."},{"Start":"02:12.725 ","End":"02:14.665","Text":"When we draw the tree,"},{"Start":"02:14.665 ","End":"02:19.190","Text":"we don\u0027t draw z at the top of the tree,"},{"Start":"02:19.190 ","End":"02:21.770","Text":"we choose f at the top."},{"Start":"02:21.770 ","End":"02:28.325","Text":"We always at the top of the tree put the function for which we did a substitution."},{"Start":"02:28.325 ","End":"02:32.090","Text":"If I substituted the variables of f, which is what I did,"},{"Start":"02:32.090 ","End":"02:34.280","Text":"I replaced x over y by t,"},{"Start":"02:34.280 ","End":"02:36.485","Text":"f stands at the top of the tree."},{"Start":"02:36.485 ","End":"02:43.355","Text":"F depends at the moment on t. I\u0027m looking at this here."},{"Start":"02:43.355 ","End":"02:46.085","Text":"F depends on t,"},{"Start":"02:46.085 ","End":"02:50.055","Text":"but t, which is x/y,"},{"Start":"02:50.055 ","End":"02:57.360","Text":"depends both on x and on y. F indirectly depends on x and y,"},{"Start":"02:57.360 ","End":"03:00.085","Text":"and we can use the chain rule."},{"Start":"03:00.085 ","End":"03:06.760","Text":"Let\u0027s first of all do the derivative of f by x. I\u0027ll take this path"},{"Start":"03:06.760 ","End":"03:14.540","Text":"down and I can say that f by x is equal to f by t,"},{"Start":"03:14.540 ","End":"03:19.550","Text":"t by x. I don\u0027t know what the function f is."},{"Start":"03:19.550 ","End":"03:20.960","Text":"There\u0027s nothing I can do there."},{"Start":"03:20.960 ","End":"03:25.520","Text":"But t by x is the derivative of t with respect to x."},{"Start":"03:25.520 ","End":"03:28.340","Text":"It\u0027s just 1/y."},{"Start":"03:28.340 ","End":"03:29.550","Text":"Y is a constant,"},{"Start":"03:29.550 ","End":"03:31.505","Text":"as far as that is concerned."},{"Start":"03:31.505 ","End":"03:34.140","Text":"If I want f with respect to,"},{"Start":"03:34.140 ","End":"03:35.705","Text":"I like it on the same line,"},{"Start":"03:35.705 ","End":"03:37.790","Text":"f with respect to y,"},{"Start":"03:37.790 ","End":"03:43.620","Text":"this is equal to just using a different path, we\u0027ll get f by t,"},{"Start":"03:43.620 ","End":"03:51.900","Text":"t by y, which equals f by t and t by y is x is a constant this time."},{"Start":"03:51.900 ","End":"03:55.785","Text":"The derivative of 1/y is minus 1/y squared."},{"Start":"03:55.785 ","End":"04:01.355","Text":"We get minus x/y squared."},{"Start":"04:01.355 ","End":"04:03.440","Text":"Now I\u0027ll need these in a moment."},{"Start":"04:03.440 ","End":"04:06.620","Text":"This f by x and f by y,"},{"Start":"04:06.620 ","End":"04:08.120","Text":"we\u0027ll soon need them."},{"Start":"04:08.120 ","End":"04:11.225","Text":"Because when I start to show this, you\u0027ll see."},{"Start":"04:11.225 ","End":"04:13.730","Text":"Let\u0027s start to prove this."},{"Start":"04:13.730 ","End":"04:17.765","Text":"We\u0027ll start from the left-hand side and see if we reach the right-hand side."},{"Start":"04:17.765 ","End":"04:23.115","Text":"X times z by x,"},{"Start":"04:23.115 ","End":"04:27.730","Text":"plus y times z by y,"},{"Start":"04:27.730 ","End":"04:30.440","Text":"is equal to x times,"},{"Start":"04:30.440 ","End":"04:32.975","Text":"now z by x is,"},{"Start":"04:32.975 ","End":"04:35.090","Text":"I have to differentiate this,"},{"Start":"04:35.090 ","End":"04:37.810","Text":"the partial derivative with respect to x."},{"Start":"04:37.810 ","End":"04:42.110","Text":"This is 2x, because x is the variable,"},{"Start":"04:42.110 ","End":"04:43.340","Text":"y is the constant,"},{"Start":"04:43.340 ","End":"04:47.075","Text":"plus y squared f by x."},{"Start":"04:47.075 ","End":"04:49.910","Text":"But this equals f by"},{"Start":"04:49.910 ","End":"04:58.025","Text":"t times minus x/y squared plus y."},{"Start":"04:58.025 ","End":"04:59.840","Text":"The x squared gives nothing,"},{"Start":"04:59.840 ","End":"05:02.000","Text":"but this is a product."},{"Start":"05:02.000 ","End":"05:06.770","Text":"The derivative of this times this,"},{"Start":"05:06.770 ","End":"05:11.710","Text":"abbreviated to f, plus this as is,"},{"Start":"05:11.710 ","End":"05:15.605","Text":"times the derivative of this with respect to y,"},{"Start":"05:15.605 ","End":"05:19.175","Text":"which is f with respect to t,"},{"Start":"05:19.175 ","End":"05:23.735","Text":"and minus x/y squared."},{"Start":"05:23.735 ","End":"05:30.140","Text":"I just realized I copied the wrong thing for f with respect to x. I went over here,"},{"Start":"05:30.140 ","End":"05:32.090","Text":"I should have done this this."},{"Start":"05:32.090 ","End":"05:35.540","Text":"This bit is not minus x/y squared, it\u0027s 1/y."},{"Start":"05:35.540 ","End":"05:37.385","Text":"Excuse me while I correct."},{"Start":"05:37.385 ","End":"05:41.060","Text":"I think my lesson here is not to put 2 things on the same line."},{"Start":"05:41.060 ","End":"05:43.960","Text":"Or if I do, to separate it."},{"Start":"05:43.960 ","End":"05:47.920","Text":"See what we can do about simplifying and expanding this."},{"Start":"05:47.920 ","End":"05:50.540","Text":"Let\u0027s just open up the brackets first of all,"},{"Start":"05:50.540 ","End":"05:51.785","Text":"that might be the easiest."},{"Start":"05:51.785 ","End":"05:56.075","Text":"X times 2x is 2x squared."},{"Start":"05:56.075 ","End":"06:02.105","Text":"Next term is going to be plus x times y squared times 1/y."},{"Start":"06:02.105 ","End":"06:12.200","Text":"What we get is xy times ft,"},{"Start":"06:12.200 ","End":"06:20.520","Text":"it continuing, plus 2y squared f. Let\u0027s see here."},{"Start":"06:20.520 ","End":"06:23.700","Text":"The y squared cancels with the y squared."},{"Start":"06:23.700 ","End":"06:28.545","Text":"So what we\u0027re left with is minus xyft."},{"Start":"06:28.545 ","End":"06:33.165","Text":"Something cancels this with this,"},{"Start":"06:33.165 ","End":"06:38.745","Text":"and what I\u0027m left with is just 2x squared plus"},{"Start":"06:38.745 ","End":"06:45.060","Text":"2y squared f. Now I was lazy to put what f was,"},{"Start":"06:45.060 ","End":"06:47.540","Text":"I just called it f, but it was f of t,"},{"Start":"06:47.540 ","End":"06:50.140","Text":"which is f of x/y,"},{"Start":"06:50.140 ","End":"06:55.520","Text":"and this is exactly twice x squared"},{"Start":"06:55.520 ","End":"07:01.865","Text":"plus y squared f of x/y."},{"Start":"07:01.865 ","End":"07:09.030","Text":"This thing here is exactly equal to z."},{"Start":"07:09.030 ","End":"07:11.650","Text":"It\u0027s equal to 2z,"},{"Start":"07:11.650 ","End":"07:14.515","Text":"which is what we wanted."},{"Start":"07:14.515 ","End":"07:17.260","Text":"We write QED."},{"Start":"07:18.080 ","End":"07:21.460","Text":"That\u0027s another example done,"},{"Start":"07:21.460 ","End":"07:23.910","Text":"a bit more difficult this time."},{"Start":"07:23.910 ","End":"07:26.950","Text":"Onto the last one."}],"ID":8963},{"Watched":false,"Name":"The Chain Rule 6","Duration":"10m 19s","ChapterTopicVideoID":8618,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"Continuing with exercises in the chain rule,"},{"Start":"00:03.750 ","End":"00:06.795","Text":"here\u0027s our last example exercise."},{"Start":"00:06.795 ","End":"00:16.680","Text":"We are given this time that z equals x squared"},{"Start":"00:16.680 ","End":"00:24.450","Text":"times y plus x cubed times some function"},{"Start":"00:24.450 ","End":"00:33.165","Text":"of y minus 4x over 4y plus x."},{"Start":"00:33.165 ","End":"00:42.765","Text":"What we have to show or to prove is that we have the equation that x"},{"Start":"00:42.765 ","End":"00:49.470","Text":"times the partial of z_x plus y times the"},{"Start":"00:49.470 ","End":"00:58.450","Text":"partial of z with respect to y is equal to 3 times z."},{"Start":"00:59.660 ","End":"01:03.900","Text":"The solution is as follows,"},{"Start":"01:03.900 ","End":"01:07.990","Text":"very similar to the previous exercise we did."},{"Start":"01:07.990 ","End":"01:14.390","Text":"Essentially, we substitute this whole mess with a t equals"},{"Start":"01:14.390 ","End":"01:21.030","Text":"this y minus 4x over 4y plus x."},{"Start":"01:21.030 ","End":"01:27.630","Text":"Then it simplifies z to equal x"},{"Start":"01:27.630 ","End":"01:35.465","Text":"squared y plus x cubed times f of t. Now,"},{"Start":"01:35.465 ","End":"01:44.940","Text":"what we\u0027re going to have to do is differentiate z partially by x and by y."},{"Start":"01:44.940 ","End":"01:48.430","Text":"Now, we\u0027re going to need the partial derivatives of"},{"Start":"01:48.430 ","End":"01:54.290","Text":"f. What we\u0027ll do is we\u0027ll draw our dependency tree."},{"Start":"01:55.610 ","End":"01:57.990","Text":"Remember, at the top,"},{"Start":"01:57.990 ","End":"02:01.605","Text":"we put nought z as I\u0027m attempted to do but"},{"Start":"02:01.605 ","End":"02:05.499","Text":"f. We put the function for which we replaced the variable,"},{"Start":"02:05.499 ","End":"02:07.120","Text":"we did the substitution."},{"Start":"02:07.120 ","End":"02:09.845","Text":"f is the top of the tree."},{"Start":"02:09.845 ","End":"02:15.705","Text":"f depends presently only on t. t,"},{"Start":"02:15.705 ","End":"02:20.320","Text":"in its turn, depends on x and on y."},{"Start":"02:21.580 ","End":"02:27.440","Text":"Now, I know we\u0027re going to need f by x and f by y. I like to do it before"},{"Start":"02:27.440 ","End":"02:32.615","Text":"I need it but some people prefer to compute it on demand as needed."},{"Start":"02:32.615 ","End":"02:35.930","Text":"What I\u0027ll do now is say,"},{"Start":"02:35.930 ","End":"02:40.565","Text":"f by x is equal to,"},{"Start":"02:40.565 ","End":"02:46.280","Text":"we take the branch of the tree here and here to get f by x."},{"Start":"02:46.280 ","End":"02:51.610","Text":"It\u0027s f by t, t by x."},{"Start":"02:52.820 ","End":"02:58.380","Text":"Well, I don\u0027t know what the function f is so I can\u0027t differentiate but I can"},{"Start":"02:58.380 ","End":"03:03.795","Text":"say what t by x is because I have this expression for t,"},{"Start":"03:03.795 ","End":"03:06.025","Text":"and t by x would be,"},{"Start":"03:06.025 ","End":"03:14.000","Text":"using the quotient rule to differentiate this partially with respect to x. t by x is,"},{"Start":"03:14.000 ","End":"03:18.655","Text":"from the quotient, I know I always have the denominator squared."},{"Start":"03:18.655 ","End":"03:21.410","Text":"Then I have the derivative of"},{"Start":"03:21.410 ","End":"03:26.570","Text":"the numerator and we\u0027re differentiating by x so it\u0027s minus 4,"},{"Start":"03:26.570 ","End":"03:32.045","Text":"times the denominator as is minus the numerator as is"},{"Start":"03:32.045 ","End":"03:38.165","Text":"times the derivative of the denominator with respect to x."},{"Start":"03:38.165 ","End":"03:41.615","Text":"The derivative of 4y plus x is just 1."},{"Start":"03:41.615 ","End":"03:43.175","Text":"This equal, let\u0027s see,"},{"Start":"03:43.175 ","End":"03:49.195","Text":"minus 16 minus 1 minus 17y."},{"Start":"03:49.195 ","End":"03:50.850","Text":"The x is canceled,"},{"Start":"03:50.850 ","End":"03:55.050","Text":"minus 4X plus 4X is nothing."},{"Start":"03:55.050 ","End":"03:57.900","Text":"This is this over the denominator,"},{"Start":"03:57.900 ","End":"04:03.460","Text":"the 4Y plus x squared."},{"Start":"04:03.950 ","End":"04:08.760","Text":"I know I\u0027ll need f by y."},{"Start":"04:08.760 ","End":"04:14.010","Text":"This is just f by t, t by y."},{"Start":"04:14.010 ","End":"04:23.200","Text":"Yes, I should have highlighted the other branch."},{"Start":"04:23.840 ","End":"04:27.510","Text":"We get f by t, t by y."},{"Start":"04:27.510 ","End":"04:30.375","Text":"This is equal to."},{"Start":"04:30.375 ","End":"04:34.695","Text":"Again, the ft, we don\u0027t know what the function f is so we leave it as it is."},{"Start":"04:34.695 ","End":"04:37.110","Text":"Again, we get the quotient rule,"},{"Start":"04:37.110 ","End":"04:40.750","Text":"we get the same denominator."},{"Start":"04:40.750 ","End":"04:47.465","Text":"Just a little bit different this time because I need to differentiate this,"},{"Start":"04:47.465 ","End":"04:53.410","Text":"not by x this time but by y. I get the derivative"},{"Start":"04:53.410 ","End":"04:59.940","Text":"of the numerator is 1 times denominator,"},{"Start":"04:59.940 ","End":"05:10.425","Text":"4y plus x, minus the derivative of the denominator but this time,"},{"Start":"05:10.425 ","End":"05:13.485","Text":"by y so that\u0027s 4,"},{"Start":"05:13.485 ","End":"05:23.260","Text":"and the numerator as is, y minus 4x."},{"Start":"05:27.830 ","End":"05:33.435","Text":"Let\u0027s see what it comes out to this time."},{"Start":"05:33.435 ","End":"05:38.040","Text":"I get 4y minus 4y, cancels,"},{"Start":"05:38.040 ","End":"05:43.290","Text":"plus x plus 16x plus 17x."},{"Start":"05:43.290 ","End":"05:52.980","Text":"I\u0027ve got 17x over 4y plus x squared."},{"Start":"05:52.980 ","End":"05:58.035","Text":"This is the f by x and f by y,"},{"Start":"05:58.035 ","End":"06:02.050","Text":"and we\u0027ll need them very shortly."},{"Start":"06:02.600 ","End":"06:11.865","Text":"I forgot to copy the f with respect to t and I put it 1 here and 1 here."},{"Start":"06:11.865 ","End":"06:15.485","Text":"Now, we\u0027re going to prove what we have to prove."},{"Start":"06:15.485 ","End":"06:17.885","Text":"Let\u0027s start with the left-hand side."},{"Start":"06:17.885 ","End":"06:22.739","Text":"x, z with respect to x plus y,"},{"Start":"06:22.739 ","End":"06:25.185","Text":"z with respect to y is equal."},{"Start":"06:25.185 ","End":"06:29.040","Text":"We\u0027ll keep working on it till we get the right-hand side."},{"Start":"06:29.150 ","End":"06:32.160","Text":"It\u0027s x. Now, here,"},{"Start":"06:32.160 ","End":"06:34.860","Text":"I\u0027ll see what z with respect to x is."},{"Start":"06:34.860 ","End":"06:38.940","Text":"I look at this and differentiate it with respect to x."},{"Start":"06:38.940 ","End":"06:42.915","Text":"This term gives me just 2xy,"},{"Start":"06:42.915 ","End":"06:47.655","Text":"y is a constant."},{"Start":"06:47.655 ","End":"06:50.685","Text":"Now, I have a product."},{"Start":"06:50.685 ","End":"06:56.370","Text":"It\u0027s 3x squared times this."},{"Start":"06:56.370 ","End":"06:59.110","Text":"I\u0027ll just call it f for short."},{"Start":"06:59.240 ","End":"07:02.290","Text":"Leave the brackets there."},{"Start":"07:02.660 ","End":"07:08.775","Text":"Then x cubed times this"},{"Start":"07:08.775 ","End":"07:12.270","Text":"differentiated but with respect to"},{"Start":"07:12.270 ","End":"07:17.465","Text":"x. I have that here and I go across here, and there it is."},{"Start":"07:17.465 ","End":"07:22.690","Text":"It\u0027s times minus 17y"},{"Start":"07:22.690 ","End":"07:33.525","Text":"over 4y plus x squared times ft plus y,"},{"Start":"07:33.525 ","End":"07:35.205","Text":"that\u0027s this y here,"},{"Start":"07:35.205 ","End":"07:39.000","Text":"times z with respect to y."},{"Start":"07:39.000 ","End":"07:41.640","Text":"With respect to y, here,"},{"Start":"07:41.640 ","End":"07:45.040","Text":"I get just x squared."},{"Start":"07:45.230 ","End":"07:52.880","Text":"Here, I get x cubed times the derivative of this with respect to y."},{"Start":"07:53.360 ","End":"07:57.565","Text":"Now, fy also, we have it prepared in advance."},{"Start":"07:57.565 ","End":"08:05.960","Text":"It\u0027s this thing times 17x over 4y plus x"},{"Start":"08:06.980 ","End":"08:12.810","Text":"squared times f with respect to t. What I"},{"Start":"08:12.810 ","End":"08:21.165","Text":"claim is that this term and this term, these cancel."},{"Start":"08:21.165 ","End":"08:24.155","Text":"Now, here and here,"},{"Start":"08:24.155 ","End":"08:28.820","Text":"I have the same thing with the following difference."},{"Start":"08:28.820 ","End":"08:30.230","Text":"Here\u0027s a minus and here,"},{"Start":"08:30.230 ","End":"08:32.015","Text":"there is no minus."},{"Start":"08:32.015 ","End":"08:35.125","Text":"Here\u0027s a y and here\u0027s an x."},{"Start":"08:35.125 ","End":"08:39.660","Text":"But look, the business with the x and y works out because this x is really xy,"},{"Start":"08:39.660 ","End":"08:41.295","Text":"because there\u0027s a y here."},{"Start":"08:41.295 ","End":"08:46.295","Text":"This y is also really xy because there\u0027s an x there."},{"Start":"08:46.295 ","End":"08:48.170","Text":"Because it\u0027s plus and minus,"},{"Start":"08:48.170 ","End":"08:49.655","Text":"they cancel each other out."},{"Start":"08:49.655 ","End":"08:52.215","Text":"Let\u0027s see what we have left now."},{"Start":"08:52.215 ","End":"08:56.355","Text":"From here, we have 2x squared y."},{"Start":"08:56.355 ","End":"08:59.795","Text":"From here, we have x squared y."},{"Start":"08:59.795 ","End":"09:06.420","Text":"This term and this term gives us 3x squared y."},{"Start":"09:06.420 ","End":"09:11.675","Text":"All that\u0027s left is this term which we have to multiply by x."},{"Start":"09:11.675 ","End":"09:17.915","Text":"What we get is plus 3x cubed"},{"Start":"09:17.915 ","End":"09:25.790","Text":"times f. If I take 3 outside the brackets,"},{"Start":"09:25.790 ","End":"09:34.070","Text":"I get x squared y plus x cubed f. This time,"},{"Start":"09:34.070 ","End":"09:44.105","Text":"I\u0027ll not be lazy and I\u0027ll put what\u0027s in the brackets which is y minus 4x over 4y plus x."},{"Start":"09:44.105 ","End":"09:48.235","Text":"Now, I\u0027d like you to take a look at something."},{"Start":"09:48.235 ","End":"09:54.255","Text":"What\u0027s inside the brackets here and this expression here."},{"Start":"09:54.255 ","End":"09:57.360","Text":"I\u0027d say they\u0027re pretty much the same."},{"Start":"09:57.360 ","End":"09:59.240","Text":"If that\u0027s the case,"},{"Start":"09:59.240 ","End":"10:07.820","Text":"this is equal to z so we finally end up with 3z which is this right-hand side."},{"Start":"10:07.820 ","End":"10:09.200","Text":"We started with the left-hand side,"},{"Start":"10:09.200 ","End":"10:10.730","Text":"ended up with the right-hand side."},{"Start":"10:10.730 ","End":"10:12.560","Text":"This is very good."},{"Start":"10:12.560 ","End":"10:19.470","Text":"This is QED, what we had to show. We are done."}],"ID":8964},{"Watched":false,"Name":"Exercise 1","Duration":"5m 53s","ChapterTopicVideoID":8619,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.899","Text":"Before I start, I\u0027d just like to make a general note that"},{"Start":"00:03.899 ","End":"00:13.170","Text":"this letter Z in America is called zed in England,"},{"Start":"00:13.170 ","End":"00:16.320","Text":"and the American 1 is more common,"},{"Start":"00:16.320 ","End":"00:17.760","Text":"I\u0027m going to say z."},{"Start":"00:17.760 ","End":"00:22.845","Text":"Also, notice that I put a little line through it when I write to distinguish it from a 2,"},{"Start":"00:22.845 ","End":"00:24.270","Text":"and it\u0027s done in Europe."},{"Start":"00:24.270 ","End":"00:26.095","Text":"That\u0027s just to remark."},{"Start":"00:26.095 ","End":"00:28.325","Text":"Now to the exercise,"},{"Start":"00:28.325 ","End":"00:33.055","Text":"we\u0027re given z as a function of x and y here,"},{"Start":"00:33.055 ","End":"00:36.090","Text":"and x is a function of u and v,"},{"Start":"00:36.090 ","End":"00:39.550","Text":"and y is also a function of u and v as given here."},{"Start":"00:39.550 ","End":"00:44.480","Text":"We want to compute the partial derivatives of z with respect to u and"},{"Start":"00:44.480 ","End":"00:51.590","Text":"v. What we\u0027re going to do is use the chain rule in multivariable."},{"Start":"00:51.590 ","End":"00:55.450","Text":"What I\u0027ll do is draw a dependency diagram first."},{"Start":"00:55.450 ","End":"00:58.484","Text":"At the top level, we have z,"},{"Start":"00:58.484 ","End":"01:00.884","Text":"and it depends on 2 things,"},{"Start":"01:00.884 ","End":"01:04.370","Text":"on x, 2 variables on x, and on y."},{"Start":"01:04.370 ","End":"01:06.980","Text":"But x depends on u and v,"},{"Start":"01:06.980 ","End":"01:10.679","Text":"so I draw, this is a tree diagram,"},{"Start":"01:10.679 ","End":"01:12.630","Text":"x depends on u and v depends,"},{"Start":"01:12.630 ","End":"01:16.815","Text":"it means it\u0027s below x depends on u,"},{"Start":"01:16.815 ","End":"01:21.940","Text":"y also depends on u and on v. Indirectly,"},{"Start":"01:21.940 ","End":"01:24.845","Text":"z depends on u and v. Now,"},{"Start":"01:24.845 ","End":"01:30.620","Text":"if we want to compute z partial derivative with respect to u,"},{"Start":"01:30.620 ","End":"01:38.935","Text":"and there\u0027s 2 ways I can get to u. I can go through x and then from x to u,"},{"Start":"01:38.935 ","End":"01:44.090","Text":"or I can go from z to y and then from y to u."},{"Start":"01:44.090 ","End":"01:50.539","Text":"Using this as mnemonic will get that z with respect to u."},{"Start":"01:50.539 ","End":"01:58.190","Text":"First of all, we can get to u through x or z with respect to x times x with respect to u."},{"Start":"01:58.190 ","End":"02:00.935","Text":"Plus the other branch, the path,"},{"Start":"02:00.935 ","End":"02:03.650","Text":"z with respect to y,"},{"Start":"02:03.650 ","End":"02:05.965","Text":"y with respect to u."},{"Start":"02:05.965 ","End":"02:08.960","Text":"Now, of course, we need to compute all of these."},{"Start":"02:08.960 ","End":"02:12.020","Text":"We get z with respect to x."},{"Start":"02:12.020 ","End":"02:14.555","Text":"First of all, now,"},{"Start":"02:14.555 ","End":"02:18.840","Text":"z is natural log of x squared minus y squared,"},{"Start":"02:18.840 ","End":"02:22.805","Text":"and y squared is a constant for the purposes of this."},{"Start":"02:22.805 ","End":"02:25.880","Text":"Because of the log, natural log,"},{"Start":"02:25.880 ","End":"02:29.900","Text":"we get 1 over x squared minus"},{"Start":"02:29.900 ","End":"02:34.405","Text":"y squared times the inner derivative which I put on the top,"},{"Start":"02:34.405 ","End":"02:39.050","Text":"and that\u0027s just 2x because this is y squared is a constant."},{"Start":"02:39.050 ","End":"02:41.270","Text":"Now, x with respect to u,"},{"Start":"02:41.270 ","End":"02:43.690","Text":"I look at this 1, and clearly,"},{"Start":"02:43.690 ","End":"02:52.155","Text":"the answer is just 2 times 2 plus z with respect to y."},{"Start":"02:52.155 ","End":"02:55.609","Text":"From here, again, because of the natural logarithm,"},{"Start":"02:55.609 ","End":"03:00.140","Text":"I get 1 over x squared minus y squared,"},{"Start":"03:00.140 ","End":"03:04.945","Text":"but this time the inner derivative is minus 2y."},{"Start":"03:04.945 ","End":"03:12.780","Text":"Finally, y with respect 2u is just 2u."},{"Start":"03:13.610 ","End":"03:23.135","Text":"Then at the end, we just substitute what x is in terms of u and v. I would write 2,"},{"Start":"03:23.135 ","End":"03:24.530","Text":"and then everywhere I see x,"},{"Start":"03:24.530 ","End":"03:33.290","Text":"I put 2u minus v over 2u minus v squared minus,"},{"Start":"03:33.290 ","End":"03:38.315","Text":"and then y is u squared, not minus, sorry."},{"Start":"03:38.315 ","End":"03:44.685","Text":"Plus v cubed, and this squared,"},{"Start":"03:44.685 ","End":"03:46.800","Text":"all these times 2,"},{"Start":"03:46.800 ","End":"03:49.005","Text":"so I\u0027ll put the 2 here."},{"Start":"03:49.005 ","End":"03:52.020","Text":"Then I\u0027ve got a plus and a minus,"},{"Start":"03:52.020 ","End":"03:54.790","Text":"so I\u0027ll just make it as a minus,"},{"Start":"03:55.040 ","End":"04:06.540","Text":"and then I have 2 times 2 times y is,"},{"Start":"04:06.540 ","End":"04:08.175","Text":"where is why here it is,"},{"Start":"04:08.175 ","End":"04:11.270","Text":"u squared plus v cubed,"},{"Start":"04:11.270 ","End":"04:17.685","Text":"and then u over the same thing here,"},{"Start":"04:17.685 ","End":"04:20.340","Text":"which I just copied."},{"Start":"04:20.340 ","End":"04:23.895","Text":"You could keep going and simplifying it,"},{"Start":"04:23.895 ","End":"04:26.825","Text":"make it nicer, but I\u0027m stopping here."},{"Start":"04:26.825 ","End":"04:30.425","Text":"In fact, in many cases in the future I might even stop here."},{"Start":"04:30.425 ","End":"04:32.420","Text":"Now, the other one,"},{"Start":"04:32.420 ","End":"04:35.900","Text":"z with respect to v. Once again,"},{"Start":"04:35.900 ","End":"04:40.914","Text":"I can get from z to v along 2 different paths."},{"Start":"04:40.914 ","End":"04:42.710","Text":"When I get from z to x,"},{"Start":"04:42.710 ","End":"04:45.065","Text":"then I make a turn from x to v,"},{"Start":"04:45.065 ","End":"04:48.335","Text":"and similarly from z to y and y to v,"},{"Start":"04:48.335 ","End":"04:52.440","Text":"and so here I have Z_x X_v,"},{"Start":"04:52.850 ","End":"04:59.445","Text":"and from the other path Z_y, Y_v."},{"Start":"04:59.445 ","End":"05:03.810","Text":"When you look at this, I could say the x\u0027s cancel goes from z to v,"},{"Start":"05:03.810 ","End":"05:07.050","Text":"the y\u0027s cancel goes from z to v. Now,"},{"Start":"05:07.050 ","End":"05:09.135","Text":"we just need to compute these."},{"Start":"05:09.135 ","End":"05:11.899","Text":"This one we already have."},{"Start":"05:11.899 ","End":"05:16.140","Text":"In fact, these two I can just copy from here and here."},{"Start":"05:16.140 ","End":"05:19.850","Text":"Because this part and this part is the same as this and this,"},{"Start":"05:19.850 ","End":"05:20.950","Text":"so I just copied them."},{"Start":"05:20.950 ","End":"05:27.765","Text":"Now all I need is these two take x with respect to v this time,"},{"Start":"05:27.765 ","End":"05:35.135","Text":"and that from here will give me minus 1 and also y with respect to v this time,"},{"Start":"05:35.135 ","End":"05:39.270","Text":"that will give me here 3v squared."},{"Start":"05:40.250 ","End":"05:45.695","Text":"Then we simplify, I mean, just substitute back."},{"Start":"05:45.695 ","End":"05:49.460","Text":"We substitute x and y from here and so on."},{"Start":"05:49.460 ","End":"05:53.040","Text":"I\u0027m not going to do that. Now we\u0027re done."}],"ID":8965},{"Watched":false,"Name":"Exercise 2","Duration":"7m 28s","ChapterTopicVideoID":8620,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.670","Text":"In this exercise, we have v as a function of t and k,"},{"Start":"00:05.670 ","End":"00:10.500","Text":"and we have u as a function of t and m. Then we have"},{"Start":"00:10.500 ","End":"00:17.474","Text":"z as a function of u and v. We have to compute the partial derivatives of z,"},{"Start":"00:17.474 ","End":"00:20.625","Text":"it\u0027s z everywhere with respect to t, m,"},{"Start":"00:20.625 ","End":"00:28.710","Text":"and k. Notice that we\u0027re using the notation with the funny d. Of course,"},{"Start":"00:28.710 ","End":"00:31.680","Text":"we could have written it z with respect to t,"},{"Start":"00:31.680 ","End":"00:33.495","Text":"z with respect to m,"},{"Start":"00:33.495 ","End":"00:36.260","Text":"z with respect to k partial derivatives,"},{"Start":"00:36.260 ","End":"00:38.955","Text":"but you shouldn\u0027t get used to one notation too much,"},{"Start":"00:38.955 ","End":"00:40.920","Text":"this is an alternative notation."},{"Start":"00:40.920 ","End":"00:43.460","Text":"Let\u0027s keep to this notation with the funny"},{"Start":"00:43.460 ","End":"00:51.085","Text":"d. One thing that helps us in this situation is a dependency tree."},{"Start":"00:51.085 ","End":"00:54.340","Text":"At the top level actually is the z,"},{"Start":"00:54.340 ","End":"01:03.230","Text":"because the z depends on u and v and each of these depends on t,"},{"Start":"01:03.230 ","End":"01:07.310","Text":"m, and k, but not always all of them."},{"Start":"01:07.310 ","End":"01:12.950","Text":"For example, this one depends on t and m. Theoretically,"},{"Start":"01:12.950 ","End":"01:17.330","Text":"you could say it depends on k implicitly,"},{"Start":"01:17.330 ","End":"01:18.980","Text":"but k doesn\u0027t appear."},{"Start":"01:18.980 ","End":"01:28.620","Text":"Similarly, v depends on t and k and m does not appear."},{"Start":"01:28.620 ","End":"01:31.740","Text":"Although theoretically, v could be a function of t, m,"},{"Start":"01:31.740 ","End":"01:36.630","Text":"and k. Anyway, we want these 3,"},{"Start":"01:36.630 ","End":"01:40.395","Text":"so what we do is we say that"},{"Start":"01:40.395 ","End":"01:48.040","Text":"Z with respect to t or rather dz by dt,"},{"Start":"01:48.650 ","End":"01:54.890","Text":"is the sum of 2 things because I can get from z to t first along here,"},{"Start":"01:54.890 ","End":"01:56.810","Text":"and then along here,"},{"Start":"01:56.810 ","End":"02:01.410","Text":"or this way and this way."},{"Start":"02:02.680 ","End":"02:12.995","Text":"This equals dz by du times du by dt."},{"Start":"02:12.995 ","End":"02:16.640","Text":"It\u0027s a bit more writing to do when you use this notation."},{"Start":"02:16.640 ","End":"02:19.850","Text":"This of course is more compact if you get used to this also."},{"Start":"02:19.850 ","End":"02:25.510","Text":"Then get to t the other way through v. We get"},{"Start":"02:34.250 ","End":"02:41.910","Text":"v with respect to t. Let\u0027s see what this equals,"},{"Start":"02:41.910 ","End":"02:45.410","Text":"z with respect to u, I look here."},{"Start":"02:45.410 ","End":"02:47.915","Text":"First of all, I see an exponent."},{"Start":"02:47.915 ","End":"02:54.440","Text":"It\u0027s just e to the power of u minus v times the inner derivative."},{"Start":"02:54.440 ","End":"02:56.600","Text":"Remember u is the variable, v is the constant,"},{"Start":"02:56.600 ","End":"02:59.450","Text":"but the derivative of u minus v is just 1."},{"Start":"02:59.450 ","End":"03:04.665","Text":"Now all this is just this factor here in the first term."},{"Start":"03:04.665 ","End":"03:09.785","Text":"Then du by dt, I have to look over here and I see that it\u0027s 2t,"},{"Start":"03:09.785 ","End":"03:12.125","Text":"because here m is a constant,"},{"Start":"03:12.125 ","End":"03:16.305","Text":"plus z with respect to v,"},{"Start":"03:16.305 ","End":"03:18.065","Text":"so I look here."},{"Start":"03:18.065 ","End":"03:23.780","Text":"Again, I have the e to the power of u minus v. Only this time the v is the variable,"},{"Start":"03:23.780 ","End":"03:27.590","Text":"so the inner derivative is not 1, it\u0027s minus 1."},{"Start":"03:27.590 ","End":"03:34.290","Text":"Then I need still v with respect to t. I look over here and I see that,"},{"Start":"03:34.290 ","End":"03:37.060","Text":"that is equal to 4."},{"Start":"03:37.310 ","End":"03:46.980","Text":"Now, I can simplify it and get e to the power of u minus v. First of all,"},{"Start":"03:46.980 ","End":"03:49.340","Text":"I can take this outside the brackets and what do I get?"},{"Start":"03:49.340 ","End":"03:52.115","Text":"1 times 2t is 2t."},{"Start":"03:52.115 ","End":"03:59.670","Text":"Then minus 1 times 4 is minus 4."},{"Start":"03:59.670 ","End":"04:02.805","Text":"Then at the end,"},{"Start":"04:02.805 ","End":"04:12.740","Text":"I would put u and v from here and here in terms of"},{"Start":"04:12.740 ","End":"04:19.130","Text":"t and m and k. I get e to the power of"},{"Start":"04:19.130 ","End":"04:26.795","Text":"u minus v is this minus this is t squared."},{"Start":"04:26.795 ","End":"04:30.535","Text":"Let me write the minus 4t first."},{"Start":"04:30.535 ","End":"04:39.100","Text":"Then I have the 4m and then minus the k squared."},{"Start":"04:40.100 ","End":"04:46.060","Text":"Here I have 2t minus 4."},{"Start":"04:46.270 ","End":"04:49.010","Text":"That\u0027s just to t minus 4."},{"Start":"04:49.010 ","End":"04:54.500","Text":"Of course, I could take 2 outside and put it as t minus 2."},{"Start":"04:54.500 ","End":"04:57.770","Text":"I don\u0027t have to do that and write it as 2t minus 4."},{"Start":"04:57.770 ","End":"05:01.100","Text":"Anyway, that\u0027s the first one."},{"Start":"05:01.100 ","End":"05:04.460","Text":"The second one, I won\u0027t do all the stages."},{"Start":"05:04.460 ","End":"05:07.595","Text":"Well, and the third also,"},{"Start":"05:07.595 ","End":"05:11.700","Text":"the second one is dz by dm."},{"Start":"05:12.890 ","End":"05:16.335","Text":"We need to modify the path."},{"Start":"05:16.335 ","End":"05:19.730","Text":"There is only one way to get to m,"},{"Start":"05:19.730 ","End":"05:29.465","Text":"it\u0027s through u and then from u to m. This is equal to just z with respect to u"},{"Start":"05:29.465 ","End":"05:32.965","Text":"times u with respect to"},{"Start":"05:32.965 ","End":"05:41.090","Text":"m. We already have z with respect to u, that\u0027s over here."},{"Start":"05:41.090 ","End":"05:49.415","Text":"That\u0027s this bit, it\u0027s just e to the power of u minus v times the 1 which I won\u0027t write."},{"Start":"05:49.415 ","End":"05:52.295","Text":"Then u with respect to m,"},{"Start":"05:52.295 ","End":"05:56.690","Text":"I look over here and see that t is a constant now,"},{"Start":"05:56.690 ","End":"06:01.965","Text":"so it\u0027s just 4 times 4 and put the 4 in front."},{"Start":"06:01.965 ","End":"06:04.265","Text":"Instead of u minus v,"},{"Start":"06:04.265 ","End":"06:07.190","Text":"I can do what I did before."},{"Start":"06:07.190 ","End":"06:11.465","Text":"I\u0027m just not going to continue because it\u0027s just the same,"},{"Start":"06:11.465 ","End":"06:16.290","Text":"just routine, replacing u minus v by this."},{"Start":"06:16.290 ","End":"06:19.425","Text":"Then we have a third one,"},{"Start":"06:19.425 ","End":"06:23.295","Text":"z by k. I\u0027ll just do it over here,"},{"Start":"06:23.295 ","End":"06:30.665","Text":"dz by dk is equal to,"},{"Start":"06:30.665 ","End":"06:33.840","Text":"well, I only have one path here and here."},{"Start":"06:36.550 ","End":"06:41.365","Text":"See the path is here and then here,"},{"Start":"06:41.365 ","End":"06:51.120","Text":"so dz by dv times dv by dk."},{"Start":"06:51.880 ","End":"06:55.130","Text":"We already have dz by dv,"},{"Start":"06:55.130 ","End":"06:59.240","Text":"which is here, which is e to the u minus v minus 1,"},{"Start":"06:59.240 ","End":"07:00.575","Text":"I\u0027ll put the minus in front,"},{"Start":"07:00.575 ","End":"07:08.610","Text":"minus e to the u minus v. Then v with respect to k, and where are we?"},{"Start":"07:09.650 ","End":"07:13.065","Text":"Here we are, v with respect to k,"},{"Start":"07:13.065 ","End":"07:16.050","Text":"t is a constant, so it\u0027s just 2k."},{"Start":"07:16.050 ","End":"07:20.430","Text":"We can bring the 2 in front instead of u minus v,"},{"Start":"07:20.430 ","End":"07:22.200","Text":"do the same thing as here."},{"Start":"07:22.200 ","End":"07:27.520","Text":"I\u0027ll just say and so on and we\u0027re done."}],"ID":8966},{"Watched":false,"Name":"Exercise 3","Duration":"3m 41s","ChapterTopicVideoID":8621,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"In this exercise, we\u0027re given z"},{"Start":"00:03.120 ","End":"00:09.105","Text":"indirectly as a function of x and y but as some function of one variable."},{"Start":"00:09.105 ","End":"00:13.004","Text":"But we substitute x squared minus y squared for that variable,"},{"Start":"00:13.004 ","End":"00:14.970","Text":"where f is some unknown function."},{"Start":"00:14.970 ","End":"00:20.820","Text":"We have to prove that the following equation holds even without knowing what f is."},{"Start":"00:20.820 ","End":"00:25.095","Text":"So, let\u0027s get started on that."},{"Start":"00:25.095 ","End":"00:28.410","Text":"I\u0027d like to just introduce an extra variable."},{"Start":"00:28.410 ","End":"00:31.680","Text":"Let\u0027s call this x squared minus y squared,"},{"Start":"00:31.680 ","End":"00:34.950","Text":"the variable t, because f is a function of something,"},{"Start":"00:34.950 ","End":"00:41.010","Text":"let f be a function of t. We let t equals x squared minus y squared and then we get that"},{"Start":"00:41.010 ","End":"00:48.665","Text":"z is f of t. Now we can draw a dependency tree."},{"Start":"00:48.665 ","End":"00:53.240","Text":"We have z at the top and"},{"Start":"00:53.240 ","End":"00:59.945","Text":"z is equal to f of t so it depends on just t,"},{"Start":"00:59.945 ","End":"01:05.099","Text":"but t depends on x and on y."},{"Start":"01:06.890 ","End":"01:09.045","Text":"If I want to say,"},{"Start":"01:09.045 ","End":"01:12.755","Text":"what is this bit here?"},{"Start":"01:12.755 ","End":"01:15.260","Text":"Partial derivative of z with respect to x,"},{"Start":"01:15.260 ","End":"01:22.140","Text":"I need to take a path down to x and that would be down here and along here."},{"Start":"01:22.870 ","End":"01:28.160","Text":"This is equal to derivative of z with"},{"Start":"01:28.160 ","End":"01:33.260","Text":"respect to t. It\u0027s actually not a partial derivative."},{"Start":"01:33.260 ","End":"01:39.319","Text":"In fact, it\u0027s just a regular derivative but that\u0027s okay because it\u0027s only one variable."},{"Start":"01:39.319 ","End":"01:42.620","Text":"Then t with respect to x, which is,"},{"Start":"01:42.620 ","End":"01:46.420","Text":"I can just compute it, that\u0027s just 2x."},{"Start":"01:46.420 ","End":"01:50.720","Text":"Now, similarly, z with respect to y,"},{"Start":"01:50.720 ","End":"01:56.180","Text":"we need to go along here so delete that and put to highlight here instead."},{"Start":"01:56.180 ","End":"02:02.970","Text":"So, z depends on y through t. We get z with respect to"},{"Start":"02:02.970 ","End":"02:10.765","Text":"t times t with respect to y is minus 2y."},{"Start":"02:10.765 ","End":"02:14.390","Text":"Now, we can\u0027t compute the derivative of z"},{"Start":"02:14.390 ","End":"02:17.270","Text":"with respect to t because we don\u0027t know what the function f is."},{"Start":"02:17.270 ","End":"02:23.704","Text":"I mean, I could write f prime of t but it\u0027s still going to be not specific."},{"Start":"02:23.704 ","End":"02:29.955","Text":"Let\u0027s just see if we have enough to prove the formula. Let\u0027s compute."},{"Start":"02:29.955 ","End":"02:32.025","Text":"To compute an equality,"},{"Start":"02:32.025 ","End":"02:35.010","Text":"we just start with one side and reach the other side."},{"Start":"02:35.010 ","End":"02:36.905","Text":"Let\u0027s start with the left-hand side."},{"Start":"02:36.905 ","End":"02:42.710","Text":"Y times z with respect to x plus x times z with respect to y,"},{"Start":"02:42.710 ","End":"02:44.210","Text":"which is the left-hand side,"},{"Start":"02:44.210 ","End":"02:48.110","Text":"is equal to y times now zx."},{"Start":"02:48.110 ","End":"02:57.230","Text":"Here, it\u0027s zt times 2x plus x times z with respect to y,"},{"Start":"02:57.230 ","End":"03:04.915","Text":"which is from here, which is zt times minus 2y."},{"Start":"03:04.915 ","End":"03:10.180","Text":"Now if we open the brackets, here we get 2xyzt."},{"Start":"03:12.140 ","End":"03:16.370","Text":"Just rearranged here, but here I just take"},{"Start":"03:16.370 ","End":"03:20.135","Text":"the minus out the brackets and then we get minus."},{"Start":"03:20.135 ","End":"03:22.385","Text":"I can also take the 2 first,"},{"Start":"03:22.385 ","End":"03:24.380","Text":"then the x. I can pick the order,"},{"Start":"03:24.380 ","End":"03:27.355","Text":"then the y and then the zt."},{"Start":"03:27.355 ","End":"03:33.805","Text":"Look, this term equals this term just cancels out,"},{"Start":"03:33.805 ","End":"03:37.250","Text":"leaves us with 0, which is the right-hand side."},{"Start":"03:37.250 ","End":"03:42.330","Text":"So yes, we\u0027ve proven it. I\u0027m done."}],"ID":8967},{"Watched":false,"Name":"Exercise 4","Duration":"2m 48s","ChapterTopicVideoID":8622,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"What we have here is a function f and Z"},{"Start":"00:04.860 ","End":"00:08.385","Text":"is that function applied to x, y."},{"Start":"00:08.385 ","End":"00:10.920","Text":"In this case, we have to prove that"},{"Start":"00:10.920 ","End":"00:12.870","Text":"the following equality holds even"},{"Start":"00:12.870 ","End":"00:15.375","Text":"without knowing what the function f is."},{"Start":"00:15.375 ","End":"00:19.710","Text":"Really, f is a function of 1 variable."},{"Start":"00:19.710 ","End":"00:26.960","Text":"Z might be some function of t but instead of t, we substitute x,"},{"Start":"00:26.960 ","End":"00:32.310","Text":"y so we can go via an intermediate variable t."},{"Start":"00:32.310 ","End":"00:35.115","Text":"Now, we want to start proving this."},{"Start":"00:35.115 ","End":"00:40.010","Text":"We need partial derivative of Z with respect to x."},{"Start":"00:40.010 ","End":"00:43.280","Text":"What really helps is a dependency tree."},{"Start":"00:43.280 ","End":"00:48.079","Text":"We see that Z at the top depends only on t,"},{"Start":"00:48.079 ","End":"00:54.065","Text":"but t depends on both x and on y."},{"Start":"00:54.065 ","End":"00:57.320","Text":"If I want Z with respect to x,"},{"Start":"00:57.320 ","End":"01:02.635","Text":"then I go down here and across here."},{"Start":"01:02.635 ","End":"01:07.459","Text":"So I get Z with respect to t,"},{"Start":"01:07.459 ","End":"01:09.410","Text":"which I don\u0027t know."},{"Start":"01:09.410 ","End":"01:11.120","Text":"I could say it\u0027s f prime of t,"},{"Start":"01:11.120 ","End":"01:12.560","Text":"but that doesn\u0027t help."},{"Start":"01:12.560 ","End":"01:14.330","Text":"Then t with respect to x,"},{"Start":"01:14.330 ","End":"01:16.175","Text":"well, I get that from here."},{"Start":"01:16.175 ","End":"01:18.275","Text":"T with respect to x,"},{"Start":"01:18.275 ","End":"01:19.789","Text":"y is like a constant."},{"Start":"01:19.789 ","End":"01:22.050","Text":"It could have been like x times 7 or 7x"},{"Start":"01:22.050 ","End":"01:24.530","Text":"so then we would say that derivative is just 7."},{"Start":"01:24.530 ","End":"01:27.805","Text":"In this case, it would be just y."},{"Start":"01:27.805 ","End":"01:30.005","Text":"That\u0027s 1 part of it."},{"Start":"01:30.005 ","End":"01:31.985","Text":"Now z with respect to y,"},{"Start":"01:31.985 ","End":"01:35.045","Text":"same dependence tree, but we need a different path."},{"Start":"01:35.045 ","End":"01:39.480","Text":"This time we go from Z to t and from t to y."},{"Start":"01:39.710 ","End":"01:49.095","Text":"We get Z with respect to t. Then t with respect to y is just x."},{"Start":"01:49.095 ","End":"01:51.090","Text":"Because x is a constant,"},{"Start":"01:51.090 ","End":"01:53.445","Text":"constant times y so it\u0027s just the constant."},{"Start":"01:53.445 ","End":"01:56.320","Text":"Even though we don\u0027t know what this quantity is,"},{"Start":"01:56.320 ","End":"02:01.500","Text":"we can still substitute because we get x times Z_x,"},{"Start":"02:01.500 ","End":"02:03.400","Text":"I\u0027m checking in the left-hand side to see"},{"Start":"02:03.400 ","End":"02:05.050","Text":"if we get the right-hand side,"},{"Start":"02:05.050 ","End":"02:11.360","Text":"minus y, Z_y, this equals x times."},{"Start":"02:11.360 ","End":"02:18.875","Text":"Now from here I get Z_t times y minus, we get y."},{"Start":"02:18.875 ","End":"02:26.480","Text":"Then Z_y is Z_t times x. This equals 0."},{"Start":"02:26.480 ","End":"02:27.890","Text":"But just in case you can\u0027t see it,"},{"Start":"02:27.890 ","End":"02:30.165","Text":"let me just rearrange the order."},{"Start":"02:30.165 ","End":"02:32.580","Text":"The second write xy times Z_t."},{"Start":"02:32.580 ","End":"02:36.380","Text":"This also I can bring the x in front, xyZ_t."},{"Start":"02:37.490 ","End":"02:42.335","Text":"Then we have something minus itself, so it cancels."},{"Start":"02:42.335 ","End":"02:45.580","Text":"This is equal to 0 and that is the right-hand side."},{"Start":"02:45.580 ","End":"02:48.080","Text":"This, we\u0027ve proven it."}],"ID":8968},{"Watched":false,"Name":"Exercise 5","Duration":"4m 6s","ChapterTopicVideoID":8623,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise, we\u0027re given that z is some function f we don\u0027t know,"},{"Start":"00:04.590 ","End":"00:06.690","Text":"but of x over y,"},{"Start":"00:06.690 ","End":"00:09.340","Text":"where x and y are 2 variables."},{"Start":"00:09.340 ","End":"00:13.890","Text":"We have to prove that the following equality holds."},{"Start":"00:13.890 ","End":"00:16.350","Text":"Let\u0027s give a name,"},{"Start":"00:16.350 ","End":"00:20.250","Text":"a letter for the independent variable of f, we say,"},{"Start":"00:20.250 ","End":"00:28.740","Text":"t. We can say that z is some function of dummy variable t. In our case,"},{"Start":"00:28.740 ","End":"00:32.380","Text":"t is equal to x over y."},{"Start":"00:32.720 ","End":"00:36.990","Text":"Now we need the partial derivatives."},{"Start":"00:36.990 ","End":"00:42.015","Text":"We need z_x and we need z_y so let\u0027s draw a dependency tree."},{"Start":"00:42.015 ","End":"00:45.060","Text":"At the top, we put z."},{"Start":"00:45.060 ","End":"00:47.919","Text":"It depends only on t,"},{"Start":"00:47.919 ","End":"00:53.650","Text":"and t depends on x and y."},{"Start":"00:54.080 ","End":"01:01.505","Text":"At first we need to go along here and then along here to get from z to x."},{"Start":"01:01.505 ","End":"01:07.705","Text":"We get z with respect to t and then t with respect to x."},{"Start":"01:07.705 ","End":"01:10.010","Text":"This we can\u0027t compute the t with respect to x,"},{"Start":"01:10.010 ","End":"01:17.205","Text":"we can\u0027t compute because this is x over a constant so it\u0027s just that constant."},{"Start":"01:17.205 ","End":"01:21.000","Text":"It\u0027s just z_t times 1 over y."},{"Start":"01:21.000 ","End":"01:24.100","Text":"If y is a constant and the x is the variable,"},{"Start":"01:24.100 ","End":"01:26.110","Text":"this is the derivative."},{"Start":"01:26.110 ","End":"01:29.660","Text":"Then we need z_y."},{"Start":"01:31.100 ","End":"01:34.150","Text":"We take a different path again down to t,"},{"Start":"01:34.150 ","End":"01:36.910","Text":"but this time to y."},{"Start":"01:36.910 ","End":"01:41.410","Text":"We get once again z with respect to t, same as above."},{"Start":"01:41.410 ","End":"01:44.815","Text":"But the derivative of this with respect to y,"},{"Start":"01:44.815 ","End":"01:46.330","Text":"y is the variable."},{"Start":"01:46.330 ","End":"01:54.200","Text":"Remember the derivative of 1 over y would be minus 1 over y squared."},{"Start":"01:58.130 ","End":"02:02.985","Text":"But because it\u0027s an x times, the x sticks."},{"Start":"02:02.985 ","End":"02:11.020","Text":"Let me just rewrite this as z_t."},{"Start":"02:12.140 ","End":"02:15.730","Text":"I\u0027ll put the minus in front."},{"Start":"02:15.860 ","End":"02:21.920","Text":"Then x over y squared."},{"Start":"02:21.920 ","End":"02:24.220","Text":"I don\u0027t know what z_t is."},{"Start":"02:24.220 ","End":"02:26.105","Text":"I know it\u0027s just f prime of t,"},{"Start":"02:26.105 ","End":"02:27.920","Text":"but that\u0027s no help."},{"Start":"02:27.920 ","End":"02:30.800","Text":"Let\u0027s just go ahead and do the equality anyway."},{"Start":"02:30.800 ","End":"02:32.840","Text":"We don\u0027t need to know what f is."},{"Start":"02:32.840 ","End":"02:38.395","Text":"X times z_x plus y times z_y."},{"Start":"02:38.395 ","End":"02:43.190","Text":"To prove something, we start with 1 side and end up in the other side hopefully."},{"Start":"02:43.190 ","End":"02:45.050","Text":"I\u0027ve put a question mark meanwhile."},{"Start":"02:45.050 ","End":"02:47.450","Text":"This is equal to x times."},{"Start":"02:47.450 ","End":"02:49.740","Text":"Z_x is this."},{"Start":"02:50.690 ","End":"02:55.540","Text":"I can just say that it\u0027s x over y."},{"Start":"02:55.540 ","End":"02:57.935","Text":"Maybe not, maybe I won\u0027t skip a step."},{"Start":"02:57.935 ","End":"03:05.445","Text":"I\u0027ll just write that full x times z_t times 1 over y."},{"Start":"03:05.445 ","End":"03:08.505","Text":"Then plus and then y."},{"Start":"03:08.505 ","End":"03:11.474","Text":"Now z_y is from here,"},{"Start":"03:11.474 ","End":"03:20.895","Text":"which is minus z_t x"},{"Start":"03:20.895 ","End":"03:23.475","Text":"over y squared."},{"Start":"03:23.475 ","End":"03:25.520","Text":"If I simplify it,"},{"Start":"03:25.520 ","End":"03:30.600","Text":"I can just say this is x over y times z_t."},{"Start":"03:32.390 ","End":"03:38.230","Text":"Note that y cancels into y squared just y times."},{"Start":"03:39.110 ","End":"03:43.050","Text":"I just get x over y."},{"Start":"03:43.050 ","End":"03:46.425","Text":"But also I\u0027ll make this a minus."},{"Start":"03:46.425 ","End":"03:48.455","Text":"That takes care of that minus."},{"Start":"03:48.455 ","End":"03:53.790","Text":"Just x over y, z_t."},{"Start":"03:53.930 ","End":"04:01.490","Text":"This term and this term are equal and so I end up with 0,"},{"Start":"04:01.490 ","End":"04:06.180","Text":"which is exactly what this is. It\u0027s proven."}],"ID":8969},{"Watched":false,"Name":"Exercise 6","Duration":"3m 14s","ChapterTopicVideoID":8624,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this exercise, we\u0027re given z as a function of x minus y and y minus"},{"Start":"00:05.250 ","End":"00:10.110","Text":"x and we have to prove this equality. What does this mean?"},{"Start":"00:10.110 ","End":"00:12.990","Text":"This actually means that f is a function of 2 variables."},{"Start":"00:12.990 ","End":"00:15.870","Text":"I want to take 2 letters"},{"Start":"00:15.870 ","End":"00:21.015","Text":"and I\u0027ll take u and v. Z is actually a function of 2 variables,"},{"Start":"00:21.015 ","End":"00:27.695","Text":"u and v. Here we are substituting that u"},{"Start":"00:27.695 ","End":"00:34.565","Text":"is equal to x minus y and v is equal to y minus x."},{"Start":"00:34.565 ","End":"00:37.405","Text":"We have to prove this."},{"Start":"00:37.405 ","End":"00:42.680","Text":"Let\u0027s do a dependency tree which will help us to find each of these."},{"Start":"00:42.680 ","End":"00:45.185","Text":"We start at the top with the z."},{"Start":"00:45.185 ","End":"00:55.130","Text":"Z depends initially on u and v. Each of u and v depends on x and y. I have an x and"},{"Start":"00:55.130 ","End":"01:05.270","Text":"a y here for u and an x and a y for v. To find z_x partial derivative with respect to x,"},{"Start":"01:05.270 ","End":"01:09.110","Text":"I need to go from here down to here or to here."},{"Start":"01:09.110 ","End":"01:15.850","Text":"I\u0027ll just highlight this way and this way or this way and this way."},{"Start":"01:15.850 ","End":"01:20.090","Text":"That gives us that this is z with respect to u,"},{"Start":"01:20.090 ","End":"01:21.830","Text":"u with respect to x,"},{"Start":"01:21.830 ","End":"01:27.500","Text":"plus the other path zv, vx."},{"Start":"01:27.500 ","End":"01:30.420","Text":"Now, this equals z"},{"Start":"01:30.420 ","End":"01:33.740","Text":"with respect to u I don\u0027t know because I don\u0027t know what the function f is,"},{"Start":"01:33.740 ","End":"01:37.370","Text":"but I do know u with respect to x. Y is a constant,"},{"Start":"01:37.370 ","End":"01:39.500","Text":"so this is just 1."},{"Start":"01:39.500 ","End":"01:42.350","Text":"Z with respect to v. I don\u0027t know what it is."},{"Start":"01:42.350 ","End":"01:46.385","Text":"It\u0027s a partial derivative of f with respect to the second variable, but I don\u0027t know."},{"Start":"01:46.385 ","End":"01:47.960","Text":"But v with respect to x,"},{"Start":"01:47.960 ","End":"01:51.325","Text":"I do know that\u0027s minus 1."},{"Start":"01:51.325 ","End":"01:54.915","Text":"Then we have z with respect to y."},{"Start":"01:54.915 ","End":"01:56.780","Text":"Instead of going here or here,"},{"Start":"01:56.780 ","End":"02:01.700","Text":"I need to end up in y. I go this way and this way,"},{"Start":"02:01.700 ","End":"02:05.815","Text":"or this way and this way."},{"Start":"02:05.815 ","End":"02:08.370","Text":"This is z with respect to u,"},{"Start":"02:08.370 ","End":"02:11.055","Text":"u with respect to y."},{"Start":"02:11.055 ","End":"02:14.715","Text":"The other path zv, vy,"},{"Start":"02:14.715 ","End":"02:20.850","Text":"vvy, which is equal to as before,"},{"Start":"02:20.850 ","End":"02:22.430","Text":"I don\u0027t know what zu is,"},{"Start":"02:22.430 ","End":"02:27.230","Text":"but u with respect to y is minus 1."},{"Start":"02:27.230 ","End":"02:34.125","Text":"Then zv and v with respect to y is plus 1."},{"Start":"02:34.125 ","End":"02:39.530","Text":"Now zx plus zy,"},{"Start":"02:39.530 ","End":"02:41.990","Text":"which is the left-hand side here of what I\u0027m"},{"Start":"02:41.990 ","End":"02:44.585","Text":"trying to prove and trying to prove is this equal to 0?"},{"Start":"02:44.585 ","End":"02:46.670","Text":"Let\u0027s see if we can end up with 0."},{"Start":"02:46.670 ","End":"02:50.325","Text":"It\u0027s equal to zx is from here,"},{"Start":"02:50.325 ","End":"02:52.875","Text":"just zu I don\u0027t need the 1."},{"Start":"02:52.875 ","End":"02:55.245","Text":"Here I get minus zv,"},{"Start":"02:55.245 ","End":"02:58.965","Text":"and from here I get minus zu,"},{"Start":"02:58.965 ","End":"03:02.320","Text":"and from here plus zv."},{"Start":"03:02.440 ","End":"03:05.120","Text":"This cancels with this."},{"Start":"03:05.120 ","End":"03:07.625","Text":"Here I have a plus and a minus."},{"Start":"03:07.625 ","End":"03:09.890","Text":"Altogether I end up with 0,"},{"Start":"03:09.890 ","End":"03:11.450","Text":"which is the right-hand side."},{"Start":"03:11.450 ","End":"03:13.980","Text":"We\u0027re okay and we\u0027re done."}],"ID":8970},{"Watched":false,"Name":"Exercise 7","Duration":"4m 54s","ChapterTopicVideoID":8625,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"In this exercise, we\u0027re given the following,"},{"Start":"00:02.670 ","End":"00:05.830","Text":"that w is some function of 3 variables applied to"},{"Start":"00:05.830 ","End":"00:08.208","Text":"x minus y, y minus z, z minus x,"},{"Start":"00:08.208 ","End":"00:10.815","Text":"and we have to prove this."},{"Start":"00:10.815 ","End":"00:18.360","Text":"What this really means is that we have some function f of 3 variables."},{"Start":"00:18.360 ","End":"00:22.830","Text":"Let me think of 3 new letters, say r, s, t."},{"Start":"00:22.830 ","End":"00:27.810","Text":"If we substitute for r, x minus y,"},{"Start":"00:27.810 ","End":"00:32.775","Text":"and instead of s, we put y minus z,"},{"Start":"00:32.775 ","End":"00:37.125","Text":"and instead of t, we put z minus x."},{"Start":"00:37.125 ","End":"00:40.490","Text":"Then we have to prove that this following holds because w"},{"Start":"00:40.490 ","End":"00:44.480","Text":"is indirectly a function of x, y, and z."},{"Start":"00:44.480 ","End":"00:47.240","Text":"In fact, let\u0027s do the dependency tree."},{"Start":"00:47.240 ","End":"00:49.625","Text":"At the top, we have w."},{"Start":"00:49.625 ","End":"00:56.160","Text":"It depends on 3 variables, r, s, and t."},{"Start":"00:56.160 ","End":"00:59.724","Text":"Now, each of r, s, and t depends on x, y, z,"},{"Start":"00:59.724 ","End":"01:01.005","Text":"but not all of them;"},{"Start":"01:01.005 ","End":"01:05.205","Text":"r depends just on x and y,"},{"Start":"01:05.205 ","End":"01:08.865","Text":"s depends on y and z,"},{"Start":"01:08.865 ","End":"01:14.830","Text":"and t depends on z and x."},{"Start":"01:14.830 ","End":"01:17.410","Text":"Now we want to compute each of these 3 things."},{"Start":"01:17.410 ","End":"01:20.190","Text":"Let\u0027s start with w_x."},{"Start":"01:20.220 ","End":"01:26.725","Text":"To get from w to x, we can either go along here and then along here."},{"Start":"01:26.725 ","End":"01:32.580","Text":"Or we can go this way and then this way."},{"Start":"01:33.100 ","End":"01:40.250","Text":"This equals w with respect to r, which I don\u0027t know,"},{"Start":"01:40.250 ","End":"01:43.280","Text":"it\u0027s the partial derivative with"},{"Start":"01:43.280 ","End":"01:45.176","Text":"respect to the first variable of f."},{"Start":"01:45.176 ","End":"01:47.075","Text":"But just leave it like that."},{"Start":"01:47.075 ","End":"01:51.855","Text":"But r with respect to x,"},{"Start":"01:51.855 ","End":"01:54.570","Text":"we can say what it is."},{"Start":"01:54.570 ","End":"01:56.840","Text":"In a moment, let me just finish writing this out."},{"Start":"01:56.840 ","End":"02:03.260","Text":"Plus w with respect to t and t with respect to x."},{"Start":"02:03.260 ","End":"02:05.390","Text":"As I was saying, w with respect to r,"},{"Start":"02:05.390 ","End":"02:06.740","Text":"I don\u0027t know how to compute,"},{"Start":"02:06.740 ","End":"02:08.585","Text":"but r with respect to x,"},{"Start":"02:08.585 ","End":"02:11.015","Text":"I can see that the derivative is just 1,"},{"Start":"02:11.015 ","End":"02:12.305","Text":"y is a constant."},{"Start":"02:12.305 ","End":"02:15.470","Text":"Times 1 plus w_t."},{"Start":"02:15.470 ","End":"02:17.585","Text":"t with respect to x,"},{"Start":"02:17.585 ","End":"02:18.710","Text":"z is a constant,"},{"Start":"02:18.710 ","End":"02:21.590","Text":"so this is just minus 1."},{"Start":"02:22.160 ","End":"02:26.505","Text":"Now let\u0027s do the other 2,"},{"Start":"02:26.505 ","End":"02:31.380","Text":"w with respect to y so we changed the dependency tree."},{"Start":"02:31.380 ","End":"02:34.605","Text":"To get to y, we go here and here,"},{"Start":"02:34.605 ","End":"02:40.613","Text":"or here and here, 2 ways,"},{"Start":"02:40.613 ","End":"02:44.220","Text":"and so we get w with respect to r,"},{"Start":"02:44.220 ","End":"02:51.120","Text":"r with respect to y plus w_s, s_y and this equals."},{"Start":"02:51.120 ","End":"02:53.850","Text":"Once again, the w_r and the w_s."},{"Start":"02:53.850 ","End":"02:55.299","Text":"These w\u0027s, we can\u0027t compute"},{"Start":"02:55.299 ","End":"02:58.148","Text":"but r with respect to y is minus 1"},{"Start":"02:58.148 ","End":"03:03.970","Text":"and s with respect to y is 1."},{"Start":"03:04.190 ","End":"03:13.540","Text":"w_z has just changed the tree through s to z or through t to z,"},{"Start":"03:14.570 ","End":"03:23.717","Text":"and so w_s, s_z, w_t, t_z,"},{"Start":"03:23.717 ","End":"03:25.710","Text":"and then we get w_s."},{"Start":"03:25.710 ","End":"03:29.895","Text":"s with respect to z is minus 1"},{"Start":"03:29.895 ","End":"03:38.625","Text":"and t with respect to z is plus 1."},{"Start":"03:38.625 ","End":"03:42.685","Text":"Now we just have to do a little computation."},{"Start":"03:42.685 ","End":"03:47.110","Text":"I\u0027m trying to prove this. Let\u0027s see."},{"Start":"03:47.110 ","End":"03:50.940","Text":"w_x plus w_y plus w_z."},{"Start":"03:50.940 ","End":"03:52.769","Text":"I\u0027m just writing the left-hand side"},{"Start":"03:52.769 ","End":"03:56.015","Text":"and I have to show that it\u0027s equal to the right-hand side."},{"Start":"03:56.015 ","End":"03:58.925","Text":"Let\u0027s expand this and see what we get."},{"Start":"03:58.925 ","End":"04:04.770","Text":"Copying from here, this is just w_r minus w_t."},{"Start":"04:05.500 ","End":"04:15.470","Text":"Then the next bit, w_y, here is just minus w_r plus w_s."},{"Start":"04:15.470 ","End":"04:22.185","Text":"Make this a minus w_r plus w_s, that\u0027s this."},{"Start":"04:22.185 ","End":"04:24.240","Text":"Then finally the last bit,"},{"Start":"04:24.240 ","End":"04:31.200","Text":"it\u0027s going to be minus w_s plus w_t."},{"Start":"04:31.200 ","End":"04:32.745","Text":"Now look, we got a lot of canceling."},{"Start":"04:32.745 ","End":"04:35.350","Text":"This cancels with this;"},{"Start":"04:35.690 ","End":"04:41.970","Text":"w_r and w_r with the opposite sign and then w_t,"},{"Start":"04:41.970 ","End":"04:44.430","Text":"the minus with a plus."},{"Start":"04:44.430 ","End":"04:48.910","Text":"This whole thing just comes out to be 0."},{"Start":"04:48.910 ","End":"04:51.453","Text":"That is just the right-hand side,"},{"Start":"04:51.453 ","End":"04:54.480","Text":"so yes, we\u0027ve proven it."}],"ID":8971},{"Watched":false,"Name":"Exercise 8","Duration":"6m 33s","ChapterTopicVideoID":8626,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.320","Text":"In this exercise, we\u0027re given that u is dependent on x and y,"},{"Start":"00:07.320 ","End":"00:11.160","Text":"but in the following formula it\u0027s equal to sine x plus"},{"Start":"00:11.160 ","End":"00:17.175","Text":"some function of 1 variable where we substitute sine y minus sine x,"},{"Start":"00:17.175 ","End":"00:22.019","Text":"and we have to prove that this equality holds."},{"Start":"00:22.019 ","End":"00:25.260","Text":"I won\u0027t read it out. We can see it."},{"Start":"00:25.260 ","End":"00:27.150","Text":"Let me just spell it out,"},{"Start":"00:27.150 ","End":"00:30.155","Text":"that f is a function of a single variable,"},{"Start":"00:30.155 ","End":"00:32.915","Text":"and let\u0027s give that variable a name."},{"Start":"00:32.915 ","End":"00:35.120","Text":"I see the letter T is unused,"},{"Start":"00:35.120 ","End":"00:42.620","Text":"so let\u0027s let t equal sine y minus sine x and then that will"},{"Start":"00:42.620 ","End":"00:51.365","Text":"give us that u is equal to sine x plus f of t,"},{"Start":"00:51.365 ","End":"00:56.105","Text":"where t is dependent on y and x."},{"Start":"00:56.105 ","End":"01:01.900","Text":"Now, before we do the partial derivatives we\u0027ll need u_x and u_y."},{"Start":"01:01.900 ","End":"01:07.304","Text":"Let\u0027s do a dependence tree before I say what this equals."},{"Start":"01:07.304 ","End":"01:10.155","Text":"At the top we have u."},{"Start":"01:10.155 ","End":"01:16.170","Text":"Now u depends directly on x and"},{"Start":"01:16.170 ","End":"01:24.924","Text":"t. But t is dependent on y and x."},{"Start":"01:24.924 ","End":"01:32.600","Text":"So we actually get u dependent on x in 2 different ways and I need that for this."},{"Start":"01:32.600 ","End":"01:34.320","Text":"Let me just highlight,"},{"Start":"01:34.320 ","End":"01:39.680","Text":"I can get from u to x this way or this and this."},{"Start":"01:46.760 ","End":"01:52.900","Text":"Now it\u0027s important to note that our dependency tree is going to have f at the top,"},{"Start":"01:52.900 ","End":"01:55.705","Text":"and not u, as you might think."},{"Start":"01:55.705 ","End":"02:00.920","Text":"It\u0027s just for the part that I did the substitution for,"},{"Start":"02:00.920 ","End":"02:03.770","Text":"I did a substitution for t. So I take f at the top,"},{"Start":"02:03.770 ","End":"02:06.620","Text":"and f depends on t,"},{"Start":"02:06.620 ","End":"02:13.705","Text":"but t depends on x and on y."},{"Start":"02:13.705 ","End":"02:18.675","Text":"Now, let\u0027s get to the partial derivative."},{"Start":"02:18.675 ","End":"02:20.810","Text":"U with respect to x."},{"Start":"02:20.810 ","End":"02:22.850","Text":"Well, it starts out simple enough."},{"Start":"02:22.850 ","End":"02:27.575","Text":"First of all, I have cosine x from the derivative of sine x."},{"Start":"02:27.575 ","End":"02:32.990","Text":"Now, I need the derivative of this with respect to x,"},{"Start":"02:32.990 ","End":"02:36.450","Text":"and this is where I look at the tree and I see"},{"Start":"02:36.450 ","End":"02:40.980","Text":"we have to go down here and across here to get from f to x."},{"Start":"02:41.170 ","End":"02:43.550","Text":"From here I get,"},{"Start":"02:43.550 ","End":"02:45.880","Text":"it\u0027s sort of regular,"},{"Start":"02:45.880 ","End":"02:47.780","Text":"not exactly, a regular chain rule,"},{"Start":"02:47.780 ","End":"02:51.740","Text":"but the first part is not a partial,"},{"Start":"02:51.740 ","End":"02:57.120","Text":"it\u0027s just f\u0027 of t. But t being a function of x,"},{"Start":"02:57.120 ","End":"03:03.980","Text":"I have to also multiply by t with respect to x and so if I write it out,"},{"Start":"03:03.980 ","End":"03:07.430","Text":"it\u0027s cosine of x plus f\u0027 of t,"},{"Start":"03:07.430 ","End":"03:14.070","Text":"t with respect to x. I can get from here and this is a constant as far as x"},{"Start":"03:14.070 ","End":"03:22.640","Text":"goes and so I get minus cosine x and that\u0027s this part here."},{"Start":"03:22.640 ","End":"03:25.375","Text":"Now I also need u with respect to y,"},{"Start":"03:25.375 ","End":"03:27.010","Text":"so I can substitute it in,"},{"Start":"03:27.010 ","End":"03:30.650","Text":"and I get u with respect to y equals."},{"Start":"03:30.710 ","End":"03:33.460","Text":"I go from f down to t as before,"},{"Start":"03:33.460 ","End":"03:37.130","Text":"but this time I turn the other way to get to y."},{"Start":"03:37.250 ","End":"03:40.515","Text":"So u with respect to y."},{"Start":"03:40.515 ","End":"03:43.320","Text":"Sine x is a constant, so that\u0027s nothing."},{"Start":"03:43.320 ","End":"03:47.535","Text":"In fact the right 0 just so you\u0027ll know that comes from the sine x."},{"Start":"03:47.535 ","End":"03:51.810","Text":"Then f with respect to t is f\u0027 of t,"},{"Start":"03:51.810 ","End":"03:55.470","Text":"and then t with respect to y."},{"Start":"03:55.470 ","End":"03:58.560","Text":"Altogether I\u0027ve got, 0 is nothing,"},{"Start":"03:58.560 ","End":"04:02.259","Text":"so it\u0027s f\u0027 of t. T with respect to y,"},{"Start":"04:02.259 ","End":"04:03.740","Text":"I can get that from here."},{"Start":"04:03.740 ","End":"04:08.310","Text":"X is a constant now although we get cosine y."},{"Start":"04:09.080 ","End":"04:12.840","Text":"Now that I\u0027ve got u_x and u_y,"},{"Start":"04:12.840 ","End":"04:15.730","Text":"I can now substitute in here,"},{"Start":"04:16.190 ","End":"04:18.440","Text":"1 way to do it is to start with"},{"Start":"04:18.440 ","End":"04:22.130","Text":"the left-hand side and show that I reached the right-hand side."},{"Start":"04:22.130 ","End":"04:24.440","Text":"Let\u0027s do the left-hand side."},{"Start":"04:24.440 ","End":"04:31.940","Text":"U_x cosine y plus u_y cosine x."},{"Start":"04:31.940 ","End":"04:33.410","Text":"That\u0027s just the left-hand side,"},{"Start":"04:33.410 ","End":"04:37.940","Text":"is equal to and I\u0027m going to substitute u_x u_y from here."},{"Start":"04:37.940 ","End":"04:39.830","Text":"U_x is this."},{"Start":"04:39.830 ","End":"04:45.470","Text":"So it\u0027s cosine x plus, you know what?"},{"Start":"04:45.470 ","End":"04:48.125","Text":"Let me take the minus out."},{"Start":"04:48.125 ","End":"04:58.010","Text":"So that\u0027s a minus and then I can just take cosine x in front,"},{"Start":"04:58.010 ","End":"05:04.590","Text":"f\u0027 of t. Where am I?"},{"Start":"05:08.210 ","End":"05:11.940","Text":"All this was just to get the u_x."},{"Start":"05:11.940 ","End":"05:13.610","Text":"So I\u0027ll put the brackets surround here,"},{"Start":"05:13.610 ","End":"05:17.255","Text":"that\u0027s this part and I still need a cosine y."},{"Start":"05:17.255 ","End":"05:24.260","Text":"Then plus I need to substitute u_y this time,"},{"Start":"05:24.260 ","End":"05:31.770","Text":"which is just f\u0027 of t cosine y,"},{"Start":"05:32.980 ","End":"05:43.130","Text":"and then times cosine x and all this is just this bit here,"},{"Start":"05:43.130 ","End":"05:44.900","Text":"which is the left-hand side."},{"Start":"05:44.900 ","End":"05:48.244","Text":"Let\u0027s see. I\u0027ll expand everything."},{"Start":"05:48.244 ","End":"05:52.080","Text":"First I have cosine x, cosine y,"},{"Start":"05:53.590 ","End":"05:59.134","Text":"then minus cosine x,"},{"Start":"05:59.134 ","End":"06:04.215","Text":"cosine y, f\u0027 of t,"},{"Start":"06:04.215 ","End":"06:06.725","Text":"and then this plus,"},{"Start":"06:06.725 ","End":"06:09.590","Text":"let me write first of all the cosine of x,"},{"Start":"06:09.590 ","End":"06:12.329","Text":"then the cosine y,"},{"Start":"06:12.329 ","End":"06:16.910","Text":"and then f\u0027 of t. But look,"},{"Start":"06:16.910 ","End":"06:20.525","Text":"this term with a minus and this term with a plus,"},{"Start":"06:20.525 ","End":"06:23.360","Text":"they\u0027re the same, so these cancel out."},{"Start":"06:23.360 ","End":"06:29.360","Text":"All I\u0027m left with is this and this is indeed the right-hand side of this."},{"Start":"06:29.360 ","End":"06:32.850","Text":"So we have proven it."}],"ID":8972},{"Watched":false,"Name":"Exercise 9","Duration":"6m 49s","ChapterTopicVideoID":8627,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we\u0027re given z equals y"},{"Start":"00:03.360 ","End":"00:06.570","Text":"times some function of x squared minus y squared."},{"Start":"00:06.570 ","End":"00:08.505","Text":"We don\u0027t know what this function is."},{"Start":"00:08.505 ","End":"00:12.345","Text":"We have to prove the following equality."},{"Start":"00:12.345 ","End":"00:15.465","Text":"F is some function of 1 variable,"},{"Start":"00:15.465 ","End":"00:22.335","Text":"and let\u0027s call this variable t. We\u0027ll let t equal x squared minus y squared."},{"Start":"00:22.335 ","End":"00:28.545","Text":"We have y times some function of t. Then we have that z"},{"Start":"00:28.545 ","End":"00:34.890","Text":"is equal to y times f of t. Now,"},{"Start":"00:34.890 ","End":"00:39.420","Text":"we need to compute the 2 partial derivatives of z with respect to x and y."},{"Start":"00:39.420 ","End":"00:45.185","Text":"Z with respect to x equals z with respect to, well, what?"},{"Start":"00:45.185 ","End":"00:48.010","Text":"We need a dependency tree to help us."},{"Start":"00:48.010 ","End":"00:52.100","Text":"Now, at the top of the tree is not z as you might think,"},{"Start":"00:52.100 ","End":"00:56.585","Text":"but f. It\u0027s the place where we did the substitution."},{"Start":"00:56.585 ","End":"01:00.095","Text":"F depends on t,"},{"Start":"01:00.095 ","End":"01:04.670","Text":"and then t depends on x and y,"},{"Start":"01:04.670 ","End":"01:06.440","Text":"as you can see here."},{"Start":"01:06.440 ","End":"01:11.270","Text":"There\u0027s only 1 way to get from f to x,"},{"Start":"01:11.270 ","End":"01:15.180","Text":"this way and this way."},{"Start":"01:16.040 ","End":"01:20.875","Text":"Well, we start out with the product rule and"},{"Start":"01:20.875 ","End":"01:26.305","Text":"the product rule says that we take the derivative of the 1st."},{"Start":"01:26.305 ","End":"01:28.920","Text":"Because y doesn\u0027t depend on x,"},{"Start":"01:28.920 ","End":"01:30.665","Text":"this derivative is 0,"},{"Start":"01:30.665 ","End":"01:35.150","Text":"because x and y are independent and t is dependent on both of them,"},{"Start":"01:35.150 ","End":"01:38.540","Text":"but they don\u0027t have any common dependency, so it is a set."},{"Start":"01:38.540 ","End":"01:42.390","Text":"Derivative of the 1st times the 2nd f of t"},{"Start":"01:42.390 ","End":"01:47.660","Text":"plus the 1st as is times the derivative of the 2nd."},{"Start":"01:47.660 ","End":"01:50.270","Text":"Now, the derivative is with respect to x."},{"Start":"01:50.270 ","End":"01:53.930","Text":"We have to go about it by saying first with respect to t,"},{"Start":"01:53.930 ","End":"01:56.030","Text":"which is f prime of t,"},{"Start":"01:56.030 ","End":"01:59.665","Text":"and then t with respect to x."},{"Start":"01:59.665 ","End":"02:01.905","Text":"Let me just write that at first."},{"Start":"02:01.905 ","End":"02:04.380","Text":"Then we\u0027ll say what t with respect to x is."},{"Start":"02:04.380 ","End":"02:07.645","Text":"Well, the first bit is 0, so we get y,"},{"Start":"02:07.645 ","End":"02:14.095","Text":"f prime of t and t with respect to x is just 2x, y is a constant."},{"Start":"02:14.095 ","End":"02:16.020","Text":"That\u0027s 1 of them."},{"Start":"02:16.020 ","End":"02:19.110","Text":"Now, I also need zy before I substitute."},{"Start":"02:19.110 ","End":"02:23.970","Text":"Zy is equal to for the dependency on f and y,"},{"Start":"02:23.970 ","End":"02:29.570","Text":"I need to go this way and this way and so we get z."},{"Start":"02:29.570 ","End":"02:32.180","Text":"Well, first of all, again, we have a product rule."},{"Start":"02:32.180 ","End":"02:35.720","Text":"We take the derivative of the first,"},{"Start":"02:35.720 ","End":"02:38.360","Text":"but it\u0027s with respect to y this time, so it\u0027s not 0,"},{"Start":"02:38.360 ","End":"02:43.920","Text":"it\u0027s 1 times f of t plus y as is,"},{"Start":"02:43.920 ","End":"02:46.385","Text":"and then derivative of this with respect to y."},{"Start":"02:46.385 ","End":"02:48.905","Text":"Once again it\u0027s f prime of t,"},{"Start":"02:48.905 ","End":"02:52.960","Text":"but this time not tx but ty."},{"Start":"02:52.960 ","End":"02:57.919","Text":"T with respect to y would be minus 2y,"},{"Start":"02:57.919 ","End":"03:02.510","Text":"so we get f of t plus y,"},{"Start":"03:02.510 ","End":"03:07.290","Text":"f prime of t,"},{"Start":"03:07.290 ","End":"03:11.505","Text":"and then minus 2y."},{"Start":"03:11.505 ","End":"03:19.444","Text":"Now, we have to take these 2 expressions and plug them in to this formula."},{"Start":"03:19.444 ","End":"03:23.190","Text":"Perhaps I\u0027ll just rewrite this a bit."},{"Start":"03:25.710 ","End":"03:27.775","Text":"Maybe it\u0027s not important."},{"Start":"03:27.775 ","End":"03:32.140","Text":"I was going to say that the y with the minus 2y is minus 2y squared,"},{"Start":"03:32.140 ","End":"03:34.400","Text":"never mind, when we come to it."},{"Start":"03:34.520 ","End":"03:37.030","Text":"Let\u0027s see, to prove something,"},{"Start":"03:37.030 ","End":"03:42.580","Text":"we start with the left hand side and try to end up with the right hand side,"},{"Start":"03:42.580 ","End":"03:44.470","Text":"at least that\u0027s 1 way of going about it."},{"Start":"03:44.470 ","End":"03:52.095","Text":"Let me start with 1 over x times zx plus 1 over y, zy."},{"Start":"03:52.095 ","End":"03:54.150","Text":"Now, I plug in."},{"Start":"03:54.150 ","End":"03:56.290","Text":"For the zx, what I have here,"},{"Start":"03:56.290 ","End":"04:00.505","Text":"1 over x times y,"},{"Start":"04:00.505 ","End":"04:08.270","Text":"f prime of t times 2x plus 1 over y,"},{"Start":"04:08.270 ","End":"04:10.520","Text":"z with respect to y is this."},{"Start":"04:10.520 ","End":"04:13.050","Text":"I\u0027ll need a bracket."},{"Start":"04:13.240 ","End":"04:24.960","Text":"Then we get f of t. Let\u0027s see,"},{"Start":"04:27.890 ","End":"04:32.285","Text":"f of t times,"},{"Start":"04:32.285 ","End":"04:34.010","Text":"I\u0027m sorry, just this term."},{"Start":"04:34.010 ","End":"04:38.150","Text":"But now I\u0027m going to combine the minus 2y with the y to get minus 2y"},{"Start":"04:38.150 ","End":"04:45.295","Text":"squared f prime of t. Now,"},{"Start":"04:45.295 ","End":"04:47.570","Text":"what we get here is,"},{"Start":"04:47.570 ","End":"04:49.385","Text":"if we just collect things together,"},{"Start":"04:49.385 ","End":"04:54.050","Text":"this x with this x just cancels."},{"Start":"04:54.050 ","End":"04:56.535","Text":"Just the x, the 2 is there."},{"Start":"04:56.535 ","End":"05:01.250","Text":"Over here, well, in the 1st 1 we can\u0027t cancel anything,"},{"Start":"05:01.250 ","End":"05:08.340","Text":"but then the 2nd we\u0027ll be able to do 1 over y with y squared when we get to it."},{"Start":"05:09.170 ","End":"05:13.095","Text":"We have put the 2 in front."},{"Start":"05:13.095 ","End":"05:22.040","Text":"2y, f prime of t. Then plus 1 over y,"},{"Start":"05:22.040 ","End":"05:26.925","Text":"f of t, or just f of t over y."},{"Start":"05:26.925 ","End":"05:31.430","Text":"Then minus, now in this bit, we do cancel."},{"Start":"05:31.430 ","End":"05:34.775","Text":"Maybe I\u0027ll indicate it with a dotted line."},{"Start":"05:34.775 ","End":"05:36.440","Text":"This cancels with this,"},{"Start":"05:36.440 ","End":"05:40.560","Text":"but only in the second multiplication."},{"Start":"05:40.560 ","End":"05:49.005","Text":"We get minus 2y times f prime of t. Now,"},{"Start":"05:49.005 ","End":"05:52.390","Text":"this whole thing cancels out with,"},{"Start":"05:52.640 ","End":"05:54.960","Text":"I mean this minus this is 0,"},{"Start":"05:54.960 ","End":"05:59.590","Text":"so all we\u0027re left with now is this bit,"},{"Start":"05:59.780 ","End":"06:05.685","Text":"which is f of t over y."},{"Start":"06:05.685 ","End":"06:10.330","Text":"But look, if I look here,"},{"Start":"06:11.170 ","End":"06:18.610","Text":"I can see that f of t could be written as z over y."},{"Start":"06:18.610 ","End":"06:20.985","Text":"If I plug that,"},{"Start":"06:20.985 ","End":"06:25.145","Text":"now down below there, here I mean,"},{"Start":"06:25.145 ","End":"06:28.975","Text":"then we\u0027ve got z over y over y,"},{"Start":"06:28.975 ","End":"06:33.735","Text":"so this is just z over y squared."},{"Start":"06:33.735 ","End":"06:37.640","Text":"This 1 here, I didn\u0027t want to put an arrow from here to here."},{"Start":"06:37.640 ","End":"06:40.999","Text":"This z over y squared,"},{"Start":"06:40.999 ","End":"06:42.395","Text":"if we look here now,"},{"Start":"06:42.395 ","End":"06:44.885","Text":"is exactly the right hand side."},{"Start":"06:44.885 ","End":"06:49.170","Text":"Yes, we\u0027ve gotten to there, so we\u0027ve proved it."}],"ID":8973},{"Watched":false,"Name":"Exercise 10","Duration":"8m 8s","ChapterTopicVideoID":8628,"CourseChapterTopicPlaylistID":4964,"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\u0027re given z in terms of x and y,"},{"Start":"00:04.620 ","End":"00:09.810","Text":"but there\u0027s also an unknown function f of y over x,"},{"Start":"00:09.810 ","End":"00:13.200","Text":"and we have to prove this equality."},{"Start":"00:13.200 ","End":"00:17.190","Text":"1 of the things to do is to first of all"},{"Start":"00:17.190 ","End":"00:19.740","Text":"replace this by a single letter."},{"Start":"00:19.740 ","End":"00:22.050","Text":"f is a function of 1 variable and we can call"},{"Start":"00:22.050 ","End":"00:23.685","Text":"this thing anything we haven\u0027t used,"},{"Start":"00:23.685 ","End":"00:29.550","Text":"let\u0027s say t. Let\u0027s say t is equal to y over x"},{"Start":"00:29.550 ","End":"00:31.680","Text":"and then this is f of t,"},{"Start":"00:31.680 ","End":"00:40.200","Text":"so we get that z equals xy plus xf of t,"},{"Start":"00:40.200 ","End":"00:42.045","Text":"where t is this."},{"Start":"00:42.045 ","End":"00:45.515","Text":"Now we have to go about proving this."},{"Start":"00:45.515 ","End":"00:49.265","Text":"In order to help us get the partial derivatives,"},{"Start":"00:49.265 ","End":"00:52.100","Text":"we\u0027re going to draw a dependency tree,"},{"Start":"00:52.100 ","End":"00:55.385","Text":"and at the top of the tree we\u0027ll put f,"},{"Start":"00:55.385 ","End":"01:00.270","Text":"which depends on the variable t,"},{"Start":"01:00.500 ","End":"01:05.595","Text":"but t depends on y and x."},{"Start":"01:05.595 ","End":"01:08.280","Text":"We have here x and here y."},{"Start":"01:08.280 ","End":"01:15.555","Text":"In that way, f depends indirectly on x and y. Let\u0027s see."},{"Start":"01:15.555 ","End":"01:16.650","Text":"By the chain rule,"},{"Start":"01:16.650 ","End":"01:20.175","Text":"we need z_x and then later we\u0027ll need z_y."},{"Start":"01:20.175 ","End":"01:23.490","Text":"We need both of these things in order to continue."},{"Start":"01:23.490 ","End":"01:26.300","Text":"This is equal to."},{"Start":"01:30.230 ","End":"01:34.280","Text":"We look at this expression and before"},{"Start":"01:34.280 ","End":"01:35.600","Text":"we use the chain rule,"},{"Start":"01:35.600 ","End":"01:39.350","Text":"we first of all differentiate xy with respect to x."},{"Start":"01:39.350 ","End":"01:43.430","Text":"Now, y is a constant as far as x goes,"},{"Start":"01:43.430 ","End":"01:45.680","Text":"so x times a constant with respect to x,"},{"Start":"01:45.680 ","End":"01:47.900","Text":"we\u0027re just left with that constant."},{"Start":"01:47.900 ","End":"01:50.890","Text":"The next bit we have a product rule."},{"Start":"01:50.890 ","End":"01:53.420","Text":"We take x differentiated,"},{"Start":"01:53.420 ","End":"01:56.570","Text":"which is 1, times this as is,"},{"Start":"01:56.570 ","End":"01:59.060","Text":"plus the reverse, x as is,"},{"Start":"01:59.060 ","End":"02:03.735","Text":"and then this thing differentiated with respect to x."},{"Start":"02:03.735 ","End":"02:09.995","Text":"The derivative of this with respect to x is going to be,"},{"Start":"02:09.995 ","End":"02:12.185","Text":"I\u0027ll just highlight the path,"},{"Start":"02:12.185 ","End":"02:15.380","Text":"f with respect to x goes via t,"},{"Start":"02:15.380 ","End":"02:16.640","Text":"so we have f with respect to t"},{"Start":"02:16.640 ","End":"02:19.290","Text":"and then t with respect to x. f with"},{"Start":"02:19.290 ","End":"02:23.580","Text":"respect to t is f prime of t and t with respect to x,"},{"Start":"02:23.580 ","End":"02:26.340","Text":"I\u0027ll just leave it like that for the moment,"},{"Start":"02:26.340 ","End":"02:29.210","Text":"and now I\u0027ll expand it."},{"Start":"02:31.930 ","End":"02:35.525","Text":"Well, I\u0027ll copy the rest of it then we get to that."},{"Start":"02:35.525 ","End":"02:39.000","Text":"We have y plus"},{"Start":"02:48.890 ","End":"02:56.925","Text":"f of t plus x times f prime of t. Now,"},{"Start":"02:56.925 ","End":"02:58.760","Text":"t with respect to x,"},{"Start":"02:58.760 ","End":"03:00.590","Text":"x is on the denominator."},{"Start":"03:00.590 ","End":"03:04.299","Text":"Derivative of 1 over x is minus 1 over x squared,"},{"Start":"03:04.299 ","End":"03:06.920","Text":"but it\u0027s not minus 1 over x squared"},{"Start":"03:06.920 ","End":"03:08.795","Text":"because we have the constant y,"},{"Start":"03:08.795 ","End":"03:12.390","Text":"so it\u0027s minus y over x squared."},{"Start":"03:15.350 ","End":"03:18.615","Text":"I almost forgot the brackets."},{"Start":"03:18.615 ","End":"03:22.095","Text":"Have to have brackets here otherwise it\u0027s a subtraction."},{"Start":"03:22.095 ","End":"03:26.665","Text":"Now, let\u0027s see what is z with respect to y."},{"Start":"03:26.665 ","End":"03:30.425","Text":"Once again, we start with differentiating the first term,"},{"Start":"03:30.425 ","End":"03:32.509","Text":"and this time x is a constant."},{"Start":"03:32.509 ","End":"03:35.240","Text":"Constant times y could be like 4y,"},{"Start":"03:35.240 ","End":"03:37.010","Text":"derivative would be just 4."},{"Start":"03:37.010 ","End":"03:39.530","Text":"In this case it\u0027s just x."},{"Start":"03:39.530 ","End":"03:42.320","Text":"Once again we have a product,"},{"Start":"03:42.320 ","End":"03:44.990","Text":"but actually we don\u0027t have to look at it as a product"},{"Start":"03:44.990 ","End":"03:46.760","Text":"because x is a constant."},{"Start":"03:46.760 ","End":"03:53.390","Text":"It\u0027s just x times the derivative of f of t. Now,"},{"Start":"03:53.390 ","End":"03:57.559","Text":"the derivative of f of t with respect to y,"},{"Start":"03:57.559 ","End":"03:58.880","Text":"f with respect to y,"},{"Start":"03:58.880 ","End":"04:00.785","Text":"we need to change this."},{"Start":"04:00.785 ","End":"04:06.390","Text":"The highlighting this time is this and this and so it\u0027s"},{"Start":"04:06.390 ","End":"04:13.290","Text":"f with respect to t and then t with respect to y,"},{"Start":"04:13.290 ","End":"04:19.185","Text":"and then we just have to say that t with respect to y,"},{"Start":"04:19.185 ","End":"04:21.555","Text":"I\u0027ll plug it in when I get there."},{"Start":"04:21.555 ","End":"04:28.915","Text":"It\u0027s x plus xf prime of t. Now t with respect to y,"},{"Start":"04:28.915 ","End":"04:30.800","Text":"1 over x is a constant."},{"Start":"04:30.800 ","End":"04:32.825","Text":"It\u0027s 1 over x times y."},{"Start":"04:32.825 ","End":"04:35.825","Text":"Something times y, it\u0027s just that something."},{"Start":"04:35.825 ","End":"04:42.020","Text":"In fact, here we have just 1 over x,"},{"Start":"04:42.020 ","End":"04:44.790","Text":"that\u0027s the coefficient of y here."},{"Start":"04:44.890 ","End":"04:49.670","Text":"Notice that x cancels with x,"},{"Start":"04:49.670 ","End":"04:51.980","Text":"this with this goes,"},{"Start":"04:51.980 ","End":"04:54.020","Text":"so it\u0027s just x plus f prime of t."},{"Start":"04:54.020 ","End":"04:56.254","Text":"Now that we have these 2 pieces,"},{"Start":"04:56.254 ","End":"04:58.315","Text":"we can plug in here."},{"Start":"04:58.315 ","End":"05:01.340","Text":"If we want to prove this 1 way is to start off with"},{"Start":"05:01.340 ","End":"05:04.985","Text":"the left-hand side and see if we can reach the right-hand side."},{"Start":"05:04.985 ","End":"05:10.805","Text":"On the left we have x from here, z_x from here,"},{"Start":"05:10.805 ","End":"05:14.045","Text":"that is, let me put brackets here,"},{"Start":"05:14.045 ","End":"05:19.140","Text":"y plus f of t. I don\u0027t need the 1."},{"Start":"05:23.870 ","End":"05:26.794","Text":"Now here also something cancels."},{"Start":"05:26.794 ","End":"05:28.790","Text":"I have x over x squared,"},{"Start":"05:28.790 ","End":"05:31.100","Text":"so I can cancel this x with 1 of these x\u0027s,"},{"Start":"05:31.100 ","End":"05:33.595","Text":"and I\u0027m left with just x."},{"Start":"05:33.595 ","End":"05:36.390","Text":"The minus comes out front,"},{"Start":"05:36.390 ","End":"05:47.565","Text":"what I\u0027m left with is y over x f prime of t,"},{"Start":"05:47.565 ","End":"05:50.370","Text":"and all this is just x.z_x."},{"Start":"05:50.370 ","End":"05:55.290","Text":"I still have to do plus y, and now z_y."},{"Start":"05:55.290 ","End":"06:00.405","Text":"z_y from here is x plus,"},{"Start":"06:00.405 ","End":"06:03.610","Text":"see, just f prime of t here."},{"Start":"06:06.890 ","End":"06:10.090","Text":"Let\u0027s see what we get."},{"Start":"06:13.520 ","End":"06:17.460","Text":"I\u0027ll just expand x times y is"},{"Start":"06:17.460 ","End":"06:24.540","Text":"xy plus x times f of t minus."},{"Start":"06:24.540 ","End":"06:25.925","Text":"Now look, in this term,"},{"Start":"06:25.925 ","End":"06:27.845","Text":"the x will cancel."},{"Start":"06:27.845 ","End":"06:29.615","Text":"This will cancel, well,"},{"Start":"06:29.615 ","End":"06:30.650","Text":"not with all of this,"},{"Start":"06:30.650 ","End":"06:33.050","Text":"but part of this in this term,"},{"Start":"06:33.050 ","End":"06:38.669","Text":"so we get y f prime of t,"},{"Start":"06:38.669 ","End":"06:40.620","Text":"and now this is expanded,"},{"Start":"06:40.620 ","End":"06:46.180","Text":"now we need this, plus yx,"},{"Start":"06:46.580 ","End":"06:51.120","Text":"and then plus yf prime of"},{"Start":"06:51.120 ","End":"06:59.155","Text":"t. Now, something cancels here."},{"Start":"06:59.155 ","End":"07:05.545","Text":"This 1 cancels with this 1 and now what we\u0027re left with,"},{"Start":"07:05.545 ","End":"07:07.690","Text":"I\u0027m going to scroll a bit,"},{"Start":"07:07.690 ","End":"07:15.850","Text":"is equal to, we have xy plus,"},{"Start":"07:17.570 ","End":"07:20.590","Text":"what we\u0027re aiming for is z,"},{"Start":"07:20.590 ","End":"07:22.390","Text":"we have the xy, how do we see?"},{"Start":"07:22.390 ","End":"07:24.085","Text":"We don\u0027t see any z here."},{"Start":"07:24.085 ","End":"07:29.920","Text":"But look, the definition of z is xy plus xf of t. I"},{"Start":"07:29.920 ","End":"07:35.700","Text":"can write this yx here over here as xy again,"},{"Start":"07:35.700 ","End":"07:37.725","Text":"that\u0027s this 1 here,"},{"Start":"07:37.725 ","End":"07:40.890","Text":"and then the xf of t,"},{"Start":"07:40.890 ","End":"07:48.365","Text":"I can write xf of t. Now this bit here,"},{"Start":"07:48.365 ","End":"07:51.400","Text":"I can substitute because look,"},{"Start":"07:51.400 ","End":"07:54.300","Text":"z is xy plus xf of t,"},{"Start":"07:54.300 ","End":"07:56.840","Text":"so this bit I underlined is,"},{"Start":"07:56.840 ","End":"07:58.440","Text":"I\u0027ll just write it over here,"},{"Start":"07:58.440 ","End":"08:02.025","Text":"it\u0027s xy plus z,"},{"Start":"08:02.025 ","End":"08:04.500","Text":"and that\u0027s what we wanted to get,"},{"Start":"08:04.500 ","End":"08:08.010","Text":"so we\u0027ve proved it and we\u0027re done."}],"ID":8974},{"Watched":false,"Name":"Exercise 11","Duration":"10m 37s","ChapterTopicVideoID":8629,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.460","Text":"In this exercise, we\u0027re given u as a function of 3 variables, x, y, and z,"},{"Start":"00:05.460 ","End":"00:12.225","Text":"given as x squared times some function of 2 variables applied to y over x and x over z,"},{"Start":"00:12.225 ","End":"00:14.550","Text":"and we have to prove this equality,"},{"Start":"00:14.550 ","End":"00:16.100","Text":"I won\u0027t read it out."},{"Start":"00:16.100 ","End":"00:20.160","Text":"The thing to notice is that there is a function of 2 variables,"},{"Start":"00:20.160 ","End":"00:22.190","Text":"and I want to give the variables names,"},{"Start":"00:22.190 ","End":"00:29.115","Text":"let\u0027s say f is a function of r and t. What we can say is that if we let"},{"Start":"00:29.115 ","End":"00:38.410","Text":"r equal y over x and we let t equal z over x,"},{"Start":"00:38.410 ","End":"00:43.670","Text":"then what we have is that u of x,"},{"Start":"00:43.670 ","End":"00:45.845","Text":"y, z, I\u0027ll just skip that,"},{"Start":"00:45.845 ","End":"00:55.834","Text":"is equal to x squared times f of r and t,"},{"Start":"00:55.834 ","End":"00:59.989","Text":"where r and t are given as follows."},{"Start":"00:59.989 ","End":"01:03.920","Text":"Now let\u0027s draw a dependency tree,"},{"Start":"01:03.920 ","End":"01:08.645","Text":"and at the top of the dependency tree is not u as you might think,"},{"Start":"01:08.645 ","End":"01:13.950","Text":"but f, it\u0027s the one closest to where the ones we were substituting."},{"Start":"01:13.950 ","End":"01:15.290","Text":"We substituted r and t,"},{"Start":"01:15.290 ","End":"01:22.040","Text":"so we just go up to f. We have f which depends on r and t,"},{"Start":"01:22.040 ","End":"01:24.180","Text":"and then r and t,"},{"Start":"01:24.180 ","End":"01:26.270","Text":"well, these depend on x, y, and z,"},{"Start":"01:26.270 ","End":"01:28.160","Text":"but not everyone and everything,"},{"Start":"01:28.160 ","End":"01:36.805","Text":"r just depends on x and y and t depends on x and z."},{"Start":"01:36.805 ","End":"01:39.495","Text":"Having this dependency tree,"},{"Start":"01:39.495 ","End":"01:45.260","Text":"let\u0027s now compute the partial derivatives of u with respect to x, y,"},{"Start":"01:45.260 ","End":"01:52.635","Text":"and z, so start with u with respect to x. U with respect to x,"},{"Start":"01:52.635 ","End":"01:54.140","Text":"now I\u0027m looking here,"},{"Start":"01:54.140 ","End":"01:56.270","Text":"we need a product rule."},{"Start":"01:56.270 ","End":"02:00.200","Text":"It\u0027s x squared derived,"},{"Start":"02:00.200 ","End":"02:03.500","Text":"which is 2x times this thing as is,"},{"Start":"02:03.500 ","End":"02:05.905","Text":"f of r and t,"},{"Start":"02:05.905 ","End":"02:10.535","Text":"plus x squared as is times the derivative of this with respect to"},{"Start":"02:10.535 ","End":"02:15.655","Text":"x. I need f with respect to x. I need to go to the tree,"},{"Start":"02:15.655 ","End":"02:20.405","Text":"and notice that there\u0027s 2 paths to get to x. I can go this way and this way,"},{"Start":"02:20.405 ","End":"02:23.150","Text":"or I can go this way and this way."},{"Start":"02:23.150 ","End":"02:31.775","Text":"So I\u0027m going to need the sum of 2 terms, first of all,"},{"Start":"02:31.775 ","End":"02:36.740","Text":"f with respect to r. Now that f is a function of 2 variables,"},{"Start":"02:36.740 ","End":"02:39.100","Text":"so I just say f with respect to r,"},{"Start":"02:39.100 ","End":"02:41.140","Text":"or df by dr,"},{"Start":"02:41.140 ","End":"02:46.620","Text":"and then r with respect to x,"},{"Start":"02:46.910 ","End":"02:57.869","Text":"and then plus f with respect to t and t with respect to x."},{"Start":"02:57.880 ","End":"03:04.054","Text":"Now if I plug in r with respect to x,"},{"Start":"03:04.054 ","End":"03:06.875","Text":"well, I\u0027ll just do it on the next line."},{"Start":"03:06.875 ","End":"03:15.340","Text":"This is equal to 2x f of r and t plus x squared,"},{"Start":"03:15.340 ","End":"03:21.880","Text":"and now f with respect to r. R with respect to x, I\u0027m looking here."},{"Start":"03:21.880 ","End":"03:24.900","Text":"1 over x is minus 1 over x squared,"},{"Start":"03:24.900 ","End":"03:26.365","Text":"so we have a y in,"},{"Start":"03:26.365 ","End":"03:33.025","Text":"so it\u0027s minus y over x squared with respect to x."},{"Start":"03:33.025 ","End":"03:36.465","Text":"Then plus f with respect to t,"},{"Start":"03:36.465 ","End":"03:38.700","Text":"t with respect to x,"},{"Start":"03:38.700 ","End":"03:44.890","Text":"similarly is going to be minus z over x squared."},{"Start":"03:45.920 ","End":"03:50.210","Text":"This is what we get,"},{"Start":"03:50.210 ","End":"03:53.480","Text":"but we could simplify it because x"},{"Start":"03:53.480 ","End":"04:00.310","Text":"squared here cancels with the x squared in the denominator."},{"Start":"04:00.310 ","End":"04:02.895","Text":"I think I forgot a bracket here."},{"Start":"04:02.895 ","End":"04:10.710","Text":"Yeah. So this is equal to 2x f of r and t plus,"},{"Start":"04:10.710 ","End":"04:13.540","Text":"well, it\u0027s not plus, it\u0027s going to be minus."},{"Start":"04:13.540 ","End":"04:17.655","Text":"I\u0027ll put the y in front, y fr,"},{"Start":"04:17.655 ","End":"04:27.340","Text":"and here we\u0027re going to get also a minus z f with respect to t. That\u0027s u_x."},{"Start":"04:27.340 ","End":"04:31.070","Text":"Then we need to do u with respect to y."},{"Start":"04:31.070 ","End":"04:32.630","Text":"It\u0027s going to be a bit of work,"},{"Start":"04:32.630 ","End":"04:34.970","Text":"and then after we have all 3 of them,"},{"Start":"04:34.970 ","End":"04:36.350","Text":"we\u0027re going to have to substitute."},{"Start":"04:36.350 ","End":"04:38.315","Text":"Anyway, u with respect to y,"},{"Start":"04:38.315 ","End":"04:46.185","Text":"we do get 1 break though because there\u0027s only 1 path from f to y,"},{"Start":"04:46.185 ","End":"04:50.200","Text":"and that is here and here."},{"Start":"04:50.720 ","End":"04:53.750","Text":"Once again, using the chain rule,"},{"Start":"04:53.750 ","End":"04:56.375","Text":"we get derivative with respect to y."},{"Start":"04:56.375 ","End":"04:58.190","Text":"Well, for one thing,"},{"Start":"04:58.190 ","End":"04:59.975","Text":"x squared is a constant,"},{"Start":"04:59.975 ","End":"05:02.660","Text":"so I can just take this constant and"},{"Start":"05:02.660 ","End":"05:06.500","Text":"multiply it by the derivative of this with respect to y."},{"Start":"05:06.500 ","End":"05:10.920","Text":"Now f with respect to y is"},{"Start":"05:12.140 ","End":"05:20.130","Text":"f with respect to r and then r with respect to y,"},{"Start":"05:20.130 ","End":"05:22.665","Text":"so we have f with respect to"},{"Start":"05:22.665 ","End":"05:40.370","Text":"r times r with respect to y, and that\u0027s it."},{"Start":"05:40.370 ","End":"05:45.260","Text":"That\u0027s the only path to get to f with respect to y."},{"Start":"05:45.260 ","End":"05:48.080","Text":"We didn\u0027t need 2 terms because x squared is a constant,"},{"Start":"05:48.080 ","End":"05:50.520","Text":"so no need for a product rule."},{"Start":"05:50.990 ","End":"05:55.470","Text":"I\u0027ll just drop the brackets here,"},{"Start":"05:55.470 ","End":"06:00.660","Text":"there and there, and that\u0027s basically it for u_y,"},{"Start":"06:00.660 ","End":"06:02.940","Text":"maybe a dot here."},{"Start":"06:02.940 ","End":"06:05.885","Text":"Now u with respect to z,"},{"Start":"06:05.885 ","End":"06:10.430","Text":"once again, x squared is a constant, so adjust this."},{"Start":"06:10.430 ","End":"06:13.685","Text":"I need this with respect to z."},{"Start":"06:13.685 ","End":"06:17.620","Text":"Now, z again is only 1 path that are this and this."},{"Start":"06:17.620 ","End":"06:22.155","Text":"I get this and then this,"},{"Start":"06:22.155 ","End":"06:27.050","Text":"and so f with respect to t,"},{"Start":"06:27.050 ","End":"06:30.580","Text":"t with respect to z."},{"Start":"06:30.580 ","End":"06:32.985","Text":"But I didn\u0027t quite finish here,"},{"Start":"06:32.985 ","End":"06:36.100","Text":"r with respect to y."},{"Start":"06:38.870 ","End":"06:42.740","Text":"Move this down, so this is equal to x squared"},{"Start":"06:42.740 ","End":"06:46.625","Text":"f with respect to r. But I do know what is r with respect to y."},{"Start":"06:46.625 ","End":"06:49.190","Text":"It\u0027s y times the constant 1 over x,"},{"Start":"06:49.190 ","End":"06:52.110","Text":"so it\u0027s just 1 over x."},{"Start":"06:53.020 ","End":"07:00.030","Text":"Here what I get is, let\u0027s see."},{"Start":"07:00.460 ","End":"07:08.225","Text":"We get x squared f with respect to t and t with respect to z,"},{"Start":"07:08.225 ","End":"07:11.280","Text":"that\u0027s also 1 over x."},{"Start":"07:11.600 ","End":"07:15.360","Text":"Now I have to do some combining,"},{"Start":"07:15.360 ","End":"07:17.570","Text":"let\u0027s see, just take a look."},{"Start":"07:17.570 ","End":"07:18.770","Text":"I can do it over here."},{"Start":"07:18.770 ","End":"07:20.240","Text":"I don\u0027t want to scroll."},{"Start":"07:20.240 ","End":"07:23.200","Text":"I\u0027ve got to compute this."},{"Start":"07:23.200 ","End":"07:29.340","Text":"I\u0027ll just start with the left-hand side and see if I can reach the right-hand side."},{"Start":"07:29.980 ","End":"07:35.610","Text":"Xu_x plus yu_y plus zu_z,"},{"Start":"07:35.840 ","End":"07:39.930","Text":"let\u0027s see what this is equal to."},{"Start":"07:39.930 ","End":"07:43.330","Text":"This is equal to,"},{"Start":"07:44.320 ","End":"07:47.435","Text":"I\u0027ll start over here, so we\u0027ll have enough room."},{"Start":"07:47.435 ","End":"07:51.470","Text":"It\u0027s x times u_x,"},{"Start":"07:51.470 ","End":"07:53.465","Text":"I can read off here,"},{"Start":"07:53.465 ","End":"08:03.570","Text":"and so I get 2x times f of r and t minus y f"},{"Start":"08:03.570 ","End":"08:13.610","Text":"with respect to r minus z f with respect to t. Then the next term here,"},{"Start":"08:13.610 ","End":"08:18.985","Text":"plus y times u with respect to y, is this."},{"Start":"08:18.985 ","End":"08:21.570","Text":"Note that the 1 over x will"},{"Start":"08:21.570 ","End":"08:24.920","Text":"cancel with 1 of the x\u0027s here and just put a line through the 2,"},{"Start":"08:24.920 ","End":"08:29.835","Text":"and similarly here, the 1 over x will cancel,"},{"Start":"08:29.835 ","End":"08:35.090","Text":"so y is multiplied by this,"},{"Start":"08:35.090 ","End":"08:40.440","Text":"which is just x f with respect to r,"},{"Start":"08:40.440 ","End":"08:46.800","Text":"and then I have minus z"},{"Start":"08:46.800 ","End":"08:50.880","Text":"times f with respect to"},{"Start":"08:50.880 ","End":"08:57.860","Text":"t. I\u0027m reading the wrong place."},{"Start":"08:57.860 ","End":"09:05.170","Text":"I\u0027m sorry. I need to do a take 2 on the last bit."},{"Start":"09:05.690 ","End":"09:08.605","Text":"Next is plus yu_y,"},{"Start":"09:08.605 ","End":"09:17.930","Text":"so it\u0027s plus y and u_y I\u0027m reading off here is x times f with respect to r. Finally,"},{"Start":"09:17.930 ","End":"09:21.840","Text":"z times u_z, which is from here,"},{"Start":"09:21.840 ","End":"09:26.445","Text":"which is xf_t, f with respect to t,"},{"Start":"09:26.445 ","End":"09:30.755","Text":"and now open up the brackets and see what we get."},{"Start":"09:30.755 ","End":"09:38.490","Text":"We get this with this is 2x squared f of r and t,"},{"Start":"09:38.490 ","End":"09:44.070","Text":"then this with x minus xy f with respect to r,"},{"Start":"09:44.070 ","End":"09:49.925","Text":"minus xz f with respect to t,"},{"Start":"09:49.925 ","End":"09:53.195","Text":"plus, let\u0027s put the x first,"},{"Start":"09:53.195 ","End":"09:56.700","Text":"xy f with respect to r,"},{"Start":"09:56.700 ","End":"10:03.129","Text":"and here plus xz f with respect to t,"},{"Start":"10:03.170 ","End":"10:06.665","Text":"and now plenty of stuff cancels because look,"},{"Start":"10:06.665 ","End":"10:12.690","Text":"minus xy f_r plus xy f_r minus xz,"},{"Start":"10:12.690 ","End":"10:14.985","Text":"f_t plus xz f_t."},{"Start":"10:14.985 ","End":"10:17.415","Text":"We\u0027re just left with this."},{"Start":"10:17.415 ","End":"10:21.810","Text":"But this thing, x squared f of r and t,"},{"Start":"10:21.810 ","End":"10:25.600","Text":"look over here, that\u0027s just equal to u."},{"Start":"10:25.600 ","End":"10:30.380","Text":"So I can go down here and say this is equal to 2u,"},{"Start":"10:30.380 ","End":"10:32.315","Text":"and that is the right-hand side."},{"Start":"10:32.315 ","End":"10:36.330","Text":"Yes. We\u0027ve proved it and we\u0027re done."}],"ID":8975},{"Watched":false,"Name":"Exercise 12","Duration":"7m 24s","ChapterTopicVideoID":8630,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.635","Text":"Exercise. We\u0027re given a function h of 2 variables, x and y,"},{"Start":"00:04.635 ","End":"00:09.210","Text":"but it\u0027s given in terms of f and g and as functions"},{"Start":"00:09.210 ","End":"00:12.165","Text":"of 1 variable and as follows,"},{"Start":"00:12.165 ","End":"00:13.575","Text":"a is a parameter."},{"Start":"00:13.575 ","End":"00:16.290","Text":"We don\u0027t know what the functions f and g are."},{"Start":"00:16.290 ","End":"00:19.830","Text":"We have to prove this equality involving"},{"Start":"00:19.830 ","End":"00:22.870","Text":"the second derivatives."},{"Start":"00:24.380 ","End":"00:27.170","Text":"What we want to do is,"},{"Start":"00:27.170 ","End":"00:29.360","Text":"first of all, make a substitution."},{"Start":"00:29.360 ","End":"00:32.425","Text":"F is a function of 1 variable."},{"Start":"00:32.425 ","End":"00:37.530","Text":"Let this variable be u and this variable be v,"},{"Start":"00:37.530 ","End":"00:43.649","Text":"so we have that u is y plus ax,"},{"Start":"00:43.649 ","End":"00:47.894","Text":"v is y minus ax,"},{"Start":"00:47.894 ","End":"00:52.755","Text":"and then we get that h of x,"},{"Start":"00:52.755 ","End":"01:01.520","Text":"y is equal to f of u plus g of v."},{"Start":"01:01.520 ","End":"01:04.460","Text":"I can see already that I\u0027m going to need 2 dependency trees,"},{"Start":"01:04.460 ","End":"01:09.724","Text":"1 for f. F depends on u,"},{"Start":"01:09.724 ","End":"01:14.660","Text":"but u depends on x and y."},{"Start":"01:14.660 ","End":"01:17.664","Text":"A is a parameter, not a variable,"},{"Start":"01:17.664 ","End":"01:23.670","Text":"and also g. G is dependent on v,"},{"Start":"01:23.670 ","End":"01:30.730","Text":"but v depends on x and y also."},{"Start":"01:30.920 ","End":"01:33.080","Text":"These will come in useful when"},{"Start":"01:33.080 ","End":"01:35.810","Text":"we start doing the computation."},{"Start":"01:35.810 ","End":"01:40.130","Text":"Let\u0027s start off with h with respect to x first order,"},{"Start":"01:40.130 ","End":"01:41.270","Text":"and then we\u0027ll, again,"},{"Start":"01:41.270 ","End":"01:43.400","Text":"differentiate with respect to x."},{"Start":"01:43.400 ","End":"01:47.425","Text":"First time, what I get is,"},{"Start":"01:47.425 ","End":"01:49.815","Text":"it\u0027s the sum of 2 things."},{"Start":"01:49.815 ","End":"01:55.900","Text":"I need f with respect to x and also g with respect to x."},{"Start":"01:55.900 ","End":"02:00.590","Text":"On the tree, I\u0027ll need this path to get f with respect to x,"},{"Start":"02:00.590 ","End":"02:04.440","Text":"and I\u0027ll need this path to g with respect to x."},{"Start":"02:05.000 ","End":"02:10.850","Text":"This is equal to this with respect to u."},{"Start":"02:10.850 ","End":"02:14.040","Text":"So we get f prime of u,"},{"Start":"02:14.040 ","End":"02:16.275","Text":"that\u0027s f with respect to u,"},{"Start":"02:16.275 ","End":"02:18.150","Text":"function of 1 variable,"},{"Start":"02:18.150 ","End":"02:22.280","Text":"and then u with respect to x plus"},{"Start":"02:22.280 ","End":"02:24.050","Text":"the derivative of this with respect to v."},{"Start":"02:24.050 ","End":"02:31.595","Text":"So it\u0027s g prime of v times v with respect to x."},{"Start":"02:31.595 ","End":"02:35.115","Text":"We know these 2, this is equal to,"},{"Start":"02:35.115 ","End":"02:38.480","Text":"u with respect to x is just,"},{"Start":"02:38.480 ","End":"02:40.870","Text":"let\u0027s see from here, it\u0027s just a,"},{"Start":"02:40.870 ","End":"02:42.710","Text":"y is like a constant."},{"Start":"02:42.710 ","End":"02:45.570","Text":"This is a. I\u0027ll put the a in front."},{"Start":"02:45.570 ","End":"02:48.085","Text":"It\u0027s a f-prime of u,"},{"Start":"02:48.085 ","End":"02:51.755","Text":"and v with respect to x is going to be minus a."},{"Start":"02:51.755 ","End":"02:59.060","Text":"Let me just put it minus a g-prime of v. Now I\u0027m going"},{"Start":"02:59.060 ","End":"03:02.750","Text":"to go into the second derivative with respect to"},{"Start":"03:02.750 ","End":"03:08.250","Text":"x. I\u0027ll need a dependency tree for f-prime and g-prime."},{"Start":"03:08.570 ","End":"03:11.190","Text":"I don\u0027t have to draw a new tree,"},{"Start":"03:11.190 ","End":"03:14.510","Text":"I just have to stick a prime here and here,"},{"Start":"03:14.510 ","End":"03:17.400","Text":"and the same thing works."},{"Start":"03:17.500 ","End":"03:20.105","Text":"What we get is a."},{"Start":"03:20.105 ","End":"03:24.545","Text":"Now, this with respect to x is here and here."},{"Start":"03:24.545 ","End":"03:26.900","Text":"It\u0027s f prime with respect to u,"},{"Start":"03:26.900 ","End":"03:30.050","Text":"so it\u0027s f double prime of u."},{"Start":"03:30.050 ","End":"03:34.390","Text":"Once again, u with respect to x,"},{"Start":"03:34.390 ","End":"03:39.705","Text":"and then minus from here,"},{"Start":"03:39.705 ","End":"03:47.400","Text":"a. G-prime with respect to v is, first of all,"},{"Start":"03:47.400 ","End":"03:51.430","Text":"g double-prime because it\u0027s g prime,"},{"Start":"03:51.710 ","End":"03:58.550","Text":"first of all, with respect to v and then v with respect to x."},{"Start":"03:58.550 ","End":"04:02.485","Text":"We\u0027ve already done ux and vx."},{"Start":"04:02.485 ","End":"04:05.470","Text":"This is a and this is minus a."},{"Start":"04:05.470 ","End":"04:12.785","Text":"If this is a, we get a squared f double prime of u."},{"Start":"04:12.785 ","End":"04:15.675","Text":"Here we get minus a."},{"Start":"04:15.675 ","End":"04:19.375","Text":"The minus a makes it plus a squared."},{"Start":"04:19.375 ","End":"04:21.775","Text":"Let\u0027s write this as minus a,"},{"Start":"04:21.775 ","End":"04:28.930","Text":"plus a squared g double prime of v. That\u0027s that part."},{"Start":"04:28.930 ","End":"04:37.380","Text":"Now I have to start doing with respect to y. I need h_yy."},{"Start":"04:37.380 ","End":"04:43.590","Text":"H with respect to y is,"},{"Start":"04:43.590 ","End":"04:47.070","Text":"and this time, I\u0027ll need a tree for"},{"Start":"04:47.070 ","End":"04:50.930","Text":"f and g but with respect to y, at least the dependency."},{"Start":"04:50.930 ","End":"04:54.490","Text":"So this thing goes, this thing goes."},{"Start":"04:54.490 ","End":"04:59.134","Text":"Instead of taking the x turning,"},{"Start":"04:59.134 ","End":"05:02.440","Text":"I take the turning to y."},{"Start":"05:02.440 ","End":"05:11.810","Text":"So h is f plus g. F with respect to y is just f with respect to u,"},{"Start":"05:11.810 ","End":"05:20.469","Text":"and then u with respect to y plus g with respect to v,"},{"Start":"05:20.469 ","End":"05:22.750","Text":"v with respect to y."},{"Start":"05:22.750 ","End":"05:25.214","Text":"Now uy and vy,"},{"Start":"05:25.214 ","End":"05:27.590","Text":"I can get them from here."},{"Start":"05:27.590 ","End":"05:31.285","Text":"They\u0027re both equal to 1."},{"Start":"05:31.285 ","End":"05:35.440","Text":"U with respect to y is 1 because this is a constant, similarly."},{"Start":"05:35.440 ","End":"05:43.740","Text":"This is just equal to that c. It\u0027s 1."},{"Start":"05:43.740 ","End":"05:51.505","Text":"It\u0027s just f-prime of u plus g prime of v. Now,"},{"Start":"05:51.505 ","End":"05:57.250","Text":"as before, I can use a similar tree for the derivatives."},{"Start":"05:57.250 ","End":"05:59.575","Text":"Just stick a prime here."},{"Start":"05:59.575 ","End":"06:04.515","Text":"We\u0027ve got h_yy equals."},{"Start":"06:04.515 ","End":"06:09.920","Text":"Now the derivative of this with respect to y is, first of all,"},{"Start":"06:09.920 ","End":"06:14.160","Text":"with respect to u,"},{"Start":"06:14.810 ","End":"06:17.190","Text":"and then u with respect to y,"},{"Start":"06:17.190 ","End":"06:19.730","Text":"we\u0027ve already shown that this is equal to 1"},{"Start":"06:19.730 ","End":"06:20.840","Text":"and this is equal to 1,"},{"Start":"06:20.840 ","End":"06:23.450","Text":"we used it here, so I\u0027m not going to skip a step."},{"Start":"06:23.450 ","End":"06:29.180","Text":"Plus g-prime with respect to v is g double-prime."},{"Start":"06:29.180 ","End":"06:32.840","Text":"Again, multiplied by v with respect to y,"},{"Start":"06:32.840 ","End":"06:36.860","Text":"which is just 1, so I\u0027m not going to write it here."},{"Start":"06:36.860 ","End":"06:40.430","Text":"Let me just highlight the 2 important bits."},{"Start":"06:40.430 ","End":"06:46.340","Text":"I have h_xx here,"},{"Start":"06:46.340 ","End":"06:49.710","Text":"and I have h_yy here."},{"Start":"06:50.180 ","End":"06:56.370","Text":"Proving this is actually now easy because h_xx,"},{"Start":"06:58.310 ","End":"07:02.180","Text":"I can just take a squared outside the brackets,"},{"Start":"07:02.180 ","End":"07:07.925","Text":"is a squared times f double-prime of u plus"},{"Start":"07:07.925 ","End":"07:13.970","Text":"g double prime of v. This is equal to a squared."},{"Start":"07:13.970 ","End":"07:18.120","Text":"But look, this plus this is equal to h_yy."},{"Start":"07:18.130 ","End":"07:23.970","Text":"So this is equal to this. We are done."}],"ID":8976},{"Watched":false,"Name":"Exercise 13 part a","Duration":"15m 27s","ChapterTopicVideoID":8631,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"Here we have a function u of x and y,"},{"Start":"00:03.420 ","End":"00:05.970","Text":"which is defined by means of 2 other functions,"},{"Start":"00:05.970 ","End":"00:11.010","Text":"f and g are both functions of 1 variable,"},{"Start":"00:11.010 ","End":"00:14.775","Text":"where we, in each case take e to the x sine y,"},{"Start":"00:14.775 ","End":"00:16.920","Text":"and here e to the x sine y."},{"Start":"00:16.920 ","End":"00:20.530","Text":"We have to prove this equality."},{"Start":"00:21.410 ","End":"00:27.420","Text":"The first thing you want to do is take this e to the x sine y and give it a name,"},{"Start":"00:27.420 ","End":"00:32.715","Text":"some letter t, e to the x sine y."},{"Start":"00:32.715 ","End":"00:36.240","Text":"That way we get that u of x,"},{"Start":"00:36.240 ","End":"00:42.120","Text":"y is just f of t minus g of"},{"Start":"00:42.120 ","End":"00:49.630","Text":"t. We have a dependency that we like to sketch in a tree,"},{"Start":"00:49.630 ","End":"00:56.010","Text":"1 tree for f and 1 for g. F depends on just"},{"Start":"00:56.010 ","End":"01:04.295","Text":"t. But t depends on x and y."},{"Start":"01:04.295 ","End":"01:05.990","Text":"Similar tree for g,"},{"Start":"01:05.990 ","End":"01:07.670","Text":"I\u0027m not going to make a separate tree."},{"Start":"01:07.670 ","End":"01:15.660","Text":"I\u0027ll just put f, g meaning each 1 separately here and both together in a way."},{"Start":"01:15.910 ","End":"01:20.830","Text":"What we\u0027re going to do now is compute."},{"Start":"01:20.830 ","End":"01:25.670","Text":"We\u0027ll compute the left-hand side and we\u0027ll compute the right-hand side and see"},{"Start":"01:25.670 ","End":"01:29.810","Text":"that they come out the same or some other way,"},{"Start":"01:29.810 ","End":"01:33.245","Text":"we\u0027ll prove that the both equal to same third thing."},{"Start":"01:33.245 ","End":"01:35.880","Text":"Anyway, let\u0027s start with u_xx."},{"Start":"01:36.110 ","End":"01:42.020","Text":"For u_xx I first need u with respect to x. I don\u0027t need it here anyway,"},{"Start":"01:42.020 ","End":"01:45.290","Text":"so that\u0027s going to be our start."},{"Start":"01:45.290 ","End":"01:50.165","Text":"Now let\u0027s look at the dependency tree or rather in a moment,"},{"Start":"01:50.165 ","End":"01:53.015","Text":"let me just say that this is equal to,"},{"Start":"01:53.015 ","End":"01:59.150","Text":"because it\u0027s equal to f of t minus g of t. We\u0027ve"},{"Start":"01:59.150 ","End":"02:06.320","Text":"got that this equals the derivative of this with respect to x."},{"Start":"02:06.320 ","End":"02:08.930","Text":"Here\u0027s what I\u0027m going to need the tree."},{"Start":"02:08.930 ","End":"02:12.455","Text":"Because f doesn\u0027t directly depend on x,"},{"Start":"02:12.455 ","End":"02:19.590","Text":"it depends on t and t depends on x."},{"Start":"02:19.590 ","End":"02:26.030","Text":"What I get is the derivative of f with respect to t. F only depends on t. It\u0027s"},{"Start":"02:26.030 ","End":"02:34.655","Text":"just f prime of t times derivative of t with respect to x,"},{"Start":"02:34.655 ","End":"02:38.640","Text":"and then we have minus."},{"Start":"02:38.660 ","End":"02:47.720","Text":"Same thing for g. G also depends on x via t. So we get g prime of t. Again,"},{"Start":"02:47.720 ","End":"02:56.084","Text":"derivative of t with respect to x. I can take t with respect to x outside the brackets,"},{"Start":"02:56.084 ","End":"03:00.605","Text":"and I\u0027m left with f prime minus g prime,"},{"Start":"03:00.605 ","End":"03:02.915","Text":"both of t. Now,"},{"Start":"03:02.915 ","End":"03:05.780","Text":"I\u0027m given t in terms of x and y,"},{"Start":"03:05.780 ","End":"03:09.530","Text":"and I could compute a derivative of t with respect to x."},{"Start":"03:09.530 ","End":"03:14.490","Text":"But from experience, it\u0027s messier and best to do that later."},{"Start":"03:14.490 ","End":"03:17.110","Text":"So I\u0027m postponing that."},{"Start":"03:17.110 ","End":"03:22.355","Text":"Let\u0027s go now for the second derivative."},{"Start":"03:22.355 ","End":"03:25.460","Text":"But maybe bear in mind that,"},{"Start":"03:25.460 ","End":"03:29.240","Text":"you know what, I\u0027m going to highlight the things that I\u0027m going to need later."},{"Start":"03:29.240 ","End":"03:35.070","Text":"Here I have u with respect to x,"},{"Start":"03:35.070 ","End":"03:36.690","Text":"that\u0027s this one here."},{"Start":"03:36.690 ","End":"03:40.170","Text":"Now, I also need u_xx both here and here."},{"Start":"03:40.170 ","End":"03:44.290","Text":"Let\u0027s differentiate again. This time,"},{"Start":"03:44.290 ","End":"03:49.145","Text":"I\u0027m going to use a product rule because I have this times this, so product rule."},{"Start":"03:49.145 ","End":"03:51.100","Text":"Derivative of the first,"},{"Start":"03:51.100 ","End":"03:53.365","Text":"which is just t_xx,"},{"Start":"03:53.365 ","End":"03:57.195","Text":"second derivative of t with respect to x twice,"},{"Start":"03:57.195 ","End":"04:01.120","Text":"times the second factor as is,"},{"Start":"04:01.120 ","End":"04:08.050","Text":"just f prime of t minus g prime of t. Then again the product rule,"},{"Start":"04:08.050 ","End":"04:15.880","Text":"this one as is and I need the derivative of this with respect to x."},{"Start":"04:15.880 ","End":"04:19.330","Text":"Just going to modify the dependency tree."},{"Start":"04:19.330 ","End":"04:22.940","Text":"It works just the same if I have f prime and g prime,"},{"Start":"04:22.940 ","End":"04:30.610","Text":"it\u0027s still dependent on t and t depends on x and y. I get the derivative of f prime with"},{"Start":"04:30.610 ","End":"04:33.970","Text":"respect to x goes through t. It\u0027s f"},{"Start":"04:33.970 ","End":"04:38.665","Text":"double prime of t first with respect to t and then t with respect to x."},{"Start":"04:38.665 ","End":"04:42.065","Text":"Similarly, g double prime of t,"},{"Start":"04:42.065 ","End":"04:45.300","Text":"t with respect to x."},{"Start":"04:47.150 ","End":"04:49.810","Text":"What I get here is again,"},{"Start":"04:49.810 ","End":"04:53.780","Text":"I can take the t_x outside the brackets and I\u0027ve got"},{"Start":"04:53.780 ","End":"04:59.210","Text":"t second derivative with respect to x."},{"Start":"04:59.210 ","End":"05:06.200","Text":"I\u0027m just copying f prime minus g prime plus t_x"},{"Start":"05:06.200 ","End":"05:16.900","Text":"squared times f double-prime and minus g double-prime."},{"Start":"05:17.150 ","End":"05:19.685","Text":"Since I\u0027m going to need this,"},{"Start":"05:19.685 ","End":"05:22.470","Text":"let me highlight this."},{"Start":"05:22.920 ","End":"05:27.865","Text":"Here, maybe I\u0027ll highlight this also to say what it is."},{"Start":"05:27.865 ","End":"05:33.609","Text":"We\u0027ve got u_x, u_xx for here and here."},{"Start":"05:33.609 ","End":"05:36.140","Text":"We need still u_yy."},{"Start":"05:37.650 ","End":"05:44.700","Text":"There\u0027s not going to be any major difference between x and y, exactly the same thing."},{"Start":"05:44.700 ","End":"05:46.810","Text":"There\u0027s something we often do in mathematics."},{"Start":"05:46.810 ","End":"05:52.030","Text":"We say, similarly, I\u0027m just going to repeat the same work with y."},{"Start":"05:52.030 ","End":"05:56.285","Text":"When I have the u_yy is equal to,"},{"Start":"05:56.285 ","End":"06:00.295","Text":"and the difference is that here is I\u0027m going to have yy,"},{"Start":"06:00.295 ","End":"06:02.945","Text":"same thing exactly here,"},{"Start":"06:02.945 ","End":"06:05.265","Text":"g prime of t,"},{"Start":"06:05.265 ","End":"06:08.940","Text":"and here t with respect to y squared."},{"Start":"06:08.940 ","End":"06:11.135","Text":"Also here the same thing,"},{"Start":"06:11.135 ","End":"06:15.770","Text":"minus g double prime of t. Exactly similarly,"},{"Start":"06:15.770 ","End":"06:18.665","Text":"just a waste of time to do the same thing again."},{"Start":"06:18.665 ","End":"06:21.140","Text":"I\u0027ll highlight this also."},{"Start":"06:21.140 ","End":"06:29.490","Text":"This equals this, and that\u0027s the last quantity I need as far as partial derivatives."},{"Start":"06:29.570 ","End":"06:32.865","Text":"Now, my strategy is going to be,"},{"Start":"06:32.865 ","End":"06:36.980","Text":"and I\u0027m going to compute the left-hand side of this separately and see what we get."},{"Start":"06:36.980 ","End":"06:39.470","Text":"Then the right hand and we\u0027ll see what we get,"},{"Start":"06:39.470 ","End":"06:41.750","Text":"and make sure that these are equal."},{"Start":"06:41.750 ","End":"06:45.695","Text":"At least at the moment, we have to prove, we don\u0027t know."},{"Start":"06:45.695 ","End":"06:48.395","Text":"Let\u0027s take the left-hand side, it\u0027s easier."},{"Start":"06:48.395 ","End":"06:59.290","Text":"U_xx plus u_yy is equal to this plus this."},{"Start":"06:59.870 ","End":"07:03.810","Text":"I have them handy right one above the other."},{"Start":"07:03.810 ","End":"07:09.619","Text":"I can just take a bit of a shortcut and collect terms."},{"Start":"07:09.619 ","End":"07:11.435","Text":"I mean, take stuff out of the brackets."},{"Start":"07:11.435 ","End":"07:17.885","Text":"This plus this is t_xx plus t_yy,"},{"Start":"07:17.885 ","End":"07:25.550","Text":"all this times f prime of t minus g prime of t. Now,"},{"Start":"07:25.550 ","End":"07:34.775","Text":"this plus this will give us this plus this is t_x squared plus t_y squared,"},{"Start":"07:34.775 ","End":"07:44.070","Text":"and then f double prime of t minus g double prime of t."},{"Start":"07:44.160 ","End":"07:49.405","Text":"Now, this is the point at which I would like to compute these quantities."},{"Start":"07:49.405 ","End":"07:50.920","Text":"T with respect to x,"},{"Start":"07:50.920 ","End":"07:54.835","Text":"with respect to y all the partial derivatives that we need,"},{"Start":"07:54.835 ","End":"07:59.470","Text":"now\u0027s the time, and I\u0027m going to use this."},{"Start":"07:59.470 ","End":"08:01.270","Text":"Let me just write that again,"},{"Start":"08:01.270 ","End":"08:04.435","Text":"that t is e to the x sine y,"},{"Start":"08:04.435 ","End":"08:11.980","Text":"t equals e to the x sine y,"},{"Start":"08:11.980 ","End":"08:17.780","Text":"so t with respect to x."},{"Start":"08:18.170 ","End":"08:24.300","Text":"Now the derivative of e to the x is just e to the x,"},{"Start":"08:24.300 ","End":"08:29.070","Text":"and a constant just ticks along so it\u0027s this,"},{"Start":"08:29.070 ","End":"08:36.265","Text":"and txx is same thing still,"},{"Start":"08:36.265 ","End":"08:41.500","Text":"and derivative of this again is e to the power of x times sine y."},{"Start":"08:41.500 ","End":"08:44.695","Text":"Any number of derivatives with respect to x."},{"Start":"08:44.695 ","End":"08:47.575","Text":"Whenever you have a constant times e to the x,"},{"Start":"08:47.575 ","End":"08:52.120","Text":"the derivative is the same as the function itself."},{"Start":"08:52.120 ","End":"08:54.920","Text":"That\u0027s that part."},{"Start":"08:55.280 ","End":"09:00.985","Text":"Now let\u0027s differentiate with respect to y."},{"Start":"09:00.985 ","End":"09:03.475","Text":"We have t with respect to y."},{"Start":"09:03.475 ","End":"09:06.700","Text":"This time x is the constant and e to the x is"},{"Start":"09:06.700 ","End":"09:09.895","Text":"also a constant derivative of sine is cosine,"},{"Start":"09:09.895 ","End":"09:13.989","Text":"so we have e to the x cosine y,"},{"Start":"09:13.989 ","End":"09:21.790","Text":"and if I differentiate again with respect to y,"},{"Start":"09:21.790 ","End":"09:26.065","Text":"then I get derivative of cosine is minus sine,"},{"Start":"09:26.065 ","End":"09:30.655","Text":"so I get minus e to the x sine y."},{"Start":"09:30.655 ","End":"09:36.340","Text":"Now I pretty much have all the quantities I need and I\u0027m going to just substitute."},{"Start":"09:36.340 ","End":"09:38.350","Text":"Let\u0027s see what we get."},{"Start":"09:38.350 ","End":"09:39.940","Text":"I\u0027ll just copy this again,"},{"Start":"09:39.940 ","End":"09:45.940","Text":"uxx plus uyy equals txx,"},{"Start":"09:45.940 ","End":"09:53.920","Text":"I have from here e to the x sine y plus this is minus e to the x sine y."},{"Start":"09:53.920 ","End":"09:59.435","Text":"Notice that this bit becomes 0."},{"Start":"09:59.435 ","End":"10:04.740","Text":"See this minus this, well,"},{"Start":"10:04.740 ","End":"10:10.515","Text":"perhaps it does make a note to that txx plus tyy equals 0."},{"Start":"10:10.515 ","End":"10:12.820","Text":"That takes care of this 1."},{"Start":"10:12.820 ","End":"10:17.064","Text":"Now, for here I\u0027ll need tx squared plus ty squared,"},{"Start":"10:17.064 ","End":"10:22.120","Text":"so t squared plus ty squared equals this thing"},{"Start":"10:22.120 ","End":"10:28.105","Text":"squared is I square each 1 separately,"},{"Start":"10:28.105 ","End":"10:35.695","Text":"e to the x squared is e to the 2x This squared is sine squared y,"},{"Start":"10:35.695 ","End":"10:45.670","Text":"and ty squared will be the same thing just with cosine e to the 2x cosine squared y."},{"Start":"10:45.670 ","End":"10:50.710","Text":"But because sine squared plus cosine squared is 1,"},{"Start":"10:50.710 ","End":"10:54.265","Text":"this simplifies to just e to the 2x."},{"Start":"10:54.265 ","End":"10:56.665","Text":"I\u0027ve got a simple expression for this."},{"Start":"10:56.665 ","End":"11:00.205","Text":"This 1 is 0,"},{"Start":"11:00.205 ","End":"11:03.950","Text":"this 1 is e to the 2x,"},{"Start":"11:04.470 ","End":"11:10.300","Text":"and so altogether what we get is e to the power of"},{"Start":"11:10.300 ","End":"11:17.050","Text":"2x f double-prime minus g double-prime,"},{"Start":"11:17.050 ","End":"11:24.205","Text":"and if we go back up a second,"},{"Start":"11:24.205 ","End":"11:30.190","Text":"all this is the left-hand side of this equation."},{"Start":"11:30.190 ","End":"11:36.910","Text":"Now, what we\u0027re going to do is compute the right-hand side and hopefully,"},{"Start":"11:36.910 ","End":"11:38.785","Text":"we get the same thing."},{"Start":"11:38.785 ","End":"11:43.045","Text":"Let\u0027s just remember uxx minus ux over sine squared y,"},{"Start":"11:43.045 ","End":"11:45.640","Text":"and now scroll down."},{"Start":"11:45.640 ","End":"11:50.350","Text":"I\u0027ll just also mark this 1 in the color that it was,"},{"Start":"11:50.350 ","End":"11:56.365","Text":"and now what we want to do is do uxx minus ux"},{"Start":"11:56.365 ","End":"12:03.535","Text":"over sine squared y, and let\u0027s see."},{"Start":"12:03.535 ","End":"12:07.030","Text":"Now, we have them both written next to each other."},{"Start":"12:07.030 ","End":"12:11.110","Text":"Look ux and uxx, this minus this."},{"Start":"12:11.110 ","End":"12:14.095","Text":"The first term here,"},{"Start":"12:14.095 ","End":"12:19.370","Text":"this part, minus this will give us txx minus tx."},{"Start":"12:20.430 ","End":"12:27.470","Text":"All this times the f prime minus g prime,"},{"Start":"12:30.390 ","End":"12:35.470","Text":"but the second bit just as is nothing to subtract."},{"Start":"12:35.470 ","End":"12:40.105","Text":"We get tx squared,"},{"Start":"12:40.105 ","End":"12:46.580","Text":"f double-prime minus g double-prime."},{"Start":"12:46.620 ","End":"12:52.149","Text":"Now we have a lot of these computed already,"},{"Start":"12:52.149 ","End":"12:58.600","Text":"tx squared is this bit."},{"Start":"12:58.600 ","End":"13:02.830","Text":"That\u0027s the tx squared,"},{"Start":"13:02.830 ","End":"13:12.805","Text":"and txx minus tx would be also 0."},{"Start":"13:12.805 ","End":"13:22.269","Text":"Look from these 2, this is going to be 0,"},{"Start":"13:22.269 ","End":"13:27.940","Text":"so this is 0, this is e to"},{"Start":"13:27.940 ","End":"13:37.010","Text":"the 2x sine squared y,"},{"Start":"13:37.050 ","End":"13:43.375","Text":"and so this thing is equal to"},{"Start":"13:43.375 ","End":"13:52.810","Text":"uxx minus ux is"},{"Start":"13:52.810 ","End":"13:55.610","Text":"what I just computed here."},{"Start":"13:57.840 ","End":"14:02.770","Text":"It would have been better if I just said this"},{"Start":"14:02.770 ","End":"14:08.110","Text":"divided by sine squared y, okay, that\u0027s better,"},{"Start":"14:08.110 ","End":"14:11.200","Text":"and now continuing, we get the first part,"},{"Start":"14:11.200 ","End":"14:13.120","Text":"0 times something is 0,"},{"Start":"14:13.120 ","End":"14:16.750","Text":"so we\u0027ve got e to"},{"Start":"14:16.750 ","End":"14:21.310","Text":"the 2x sine squared y"},{"Start":"14:21.310 ","End":"14:26.920","Text":"times f double-prime minus g double-prime."},{"Start":"14:26.920 ","End":"14:29.815","Text":"I\u0027m fed up of writing the t\u0027s all the time,"},{"Start":"14:29.815 ","End":"14:37.660","Text":"divided by sine squared y."},{"Start":"14:37.660 ","End":"14:43.015","Text":"Look, this sine squared y and this sine squared y cancel,"},{"Start":"14:43.015 ","End":"14:45.655","Text":"so I\u0027m just left with,"},{"Start":"14:45.655 ","End":"14:48.475","Text":"I\u0027 going to scroll once more,"},{"Start":"14:48.475 ","End":"14:59.650","Text":"this is equal to e to the 2 x f double-prime minus g double-prime."},{"Start":"14:59.650 ","End":"15:05.830","Text":"This bit is the right-hand side of the thing we have to prove."},{"Start":"15:05.830 ","End":"15:07.540","Text":"But look, this is equal to this."},{"Start":"15:07.540 ","End":"15:10.570","Text":"Okay, I was lazy and didn\u0027t write the brackets t,"},{"Start":"15:10.570 ","End":"15:12.385","Text":"but it is the same."},{"Start":"15:12.385 ","End":"15:19.180","Text":"Going back up, I can now declare that yes,"},{"Start":"15:19.180 ","End":"15:22.870","Text":"I checked each 1 separately and they came out to be the same thing,"},{"Start":"15:22.870 ","End":"15:27.050","Text":"so they\u0027re equal, and we\u0027re finally done."}],"ID":8977},{"Watched":false,"Name":"Exercise 13 part b","Duration":"6m 29s","ChapterTopicVideoID":8632,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"This exercise is actually a continuation of the previous exercise,"},{"Start":"00:04.050 ","End":"00:05.865","Text":"where we had the same setup."},{"Start":"00:05.865 ","End":"00:07.745","Text":"But there we proved something else,"},{"Start":"00:07.745 ","End":"00:12.240","Text":"and here we\u0027re proving that u x y is u y x."},{"Start":"00:12.240 ","End":"00:16.950","Text":"The 2 mixed second-order partial derivatives are equal."},{"Start":"00:16.950 ","End":"00:21.520","Text":"Let me just copy the stuff from the previous exercise."},{"Start":"00:21.650 ","End":"00:25.230","Text":"What we did there was to let t equal this,"},{"Start":"00:25.230 ","End":"00:26.520","Text":"e to the x sine y."},{"Start":"00:26.520 ","End":"00:28.560","Text":"That appears both here and here,"},{"Start":"00:28.560 ","End":"00:35.235","Text":"and then we got the u was f of t minus g of t. We also computed this."},{"Start":"00:35.235 ","End":"00:39.210","Text":"I don\u0027t think actually did you why explicitly,"},{"Start":"00:39.210 ","End":"00:43.520","Text":"but it\u0027s similarly exactly the same,"},{"Start":"00:43.520 ","End":"00:46.475","Text":"just the only differences there\u0027ll be a y here."},{"Start":"00:46.475 ","End":"00:50.780","Text":"Also f prime of t minus g prime of t,"},{"Start":"00:50.780 ","End":"00:54.710","Text":"and we had the dependency tree with or without the primes."},{"Start":"00:54.710 ","End":"01:00.945","Text":"Either way, they depend separately on t and then through t,"},{"Start":"01:00.945 ","End":"01:04.140","Text":"x and y. Let\u0027s see."},{"Start":"01:04.140 ","End":"01:08.840","Text":"Now, we have to do the second-order mixed derivatives."},{"Start":"01:08.840 ","End":"01:11.100","Text":"I think I\u0027ll go for u y x,"},{"Start":"01:11.100 ","End":"01:13.845","Text":"first, because it\u0027s just more convenient."},{"Start":"01:13.845 ","End":"01:17.960","Text":"U y handy, and the tree is marked with respect to"},{"Start":"01:17.960 ","End":"01:23.330","Text":"x. I\u0027ll differentiate this 1 with respect to x,"},{"Start":"01:23.330 ","End":"01:25.730","Text":"and later I\u0027ll do this with respect to y."},{"Start":"01:25.730 ","End":"01:30.350","Text":"This is equal to product rule."},{"Start":"01:30.350 ","End":"01:34.745","Text":"This with respect to x is just t y,"},{"Start":"01:34.745 ","End":"01:37.415","Text":"x, and this 1 as is,"},{"Start":"01:37.415 ","End":"01:40.670","Text":"forget the primes just for convenience."},{"Start":"01:40.670 ","End":"01:43.715","Text":"Plus this 1 as is,"},{"Start":"01:43.715 ","End":"01:46.685","Text":"and now I need this with respect to x."},{"Start":"01:46.685 ","End":"01:50.180","Text":"From the tree f prime, first of all,"},{"Start":"01:50.180 ","End":"01:55.670","Text":"with respect to t gives me f double prime and then t"},{"Start":"01:55.670 ","End":"02:02.110","Text":"with respect to x minus similarly g double-prime,"},{"Start":"02:02.110 ","End":"02:05.925","Text":"and again, t with respect to x,"},{"Start":"02:05.925 ","End":"02:10.175","Text":"t y x is equal to,"},{"Start":"02:10.175 ","End":"02:15.960","Text":"we already had t with respect to y,"},{"Start":"02:18.610 ","End":"02:21.470","Text":"e to the x cosine y,"},{"Start":"02:21.470 ","End":"02:24.320","Text":"I guess I didn\u0027t need to bring it and just do it again."},{"Start":"02:24.320 ","End":"02:27.845","Text":"With respect to y, the e to the x is a constant,"},{"Start":"02:27.845 ","End":"02:31.610","Text":"and then if I differentiate this with respect to x,"},{"Start":"02:31.610 ","End":"02:38.750","Text":"now this time cosine y is the constant and it\u0027s just e to the x cosine y."},{"Start":"02:38.750 ","End":"02:45.615","Text":"It doesn\u0027t change when I differentiate with respect to x. I guess I\u0027ll need the others,"},{"Start":"02:45.615 ","End":"02:47.800","Text":"t with respect to x,"},{"Start":"02:47.800 ","End":"02:57.205","Text":"either from memory or from doing it again is e to the x sine y and t x y."},{"Start":"02:57.205 ","End":"03:03.530","Text":"This with respect to y gives us e to the x cosine y."},{"Start":"03:04.700 ","End":"03:09.925","Text":"What we have here is t y x,"},{"Start":"03:09.925 ","End":"03:11.500","Text":"that\u0027s this is e to"},{"Start":"03:11.500 ","End":"03:21.175","Text":"the x, cosine y times f prime minus g prime plus g with respect to y."},{"Start":"03:21.175 ","End":"03:27.175","Text":"Here it is e to the x, cosine y,"},{"Start":"03:27.175 ","End":"03:32.420","Text":"and then times t x,"},{"Start":"03:32.420 ","End":"03:36.500","Text":"I guess I should have said that."},{"Start":"03:36.500 ","End":"03:40.200","Text":"I\u0027m bringing TX outside the brackets."},{"Start":"03:40.660 ","End":"03:47.280","Text":"Okay. I\u0027ll just indicate this or take this away from here and here and stick it here."},{"Start":"03:48.320 ","End":"03:51.180","Text":"Now, I move this to the right,"},{"Start":"03:51.180 ","End":"03:54.630","Text":"and t x here is e to the x sine y,"},{"Start":"03:54.630 ","End":"03:57.840","Text":"so e to the x sine y,"},{"Start":"03:57.840 ","End":"03:59.570","Text":"and then in brackets,"},{"Start":"03:59.570 ","End":"04:03.889","Text":"f double-prime minus g double prime."},{"Start":"04:03.889 ","End":"04:07.910","Text":"I don\u0027t think I\u0027m going to simplify it anymore."},{"Start":"04:07.910 ","End":"04:15.785","Text":"Let\u0027s just go for u x y and then we\u0027ll see if we need to simplify u x y is equal to."},{"Start":"04:15.785 ","End":"04:22.320","Text":"This time I take this and differentiate it with respect to y."},{"Start":"04:22.960 ","End":"04:26.150","Text":"So what we get is again,"},{"Start":"04:26.150 ","End":"04:32.050","Text":"the product rule, t x with respect to y is t x y."},{"Start":"04:32.050 ","End":"04:40.535","Text":"Then the second factor as is f prime minus g prime plus the first factor as is,"},{"Start":"04:40.535 ","End":"04:46.025","Text":"and the second with respect to y. I don\u0027t need to modify the Dependency Tree."},{"Start":"04:46.025 ","End":"04:51.280","Text":"This time I need the road to y."},{"Start":"04:51.470 ","End":"04:56.975","Text":"What I get is this with respect to t is just f double-prime,"},{"Start":"04:56.975 ","End":"05:00.355","Text":"and then t with respect to y,"},{"Start":"05:00.355 ","End":"05:03.650","Text":"and then minus similarly with g,"},{"Start":"05:03.650 ","End":"05:07.385","Text":"g double prime also t with respect to y,"},{"Start":"05:07.385 ","End":"05:12.095","Text":"and then this time I\u0027ll remember to take the t y out front."},{"Start":"05:12.095 ","End":"05:19.889","Text":"It\u0027s t x y. F prime minus g prime plus t x,"},{"Start":"05:19.889 ","End":"05:27.280","Text":"and then t y. I take out f double-prime minus g double-prime,"},{"Start":"05:27.280 ","End":"05:35.790","Text":"and now I see that I actually didn\u0027t have to do this step."},{"Start":"05:36.520 ","End":"05:38.960","Text":"Wasn\u0027t a waste of time,"},{"Start":"05:38.960 ","End":"05:42.890","Text":"but we\u0027re actually not going to use this because I notice now"},{"Start":"05:42.890 ","End":"05:51.380","Text":"that I can get that this is the same as this with 1 minor difference."},{"Start":"05:51.380 ","End":"05:53.545","Text":"The minor difference."},{"Start":"05:53.545 ","End":"05:58.635","Text":"When I compare these 2 is that here I have t y x,"},{"Start":"05:58.635 ","End":"06:00.365","Text":"and here I have t x y."},{"Start":"06:00.365 ","End":"06:02.735","Text":"Other than that, the same thing."},{"Start":"06:02.735 ","End":"06:08.985","Text":"But, look, here\u0027s t y x,"},{"Start":"06:08.985 ","End":"06:10.890","Text":"and here\u0027s t x y."},{"Start":"06:10.890 ","End":"06:13.125","Text":"They are both the same."},{"Start":"06:13.125 ","End":"06:17.565","Text":"Because this and this is the same now the whole expression is the same."},{"Start":"06:17.565 ","End":"06:21.660","Text":"These 2, this and this are equal."},{"Start":"06:21.660 ","End":"06:23.895","Text":"That\u0027s what we had to show,"},{"Start":"06:23.895 ","End":"06:30.180","Text":"and so we\u0027re done and even did an extra step that we didn\u0027t need to. Okay."}],"ID":8978},{"Watched":false,"Name":"Exercise 13 part c","Duration":"4m 57s","ChapterTopicVideoID":8633,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.900","Text":"This exercise is actually a continuation of the previous couple of exercises."},{"Start":"00:06.900 ","End":"00:11.940","Text":"What we had there is we did a substitution, I\u0027ll write it here,"},{"Start":"00:11.940 ","End":"00:18.165","Text":"where t is equal to e to the x sine y,"},{"Start":"00:18.165 ","End":"00:21.270","Text":"because really, f and g are functions of 1 variable t."},{"Start":"00:21.270 ","End":"00:25.500","Text":"We also had that u is f of t minus g of t."},{"Start":"00:25.500 ","End":"00:32.384","Text":"Anyway, we did the computations there and we found that u_xy,"},{"Start":"00:32.384 ","End":"00:34.895","Text":"the mixed partial derivative,"},{"Start":"00:34.895 ","End":"00:36.335","Text":"which we\u0027re going to need,"},{"Start":"00:36.335 ","End":"00:42.590","Text":"is equal to t derivative with respect to x,"},{"Start":"00:42.590 ","End":"00:51.290","Text":"with respect to y times f prime of t minus g prime of t"},{"Start":"00:51.290 ","End":"00:57.360","Text":"plus first-order derivative t with respect to x,"},{"Start":"00:57.360 ","End":"00:59.295","Text":"then t with respect to y."},{"Start":"00:59.295 ","End":"01:04.160","Text":"Here we had second-order derivative of f"},{"Start":"01:04.160 ","End":"01:08.150","Text":"with respect to t minus same thing with g."},{"Start":"01:08.150 ","End":"01:18.120","Text":"What we need to do is to compute u_x at the point 1, Pi"},{"Start":"01:18.120 ","End":"01:20.910","Text":"and we\u0027re given 2 pieces of information."},{"Start":"01:20.910 ","End":"01:23.740","Text":"Let\u0027s see if they\u0027re going to help us."},{"Start":"01:24.650 ","End":"01:29.550","Text":"Let me say that, looking here,"},{"Start":"01:29.550 ","End":"01:38.650","Text":"we need t_x, we need t_y, and we need t_xy."},{"Start":"01:40.090 ","End":"01:43.070","Text":"We actually computed them before."},{"Start":"01:43.070 ","End":"01:45.680","Text":"But they\u0027re so easy so we\u0027ll just do it again."},{"Start":"01:45.680 ","End":"01:47.630","Text":"This with respect to x."},{"Start":"01:47.630 ","End":"01:49.400","Text":"Sine y is a constant,"},{"Start":"01:49.400 ","End":"01:53.105","Text":"is just e to the x sine y."},{"Start":"01:53.105 ","End":"01:58.365","Text":"With respect to y, e to the x cosine y,"},{"Start":"01:58.365 ","End":"02:00.045","Text":"because e to the x is a constant,"},{"Start":"02:00.045 ","End":"02:09.010","Text":"and this with respect to y is also e to the x cosine y."},{"Start":"02:11.210 ","End":"02:15.970","Text":"What I want to do now is compute these at the point 1, Pi,"},{"Start":"02:15.970 ","End":"02:21.930","Text":"so we\u0027re going to let x, y equal 1, Pi"},{"Start":"02:21.930 ","End":"02:26.150","Text":"and see what each of these comes out to be."},{"Start":"02:26.150 ","End":"02:30.570","Text":"Because I\u0027m going to need to substitute u_xy at 1, Pi."},{"Start":"02:30.570 ","End":"02:31.830","Text":"Let\u0027s see."},{"Start":"02:31.830 ","End":"02:33.030","Text":"X is 1."},{"Start":"02:33.030 ","End":"02:35.250","Text":"This is the x and this is the y, of course."},{"Start":"02:35.250 ","End":"02:38.165","Text":"X is 1, y is Pi."},{"Start":"02:38.165 ","End":"02:42.885","Text":"We need to know that sine Pi is 0,"},{"Start":"02:42.885 ","End":"02:46.785","Text":"and so this thing comes out to be 0."},{"Start":"02:46.785 ","End":"02:52.320","Text":"Again, sine Pi is 0, so this is also 0."},{"Start":"02:52.320 ","End":"03:00.690","Text":"Now here, cosine of Pi is minus 1 and e to the 1 is just e,"},{"Start":"03:00.690 ","End":"03:02.910","Text":"so altogether I get minus e."},{"Start":"03:02.910 ","End":"03:05.300","Text":"This is the same, so it\u0027s also minus e."},{"Start":"03:05.300 ","End":"03:07.985","Text":"Now I have these computed,"},{"Start":"03:07.985 ","End":"03:13.075","Text":"but then I also want to compute, there are some other quantities."},{"Start":"03:13.075 ","End":"03:17.565","Text":"I need f prime of t,"},{"Start":"03:17.565 ","End":"03:20.670","Text":"but look, t is 0 at our point."},{"Start":"03:20.670 ","End":"03:28.125","Text":"So I need f prime of 0 and I need g prime of 0."},{"Start":"03:28.125 ","End":"03:31.570","Text":"Possibly, you\u0027ll see why I say possibly,"},{"Start":"03:31.570 ","End":"03:36.695","Text":"I might need f double prime of 0 and g double prime of 0."},{"Start":"03:36.695 ","End":"03:48.570","Text":"F prime of 0 is given to us is 2, and g prime of 0 is 1."},{"Start":"03:48.570 ","End":"03:51.180","Text":"You might say, but how are we going to compute it"},{"Start":"03:51.180 ","End":"03:54.270","Text":"because we\u0027re not given these 2 quantities?"},{"Start":"03:54.270 ","End":"03:56.330","Text":"This I don\u0027t know and this I don\u0027t know."},{"Start":"03:56.330 ","End":"03:59.090","Text":"Turns out it doesn\u0027t matter because look,"},{"Start":"03:59.090 ","End":"04:03.944","Text":"I get that t with respect to x is 0."},{"Start":"04:03.944 ","End":"04:07.740","Text":"This thing is 0."},{"Start":"04:07.740 ","End":"04:11.210","Text":"Once I get a 0 in a product, the whole thing is 0,"},{"Start":"04:11.210 ","End":"04:14.330","Text":"so actually I don\u0027t even care about what these are"},{"Start":"04:14.330 ","End":"04:18.300","Text":"and I have everything I need to substitute."},{"Start":"04:18.340 ","End":"04:29.400","Text":"So u_xy at 1, Pi is equal to t_xy at 1, Pi,"},{"Start":"04:29.400 ","End":"04:40.115","Text":"which is minus e times f prime of 0,"},{"Start":"04:40.115 ","End":"04:48.280","Text":"which is 2, minus g prime of 0, which is 1."},{"Start":"04:48.280 ","End":"04:50.910","Text":"Altogether, 2 minus 1 is 1,"},{"Start":"04:50.910 ","End":"04:55.640","Text":"so we just get minus e and we are done."},{"Start":"04:55.640 ","End":"04:57.660","Text":"That\u0027s the answer."}],"ID":8979},{"Watched":false,"Name":"Exercise 14 part a","Duration":"8m 23s","ChapterTopicVideoID":8634,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.145","Text":"In this exercise, we\u0027re given u as a function of x and y."},{"Start":"00:05.145 ","End":"00:11.070","Text":"Then you might recognize these 2 as the polar substitution x is r cosine Theta,"},{"Start":"00:11.070 ","End":"00:12.675","Text":"y is r sine Theta."},{"Start":"00:12.675 ","End":"00:15.510","Text":"But never mind, we\u0027re not going to use that fact,"},{"Start":"00:15.510 ","End":"00:17.670","Text":"but it is the polar substitution."},{"Start":"00:17.670 ","End":"00:21.555","Text":"What we have to show is that the following holds,"},{"Start":"00:21.555 ","End":"00:25.450","Text":"I won\u0027t read it out, we\u0027ll just get to it."},{"Start":"00:25.450 ","End":"00:29.030","Text":"We\u0027re going to figure out"},{"Start":"00:29.030 ","End":"00:33.305","Text":"left-hand side and right-hand side separately and show that they\u0027re equal,"},{"Start":"00:33.305 ","End":"00:36.785","Text":"or we can start from 1 side and reach the other side."},{"Start":"00:36.785 ","End":"00:39.530","Text":"What I suggest is to start from the right-hand side"},{"Start":"00:39.530 ","End":"00:42.050","Text":"because we don\u0027t know what u with respect to x is"},{"Start":"00:42.050 ","End":"00:47.255","Text":"because we\u0027re not given the function f. Let\u0027s start with the right-hand side."},{"Start":"00:47.255 ","End":"00:51.700","Text":"But before that even we\u0027re going to need a dependency tree."},{"Start":"00:51.700 ","End":"00:54.975","Text":"From here we see that u depends on x and y."},{"Start":"00:54.975 ","End":"00:57.050","Text":"We start off with u at the top,"},{"Start":"00:57.050 ","End":"01:02.270","Text":"and then it depends on x and y."},{"Start":"01:02.270 ","End":"01:05.510","Text":"But from here we see that x depends on r and Theta,"},{"Start":"01:05.510 ","End":"01:06.965","Text":"and in fact so does y."},{"Start":"01:06.965 ","End":"01:14.090","Text":"Each of these splits up into 2 r Theta and r Theta."},{"Start":"01:15.170 ","End":"01:19.865","Text":"Now we can start computing the partial derivatives."},{"Start":"01:19.865 ","End":"01:24.740","Text":"First, let\u0027s do u with respect to r. Now on the tree,"},{"Start":"01:24.740 ","End":"01:27.890","Text":"we can to r in 2 ways."},{"Start":"01:27.890 ","End":"01:35.430","Text":"This way and this way to get to r or this way and this way."},{"Start":"01:35.540 ","End":"01:38.710","Text":"Using the chain rule,"},{"Start":"01:38.710 ","End":"01:41.120","Text":"this is equal to u with respect to x,"},{"Start":"01:41.120 ","End":"01:45.875","Text":"x with respect to r. If we go along the other path,"},{"Start":"01:45.875 ","End":"01:47.630","Text":"u with respect to y,"},{"Start":"01:47.630 ","End":"01:51.815","Text":"y with respect to r. Now,"},{"Start":"01:51.815 ","End":"01:55.700","Text":"we know we can compute what is y with respect to r and x with"},{"Start":"01:55.700 ","End":"02:01.965","Text":"respect to r. Let me just say this is u_x,"},{"Start":"02:01.965 ","End":"02:04.375","Text":"now x with respect to r,"},{"Start":"02:04.375 ","End":"02:08.130","Text":"I just differentiate x with respect to r,"},{"Start":"02:08.130 ","End":"02:10.719","Text":"that means the cosine Thetas are constant."},{"Start":"02:10.719 ","End":"02:15.180","Text":"We\u0027re just left with the constant cosine Theta."},{"Start":"02:15.370 ","End":"02:21.655","Text":"Similarly here, y with respect to r is just sine Theta."},{"Start":"02:21.655 ","End":"02:24.400","Text":"Now we have u_r."},{"Start":"02:24.400 ","End":"02:27.830","Text":"The next quantity we need is u Theta,"},{"Start":"02:27.830 ","End":"02:30.650","Text":"partial derivative of u with respect to Theta."},{"Start":"02:30.650 ","End":"02:39.755","Text":"We go to the tree and modify it so that we land up on Theta each time, again 2 paths."},{"Start":"02:39.755 ","End":"02:44.420","Text":"This is going to equal u with respect to x,"},{"Start":"02:44.420 ","End":"02:48.285","Text":"x with respect to Theta from the first path and the other one,"},{"Start":"02:48.285 ","End":"02:49.985","Text":"u with respect to y,"},{"Start":"02:49.985 ","End":"02:53.070","Text":"y with respect to Theta."},{"Start":"02:53.240 ","End":"02:56.445","Text":"X with respect to Theta,"},{"Start":"02:56.445 ","End":"03:00.890","Text":"this time r is the constant derivative of cosine is minus sine."},{"Start":"03:00.890 ","End":"03:05.900","Text":"We get minus r sine Theta"},{"Start":"03:05.900 ","End":"03:11.120","Text":"I brought the minus in front of the r. Here the derivative of sine is cosine,"},{"Start":"03:11.120 ","End":"03:13.740","Text":"so we get u with respect to y,"},{"Start":"03:13.740 ","End":"03:17.380","Text":"y Theta is our cosine Theta."},{"Start":"03:20.660 ","End":"03:28.200","Text":"I could take r outside the brackets and say that this is r minus"},{"Start":"03:28.200 ","End":"03:36.220","Text":"u_x sine Theta plus u_y cosine Theta."},{"Start":"03:36.590 ","End":"03:39.790","Text":"We have u_r we have u Theta."},{"Start":"03:39.790 ","End":"03:44.580","Text":"Why don\u0027t we just substitute in the right-hand side and see what we get."},{"Start":"03:45.430 ","End":"03:48.050","Text":"I\u0027d like to do them on the same page and you just move"},{"Start":"03:48.050 ","End":"03:50.915","Text":"the tree and we\u0027ll go to a different color."},{"Start":"03:50.915 ","End":"03:54.560","Text":"Let\u0027s say we need u_r squared,"},{"Start":"03:54.560 ","End":"04:00.585","Text":"just copying plus 1 over r squared u Theta squared."},{"Start":"04:00.585 ","End":"04:02.160","Text":"Let\u0027s see what we get."},{"Start":"04:02.160 ","End":"04:05.940","Text":"U_r squared is this thing."},{"Start":"04:05.940 ","End":"04:09.585","Text":"We get, well, I\u0027ll just write it,"},{"Start":"04:09.585 ","End":"04:16.695","Text":"u_x cosine Theta plus u_y sine Theta"},{"Start":"04:16.695 ","End":"04:26.265","Text":"squared plus 1 over r squared times u Theta squared,"},{"Start":"04:26.265 ","End":"04:31.010","Text":"which is r squared from here."},{"Start":"04:31.010 ","End":"04:32.885","Text":"Then this thing squared,"},{"Start":"04:32.885 ","End":"04:36.755","Text":"don\u0027t know why I have square brackets, just copying this,"},{"Start":"04:36.755 ","End":"04:41.150","Text":"minus u_x sine Theta plus"},{"Start":"04:41.150 ","End":"04:48.490","Text":"u_y cosine Theta squared equals."},{"Start":"04:48.490 ","End":"04:54.050","Text":"I\u0027ll just get rid of the tree all together. Let\u0027s expand."},{"Start":"04:54.050 ","End":"05:01.550","Text":"Remember the formula for a plus or minus b squared,"},{"Start":"05:01.550 ","End":"05:07.415","Text":"special product binomial is a squared plus or minus 2ab plus b squared,"},{"Start":"05:07.415 ","End":"05:09.020","Text":"by which I mean if I take plus here,"},{"Start":"05:09.020 ","End":"05:11.135","Text":"I take plus here and minus or minus."},{"Start":"05:11.135 ","End":"05:19.355","Text":"Using this, we get first 1 squared is u_x squared cosine squared Theta,"},{"Start":"05:19.355 ","End":"05:20.630","Text":"because when I square a product,"},{"Start":"05:20.630 ","End":"05:22.655","Text":"I square each separately."},{"Start":"05:22.655 ","End":"05:25.400","Text":"Then plus twice this times this"},{"Start":"05:25.400 ","End":"05:34.060","Text":"2u_xu_y cosine Theta sine Theta"},{"Start":"05:35.960 ","End":"05:44.305","Text":"plus u_y squared sine squared Theta."},{"Start":"05:44.305 ","End":"05:47.165","Text":"I\u0027m going to continue on the next line."},{"Start":"05:47.165 ","End":"05:51.050","Text":"Plus, well, we do get a break here"},{"Start":"05:51.050 ","End":"05:55.645","Text":"because the 1 over r squared with the r squared just cancels."},{"Start":"05:55.645 ","End":"05:58.650","Text":"Continuing we get this thing squared,"},{"Start":"05:58.650 ","End":"06:05.620","Text":"which is u_x squared sine squared Theta."},{"Start":"06:05.620 ","End":"06:11.100","Text":"This is just a continuation of this over here."},{"Start":"06:11.100 ","End":"06:14.250","Text":"Then we\u0027re continuing over here."},{"Start":"06:14.250 ","End":"06:19.560","Text":"Then minus, it\u0027s a difference."},{"Start":"06:19.560 ","End":"06:29.415","Text":"Minus 2u_x_uy sine Theta"},{"Start":"06:29.415 ","End":"06:31.680","Text":"cosine Theta."},{"Start":"06:31.680 ","End":"06:39.094","Text":"Finally, u_y squared cosine squared Theta."},{"Start":"06:39.094 ","End":"06:43.235","Text":"Now, I\u0027m not going to be so bad if we play our cards."},{"Start":"06:43.235 ","End":"06:47.420","Text":"What I suggest is just simply looking"},{"Start":"06:47.420 ","End":"06:52.175","Text":"at it like in 2 rows and just adding actually came out convenient."},{"Start":"06:52.175 ","End":"06:59.490","Text":"Because if I add let\u0027s put in a line here to make it clearer, this plus this."},{"Start":"06:59.490 ","End":"07:02.950","Text":"Let\u0027s not forget that sine squared of"},{"Start":"07:02.950 ","End":"07:07.660","Text":"whatever plus cosine squared of the same thing is equal to 1."},{"Start":"07:07.660 ","End":"07:09.460","Text":"From this plus this,"},{"Start":"07:09.460 ","End":"07:12.205","Text":"we can use the cosine squared plus sine squared."},{"Start":"07:12.205 ","End":"07:14.980","Text":"We just get u_x squared."},{"Start":"07:14.980 ","End":"07:20.240","Text":"This and this, it\u0027s a plus and a minus that 0."},{"Start":"07:21.050 ","End":"07:26.305","Text":"Here again, sine squared plus cosine squared is 1,"},{"Start":"07:26.305 ","End":"07:29.449","Text":"so it\u0027s u_y squared."},{"Start":"07:29.449 ","End":"07:32.295","Text":"The middle term, it\u0027s not quite the sine is cosine,"},{"Start":"07:32.295 ","End":"07:34.020","Text":"sine this is sine cosine,"},{"Start":"07:34.020 ","End":"07:36.175","Text":"but if we just reversed the order,"},{"Start":"07:36.175 ","End":"07:38.600","Text":"obviously it\u0027s the same thing."},{"Start":"07:39.530 ","End":"07:42.890","Text":"What we get is that this thing,"},{"Start":"07:42.890 ","End":"07:46.410","Text":"if I just summarize it, is equal to."},{"Start":"07:46.870 ","End":"07:50.540","Text":"This is 1 line, this equals this."},{"Start":"07:50.540 ","End":"07:53.675","Text":"This equals, well, just going to copy it nicer."},{"Start":"07:53.675 ","End":"07:58.310","Text":"It\u0027s just u_x squared plus u_y squared."},{"Start":"07:58.310 ","End":"08:02.154","Text":"I\u0027m just throwing out the 0."},{"Start":"08:02.154 ","End":"08:05.105","Text":"Now look, I started off with this,"},{"Start":"08:05.105 ","End":"08:07.205","Text":"which is the right-hand side,"},{"Start":"08:07.205 ","End":"08:08.870","Text":"and ended up with this,"},{"Start":"08:08.870 ","End":"08:11.165","Text":"which is the left-hand side."},{"Start":"08:11.165 ","End":"08:13.780","Text":"I could add brackets,"},{"Start":"08:14.720 ","End":"08:18.360","Text":"but it\u0027s the same thing."},{"Start":"08:18.360 ","End":"08:21.200","Text":"We have proven the equality,"},{"Start":"08:21.200 ","End":"08:24.300","Text":"and so we are done."}],"ID":8980},{"Watched":false,"Name":"Exercise 14 part b","Duration":"6m 31s","ChapterTopicVideoID":8635,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.375","Text":"This exercise is a continuation of the previous one,"},{"Start":"00:04.375 ","End":"00:08.015","Text":"where we had u as a function of x and y,"},{"Start":"00:08.015 ","End":"00:11.760","Text":"and x and y given in terms of r and Theta."},{"Start":"00:11.760 ","End":"00:17.155","Text":"In fact, these are the polar coordinates or the polar substitution."},{"Start":"00:17.155 ","End":"00:22.060","Text":"This part is new what we have to prove and I\u0027m not going to read it out."},{"Start":"00:22.060 ","End":"00:25.615","Text":"What we\u0027re going to do is compute the left-hand side and see that we reach"},{"Start":"00:25.615 ","End":"00:29.725","Text":"the right-hand side and that\u0027s how we\u0027ll prove this equality."},{"Start":"00:29.725 ","End":"00:31.945","Text":"I don\u0027t want to start from scratch."},{"Start":"00:31.945 ","End":"00:36.250","Text":"Last time we already computed u with respect to r,"},{"Start":"00:36.250 ","End":"00:40.719","Text":"which was u with respect to x times"},{"Start":"00:40.719 ","End":"00:46.930","Text":"cosine Theta plus u with respect to y sine Theta."},{"Start":"00:46.930 ","End":"00:50.060","Text":"We also had a tree, the dependence tree,"},{"Start":"00:50.060 ","End":"00:56.545","Text":"where we had u at the top and you depended on x and y."},{"Start":"00:56.545 ","End":"01:02.675","Text":"Then we had each of x and y in terms of r and Theta."},{"Start":"01:02.675 ","End":"01:09.675","Text":"This is just a dependency tree and what I want to say is that instead of u,"},{"Start":"01:09.675 ","End":"01:14.750","Text":"I could also have u with respect to x because this is"},{"Start":"01:14.750 ","End":"01:20.570","Text":"also a function of x and y and therefore the same tree and also uy,"},{"Start":"01:20.570 ","End":"01:24.770","Text":"all of these quantities have the same dependency tree."},{"Start":"01:24.770 ","End":"01:31.610","Text":"I\u0027m saying this now because I want to differentiate ur with respect to r,"},{"Start":"01:31.610 ","End":"01:34.700","Text":"and so I\u0027ll have to do each of these with respect to"},{"Start":"01:34.700 ","End":"01:40.550","Text":"r. Which means that it\u0027ll have to go here and here,"},{"Start":"01:40.550 ","End":"01:43.250","Text":"or here and here,"},{"Start":"01:43.250 ","End":"01:46.350","Text":"both for ux and for uy."},{"Start":"01:47.560 ","End":"01:52.645","Text":"Now let\u0027s get to urr,"},{"Start":"01:52.645 ","End":"01:55.250","Text":"partial derivative with respect to r,"},{"Start":"01:55.250 ","End":"02:00.280","Text":"cosine Theta and sine theta are constants, they just stay."},{"Start":"02:00.280 ","End":"02:06.770","Text":"All I have to do is differentiate this with respect to r and this take"},{"Start":"02:06.770 ","End":"02:13.205","Text":"the derivative with respect to r. What I get is ux,"},{"Start":"02:13.205 ","End":"02:14.945","Text":"talking about this one now."},{"Start":"02:14.945 ","End":"02:23.885","Text":"I think it\u0027s derivative with respect to x and then x with respect to r. It\u0027s uxx,"},{"Start":"02:23.885 ","End":"02:28.890","Text":"and then x with respect to r. Anyway,"},{"Start":"02:28.890 ","End":"02:34.455","Text":"I\u0027ll just write it then we\u0027ll expand in the next one, plus,"},{"Start":"02:34.455 ","End":"02:38.480","Text":"let\u0027s see, the other path gives me ux with respect to"},{"Start":"02:38.480 ","End":"02:43.490","Text":"y and y with respect to r. Now this is just this bit,"},{"Start":"02:43.490 ","End":"02:49.365","Text":"and all this times cosine Theta and now this bit,"},{"Start":"02:49.365 ","End":"02:52.240","Text":"well, it\u0027s the same thing basically."},{"Start":"02:52.240 ","End":"02:54.020","Text":"I\u0027m just going to write it."},{"Start":"02:54.020 ","End":"02:56.930","Text":"It\u0027s uy with respect to x,"},{"Start":"02:56.930 ","End":"03:02.345","Text":"x with respect to r plus uy with respect to y,"},{"Start":"03:02.345 ","End":"03:08.770","Text":"y with respect to r, and sine Theta."},{"Start":"03:08.770 ","End":"03:12.215","Text":"The differences that I have a y here and here,"},{"Start":"03:12.215 ","End":"03:16.890","Text":"I have an x here and here and then also this cosine and here the sine."},{"Start":"03:17.020 ","End":"03:21.235","Text":"Now we just want to simplify this."},{"Start":"03:21.235 ","End":"03:25.529","Text":"I\u0027m going to also compute xr and yr,"},{"Start":"03:25.529 ","End":"03:28.040","Text":"which appear here and here and also here and here."},{"Start":"03:28.040 ","End":"03:29.870","Text":"We did those in the previous,"},{"Start":"03:29.870 ","End":"03:31.430","Text":"but we could just do them again."},{"Start":"03:31.430 ","End":"03:34.675","Text":"X with respect to r from here,"},{"Start":"03:34.675 ","End":"03:42.290","Text":"because cosine Theta is a constant it\u0027s just cosine Theta and y with respect to r. Again,"},{"Start":"03:42.290 ","End":"03:43.864","Text":"cosine Theta is a constant,"},{"Start":"03:43.864 ","End":"03:47.315","Text":"so it\u0027s just that constant sine Theta."},{"Start":"03:47.315 ","End":"03:52.350","Text":"Now I can plug everything in and it\u0027s going to be 4 terms,"},{"Start":"03:52.350 ","End":"03:57.570","Text":"this with this. Let\u0027s do the first one."},{"Start":"03:57.570 ","End":"04:01.125","Text":"We have uxx, xr cosine Theta."},{"Start":"04:01.125 ","End":"04:04.950","Text":"That\u0027s uxx, but xr is"},{"Start":"04:04.950 ","End":"04:12.200","Text":"cosine Theta with another cosine Theta that makes it cosine squared Theta."},{"Start":"04:12.200 ","End":"04:13.970","Text":"That\u0027s these 2 with this,"},{"Start":"04:13.970 ","End":"04:15.920","Text":"replacing this with cosine Theta."},{"Start":"04:15.920 ","End":"04:19.985","Text":"Next one, we have uxy,"},{"Start":"04:19.985 ","End":"04:24.124","Text":"y with respect to r is sine Theta,"},{"Start":"04:24.124 ","End":"04:28.385","Text":"but still with cosine Theta, that\u0027s these 2."},{"Start":"04:28.385 ","End":"04:34.240","Text":"Now, this part, I get uyx,"},{"Start":"04:34.240 ","End":"04:38.250","Text":"x with respect to r is cosine Theta times sine Theta."},{"Start":"04:40.200 ","End":"04:44.980","Text":"The last one, uyy,"},{"Start":"04:44.980 ","End":"04:50.510","Text":"yr is sine Theta with a sine Theta makes it sine squared Theta."},{"Start":"04:50.510 ","End":"04:55.250","Text":"Now notice that this term and this term are very similar."},{"Start":"04:55.250 ","End":"04:57.825","Text":"The fact that sine and cosine now reverse,"},{"Start":"04:57.825 ","End":"05:03.720","Text":"it\u0027s just multiplication, but uxy as a theorem equals uyx."},{"Start":"05:03.720 ","End":"05:07.370","Text":"In fact, in one of the previous exercises we actually even computed,"},{"Start":"05:07.370 ","End":"05:10.175","Text":"but these 2 are the same and therefore,"},{"Start":"05:10.175 ","End":"05:15.575","Text":"these 2 middle terms could be combined to be just twice this one."},{"Start":"05:15.575 ","End":"05:26.950","Text":"I\u0027ve got here 2uxy sine Theta, cosine Theta."},{"Start":"05:27.750 ","End":"05:31.180","Text":"The other thing to notice is that u is just F,"},{"Start":"05:31.180 ","End":"05:37.695","Text":"u is f of x y. I can rewrite this and put an equals here,"},{"Start":"05:37.695 ","End":"05:40.305","Text":"I forgot equals here too."},{"Start":"05:40.305 ","End":"05:49.350","Text":"Instead of uxx, I\u0027ll write fxx cosine squared Theta plus now copy this,"},{"Start":"05:49.350 ","End":"05:52.585","Text":"I see, I want to cosine before the sine."},{"Start":"05:52.585 ","End":"05:56.050","Text":"I\u0027ll write it in the other order 2uxy,"},{"Start":"05:56.050 ","End":"05:59.475","Text":"I\u0027ll write the cosine before the sine,"},{"Start":"05:59.475 ","End":"06:01.980","Text":"so it will really look like this."},{"Start":"06:01.980 ","End":"06:04.080","Text":"Then the last one instead of u,"},{"Start":"06:04.080 ","End":"06:09.495","Text":"I just said I\u0027m replacing u with f and I forgot to do it."},{"Start":"06:09.495 ","End":"06:14.955","Text":"Now, plus fyy sine"},{"Start":"06:14.955 ","End":"06:25.150","Text":"squared Theta and now I\u0027m going to highlight this equals this,"},{"Start":"06:25.150 ","End":"06:31.500","Text":"and that is exactly what we had to prove here. We\u0027re done."}],"ID":8981},{"Watched":false,"Name":"Exercise 15","Duration":"12m 8s","ChapterTopicVideoID":8636,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.550","Text":"In this exercise, we\u0027re given that z is a function h of u and v and that u and v,"},{"Start":"00:08.550 ","End":"00:10.170","Text":"as in the previous exercise,"},{"Start":"00:10.170 ","End":"00:13.815","Text":"are functions of x and y satisfying the Cauchy-Riemann equations."},{"Start":"00:13.815 ","End":"00:17.190","Text":"I won\u0027t go over that because we\u0027ve done it in the previous exercise."},{"Start":"00:17.190 ","End":"00:21.270","Text":"This time we have to prove the following equality."},{"Start":"00:21.270 ","End":"00:23.580","Text":"Should we using h and not z?"},{"Start":"00:23.580 ","End":"00:24.630","Text":"It doesn\u0027t really matter."},{"Start":"00:24.630 ","End":"00:29.055","Text":"Z equals h. If I make a dependency tree,"},{"Start":"00:29.055 ","End":"00:34.220","Text":"I see that Z depends on, first of all,"},{"Start":"00:34.220 ","End":"00:39.920","Text":"u and v, and it\u0027s the same thing as H. Either way I see they want h in the results,"},{"Start":"00:39.920 ","End":"00:50.240","Text":"so I\u0027ll use h and u and v each depend on x and y. X and y, x and y."},{"Start":"00:50.240 ","End":"00:54.275","Text":"This is going to help us when we do the partial derivatives."},{"Start":"00:54.275 ","End":"00:56.005","Text":"But you know what, I\u0027ll just write H here."},{"Start":"00:56.005 ","End":"00:59.640","Text":"I\u0027m not actually going to use the letter Z anywhere."},{"Start":"01:00.460 ","End":"01:04.580","Text":"Let\u0027s see if we can compute the left-hand side."},{"Start":"01:04.580 ","End":"01:08.135","Text":"Then we\u0027ll see about equating the left to the right."},{"Start":"01:08.135 ","End":"01:12.465","Text":"Meanwhile, this is with a question mark is what we have to prove."},{"Start":"01:12.465 ","End":"01:15.775","Text":"Before we do H_xx, we need H_x."},{"Start":"01:15.775 ","End":"01:18.275","Text":"We need to go from H to x."},{"Start":"01:18.275 ","End":"01:25.875","Text":"We can get there either here and here or here and here, 2 paths."},{"Start":"01:25.875 ","End":"01:37.400","Text":"This is equal to h_uU_x plus h_vV_x."},{"Start":"01:37.400 ","End":"01:41.245","Text":"Now we\u0027re going to differentiate this again with respect to x."},{"Start":"01:41.245 ","End":"01:48.235","Text":"Notice that I\u0027m going to also need the derivative of h_u with respect to x and h_v."},{"Start":"01:48.235 ","End":"01:51.205","Text":"An actual fact that the same dependency,"},{"Start":"01:51.205 ","End":"02:00.760","Text":"because h_u and h_v are also dependent on u and v. Here I could put h or I could put h_u,"},{"Start":"02:00.760 ","End":"02:02.745","Text":"or I could put h_v,"},{"Start":"02:02.745 ","End":"02:06.890","Text":"different functions and everything but the same dependency."},{"Start":"02:06.890 ","End":"02:08.345","Text":"That\u0027s the important thing."},{"Start":"02:08.345 ","End":"02:12.140","Text":"At the moment, we\u0027re working with differentiating with respect to x."},{"Start":"02:12.140 ","End":"02:15.185","Text":"The same shading, the same paths apply."},{"Start":"02:15.185 ","End":"02:20.295","Text":"Now H_xx is equal to,"},{"Start":"02:20.295 ","End":"02:22.425","Text":"now this will do by the product rule,"},{"Start":"02:22.425 ","End":"02:28.940","Text":"it\u0027s going to be h_u derivative with"},{"Start":"02:28.940 ","End":"02:36.020","Text":"respect to x times this 1 as is plus the other way around,"},{"Start":"02:36.020 ","End":"02:41.320","Text":"h_u as is and this 1 with respect to x so U_xx."},{"Start":"02:41.320 ","End":"02:44.045","Text":"That\u0027s just the first term by the product rule."},{"Start":"02:44.045 ","End":"02:46.625","Text":"I get 2 more terms from the second."},{"Start":"02:46.625 ","End":"02:50.285","Text":"Here we get the first 1 differentiated,"},{"Start":"02:50.285 ","End":"02:57.455","Text":"h_v differentiated with respect to x and V_x as is, plus vice versa."},{"Start":"02:57.455 ","End":"03:03.475","Text":"H_v as is and V_x with respect to x is V_xx."},{"Start":"03:03.475 ","End":"03:09.330","Text":"Now these are the 2 that we want to expand according to the tree."},{"Start":"03:10.510 ","End":"03:14.675","Text":"I\u0027ll just write maybe what this is,"},{"Start":"03:14.675 ","End":"03:20.060","Text":"h_u with respect to x goes through u is h_u with respect to u,"},{"Start":"03:20.060 ","End":"03:22.400","Text":"u with respect to x."},{"Start":"03:22.400 ","End":"03:27.915","Text":"That\u0027s h_uuU_x, that\u0027s from this path."},{"Start":"03:27.915 ","End":"03:29.865","Text":"Then from this path,"},{"Start":"03:29.865 ","End":"03:38.765","Text":"we get h_uvV with respect to x."},{"Start":"03:38.765 ","End":"03:41.730","Text":"These 2 are just this bit."},{"Start":"03:41.830 ","End":"03:46.700","Text":"This also I can expand h_v as I said,"},{"Start":"03:46.700 ","End":"03:49.460","Text":"h, h_u and h_v all have the same dependency tree."},{"Start":"03:49.460 ","End":"03:51.335","Text":"With respect to x,"},{"Start":"03:51.335 ","End":"03:53.585","Text":"again, I\u0027m going to go along these paths."},{"Start":"03:53.585 ","End":"04:00.365","Text":"It\u0027s h_v with respect to u,"},{"Start":"04:00.365 ","End":"04:09.920","Text":"and then u with respect to x plus h_v with respect to v,"},{"Start":"04:09.920 ","End":"04:17.715","Text":"V with respect to x. I guess this now are just these bits."},{"Start":"04:17.715 ","End":"04:23.925","Text":"Now I\u0027m going to have to write the whole thing as equaling."},{"Start":"04:23.925 ","End":"04:29.470","Text":"This, which is this times U_x."},{"Start":"04:29.600 ","End":"04:32.160","Text":"If I multiply this by U_x,"},{"Start":"04:32.160 ","End":"04:36.315","Text":"I get h_uuU_x squared,"},{"Start":"04:36.315 ","End":"04:38.920","Text":"and this by U_x,"},{"Start":"04:41.420 ","End":"04:45.700","Text":"h_uvV_xU_x, I\u0027ll write it as U_xV_x."},{"Start":"04:46.130 ","End":"04:54.195","Text":"That\u0027s this times this plus h_uU_xx that\u0027s this 1."},{"Start":"04:54.195 ","End":"04:58.800","Text":"Now this multiply these 2 by V_x."},{"Start":"04:58.800 ","End":"05:10.660","Text":"I get h_vuU_x times V_x plus this 1,"},{"Start":"05:11.270 ","End":"05:14.945","Text":"h_vvV_xV_x is just V_x squared."},{"Start":"05:14.945 ","End":"05:20.420","Text":"Finally, we have this thing, h_vV_xx."},{"Start":"05:20.420 ","End":"05:23.129","Text":"We have 6 terms altogether."},{"Start":"05:23.129 ","End":"05:26.980","Text":"There\u0027s a little simplification that can be done here."},{"Start":"05:26.980 ","End":"05:31.300","Text":"Notice that h_uv and"},{"Start":"05:31.300 ","End":"05:34.489","Text":"h_vu are the same"},{"Start":"05:34.489 ","End":"05:38.960","Text":"and these 2 terms are the same because afterwards we have U_xV_x and here U_xV_x."},{"Start":"05:38.960 ","End":"05:47.010","Text":"Instead of that, I can erase 1 of them and put 2 in front of the other, slightly simpler."},{"Start":"05:47.010 ","End":"05:52.770","Text":"Now, the next thing I want to do is figure out h_yy."},{"Start":"05:52.770 ","End":"05:54.090","Text":"You might think, \"Oh,"},{"Start":"05:54.090 ","End":"05:56.970","Text":"it\u0027s going to be all that work again.\" Not really."},{"Start":"05:56.970 ","End":"06:03.550","Text":"All I have to do is change the dependency to go the other way."},{"Start":"06:03.550 ","End":"06:06.200","Text":"Basically, if you follow it all,"},{"Start":"06:06.200 ","End":"06:08.900","Text":"it\u0027s going to be exactly the same except that everywhere we see x,"},{"Start":"06:08.900 ","End":"06:12.890","Text":"we\u0027re going to have y. I\u0027m just going to copy the result and"},{"Start":"06:12.890 ","End":"06:17.330","Text":"say that h_yy is whatever we have here,"},{"Start":"06:17.330 ","End":"06:19.550","Text":"with x replaced by y."},{"Start":"06:19.550 ","End":"06:29.930","Text":"So we have h_uuU_y squared plus twice h_uv then U_yV_y,"},{"Start":"06:29.930 ","End":"06:37.730","Text":"just replacing plus h_uU_yy."},{"Start":"06:37.730 ","End":"06:48.960","Text":"Then this one\u0027s not plus h_vvV_y squared plus h_vV_yy."},{"Start":"06:50.800 ","End":"06:54.290","Text":"Now it\u0027s going to be easy to add these 2 together."},{"Start":"06:54.290 ","End":"06:55.460","Text":"Why do I want to add these?"},{"Start":"06:55.460 ","End":"06:58.760","Text":"Because look, that\u0027s what the left-hand side here is."},{"Start":"06:58.760 ","End":"07:01.805","Text":"If I do an addition,"},{"Start":"07:01.805 ","End":"07:08.955","Text":"I\u0027ll get that h_xx plus h_yy equals,"},{"Start":"07:08.955 ","End":"07:12.350","Text":"now everything\u0027s aligned, so it\u0027s going to be easy to do the addition."},{"Start":"07:12.350 ","End":"07:18.035","Text":"Because this with this, I can see that the common factor is h_uu,"},{"Start":"07:18.035 ","End":"07:21.660","Text":"and I get U_x squared plus U_y squared."},{"Start":"07:24.140 ","End":"07:27.765","Text":"Next I get this with this."},{"Start":"07:27.765 ","End":"07:33.790","Text":"I got 2h_uv in common so 2h_uv."},{"Start":"07:34.790 ","End":"07:43.090","Text":"Then I\u0027ve got this plus this, U_xV_x plus U_yV_y."},{"Start":"07:44.090 ","End":"07:50.160","Text":"Then these 2, h_u is the common thing and then U_xx plus"},{"Start":"07:50.160 ","End":"07:59.550","Text":"Uyy plus then this with this."},{"Start":"07:59.550 ","End":"08:03.250","Text":"We\u0027ve got h_vv is the common."},{"Start":"08:03.260 ","End":"08:11.775","Text":"We have V_x squared plus V_y squared plus,"},{"Start":"08:11.775 ","End":"08:21.640","Text":"last 1, h_v is in common and here we have V_xx plus V_yy."},{"Start":"08:23.640 ","End":"08:28.629","Text":"Now this exercise is a continuation of the previous exercise."},{"Start":"08:28.629 ","End":"08:30.190","Text":"In the previous exercise,"},{"Start":"08:30.190 ","End":"08:35.950","Text":"we showed that if u and v satisfy the Cauchy-Riemann equations,"},{"Start":"08:35.950 ","End":"08:43.330","Text":"I\u0027ll just quote it, then we got that U_xx plus U_yy was 0."},{"Start":"08:43.330 ","End":"08:45.865","Text":"That\u0027s the Laplace equation for you."},{"Start":"08:45.865 ","End":"08:48.700","Text":"Also the Laplace equation for V,"},{"Start":"08:48.700 ","End":"08:52.210","Text":"V_xx plus V_yy is also equal to 0."},{"Start":"08:52.210 ","End":"08:56.235","Text":"If I put these 2 facts and let\u0027s see where."},{"Start":"08:56.235 ","End":"08:59.865","Text":"Here, this plus this equals 0."},{"Start":"08:59.865 ","End":"09:05.015","Text":"This whole term disappears because this plus this is 0 from here."},{"Start":"09:05.015 ","End":"09:08.925","Text":"For V, this bit in the bracket is 0 here."},{"Start":"09:08.925 ","End":"09:12.949","Text":"This term also disappears because of the Laplace equation."},{"Start":"09:12.949 ","End":"09:17.855","Text":"Now we have 3 terms,1, 2, and 3."},{"Start":"09:17.855 ","End":"09:20.670","Text":"Let\u0027s continue."},{"Start":"09:21.380 ","End":"09:28.490","Text":"What I\u0027m going to do next is use the Cauchy-Riemann equations."},{"Start":"09:28.490 ","End":"09:31.190","Text":"I\u0027m going to use this and this."},{"Start":"09:31.190 ","End":"09:35.120","Text":"What I want to do is looking at the right-hand side,"},{"Start":"09:35.120 ","End":"09:39.230","Text":"I see everything in terms of x. I want to use these to"},{"Start":"09:39.230 ","End":"09:44.875","Text":"convert every derivative of U or V with respect to y to something with respect to x."},{"Start":"09:44.875 ","End":"09:47.630","Text":"What we get is h_uu."},{"Start":"09:47.630 ","End":"09:52.485","Text":"Now this one\u0027s fine is with respect to x. U with respect to y, I look it up."},{"Start":"09:52.485 ","End":"09:58.790","Text":"It\u0027s minus V_x, but minus V_x when it\u0027s squared still comes out plus."},{"Start":"09:58.790 ","End":"10:02.970","Text":"This is plus V_x squared to the minus all squared."},{"Start":"10:02.970 ","End":"10:11.920","Text":"Now in this 1, we get 2h_uv, U_xV_x is fine."},{"Start":"10:11.920 ","End":"10:23.420","Text":"But here U_y is minus V_x and V_y is U_x."},{"Start":"10:23.910 ","End":"10:26.425","Text":"That\u0027s the middle 1."},{"Start":"10:26.425 ","End":"10:36.070","Text":"The last 1, we get h_vv times V_x squared. That\u0027s fine."},{"Start":"10:36.070 ","End":"10:43.370","Text":"It\u0027s x, V_y have to replace with U_x, so U_x squared."},{"Start":"10:43.370 ","End":"10:49.010","Text":"Now look, the middle term is what I want you to look at, U_xV_x minus V_xU_x."},{"Start":"10:49.010 ","End":"10:51.140","Text":"Well, multiplication doesn\u0027t matter."},{"Start":"10:51.140 ","End":"10:55.169","Text":"This whole thing, these 2 are equal."},{"Start":"10:55.270 ","End":"11:02.300","Text":"This whole middle term now cancels out because this bit is 0 minus itself."},{"Start":"11:02.300 ","End":"11:05.340","Text":"All we\u0027re left with is this."},{"Start":"11:07.220 ","End":"11:11.180","Text":"Note that this bracket is the same as this bracket again,"},{"Start":"11:11.180 ","End":"11:14.030","Text":"just a matter of an order change."},{"Start":"11:14.030 ","End":"11:15.725","Text":"Just switch, it\u0027s a plus."},{"Start":"11:15.725 ","End":"11:22.085","Text":"Just put the U before the v. If I take the common bit outside the brackets,"},{"Start":"11:22.085 ","End":"11:27.005","Text":"then we\u0027re going to get a common bit."},{"Start":"11:27.005 ","End":"11:28.130","Text":"Let\u0027s use this form,"},{"Start":"11:28.130 ","End":"11:34.610","Text":"the U_x squared plus V_x squared."},{"Start":"11:34.610 ","End":"11:43.270","Text":"What we\u0027re left with is h_uu here and h_vv from here."},{"Start":"11:43.270 ","End":"11:45.950","Text":"Just to make it look exactly like this,"},{"Start":"11:45.950 ","End":"11:49.145","Text":"how about let\u0027s use some extra brackets here and here,"},{"Start":"11:49.145 ","End":"11:51.035","Text":"which certainly doesn\u0027t hurt."},{"Start":"11:51.035 ","End":"11:58.350","Text":"Now, we clearly see that this equals this."},{"Start":"11:58.350 ","End":"12:04.005","Text":"This is exactly what it says here."},{"Start":"12:04.005 ","End":"12:07.190","Text":"We have proved it that it does equal,"},{"Start":"12:07.190 ","End":"12:09.480","Text":"and then we\u0027re done."}],"ID":8982},{"Watched":false,"Name":"Exercise 16","Duration":"11m 8s","ChapterTopicVideoID":8637,"CourseChapterTopicPlaylistID":4964,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:04.050","Text":"In this exercise, we have u as a function of x and"},{"Start":"00:04.050 ","End":"00:09.690","Text":"y. X and y are given in terms of r and s,"},{"Start":"00:09.690 ","End":"00:12.960","Text":"r cosine hyperbolic of s,"},{"Start":"00:12.960 ","End":"00:17.310","Text":"and r sine hyperbolic of s. Notice the h here,"},{"Start":"00:17.310 ","End":"00:19.230","Text":"it\u0027s not the regular cosine and sine,"},{"Start":"00:19.230 ","End":"00:21.630","Text":"these are the hyperbolic cosine and sine."},{"Start":"00:21.630 ","End":"00:23.520","Text":"If you\u0027ve forgotten about them,"},{"Start":"00:23.520 ","End":"00:27.370","Text":"then I suggest you go and read up on them."},{"Start":"00:28.070 ","End":"00:31.395","Text":"I\u0027ll just assume that you do know what they are."},{"Start":"00:31.395 ","End":"00:33.330","Text":"People pronounce them variously."},{"Start":"00:33.330 ","End":"00:35.190","Text":"Sometimes, this is called cosine,"},{"Start":"00:35.190 ","End":"00:41.910","Text":"this is pronounced shine or sinch or various names,"},{"Start":"00:41.910 ","End":"00:45.055","Text":"whatever, I\u0027ll use any 1 of them."},{"Start":"00:45.055 ","End":"00:48.995","Text":"We have to prove this equality."},{"Start":"00:48.995 ","End":"00:52.100","Text":"We had a similar 1 actually previously with"},{"Start":"00:52.100 ","End":"00:55.685","Text":"regular cosine and sine and then we had pluses here."},{"Start":"00:55.685 ","End":"00:58.790","Text":"Anyway, it\u0027s a variation of a previous exercise in"},{"Start":"00:58.790 ","End":"01:02.465","Text":"case you want to go back and look at the similar 1."},{"Start":"01:02.465 ","End":"01:05.830","Text":"I\u0027m going to start with the dependency tree,"},{"Start":"01:05.830 ","End":"01:07.560","Text":"where at the top of the tree,"},{"Start":"01:07.560 ","End":"01:09.810","Text":"we have u or f, whatever."},{"Start":"01:09.810 ","End":"01:15.150","Text":"Let\u0027s take u and u depends on x and"},{"Start":"01:15.150 ","End":"01:21.434","Text":"y. X depends on r and s,"},{"Start":"01:21.434 ","End":"01:23.640","Text":"r here, s here."},{"Start":"01:23.640 ","End":"01:32.560","Text":"Same thing for y, also depends on r and depends on s. To prove this equality,"},{"Start":"01:32.560 ","End":"01:35.395","Text":"I want to start with the right-hand side,"},{"Start":"01:35.395 ","End":"01:39.015","Text":"and by a series of steps,"},{"Start":"01:39.015 ","End":"01:41.505","Text":"reach the left-hand side."},{"Start":"01:41.505 ","End":"01:45.250","Text":"To do that, I\u0027m going to get the building block first,"},{"Start":"01:45.250 ","End":"01:47.495","Text":"let\u0027s see what u_r and u_s are."},{"Start":"01:47.495 ","End":"01:51.855","Text":"I\u0027ll start with u_r, and let\u0027s see what this equals."},{"Start":"01:51.855 ","End":"01:53.830","Text":"To get from u to r,"},{"Start":"01:53.830 ","End":"01:56.815","Text":"I can go this way via x,"},{"Start":"01:56.815 ","End":"02:00.970","Text":"or I can go this way via y."},{"Start":"02:00.970 ","End":"02:11.790","Text":"We get, this is u_x x_r plus u_y y_r."},{"Start":"02:11.790 ","End":"02:16.025","Text":"Now, u_x, I can\u0027t compute because I don\u0027t know what the function f is."},{"Start":"02:16.025 ","End":"02:17.870","Text":"But x with respect to r,"},{"Start":"02:17.870 ","End":"02:25.740","Text":"I do know r is the variable and s is a constant,"},{"Start":"02:25.740 ","End":"02:28.260","Text":"so cosine s is also a constant,"},{"Start":"02:28.260 ","End":"02:34.005","Text":"so I just have cosh s. Here,"},{"Start":"02:34.005 ","End":"02:37.905","Text":"u_y, I can\u0027t compute with y with respect to r similarly."},{"Start":"02:37.905 ","End":"02:44.960","Text":"Sinh of s, that\u0027s this u_r."},{"Start":"02:44.960 ","End":"02:47.820","Text":"Now, I\u0027ll need u_s."},{"Start":"02:51.710 ","End":"02:59.245","Text":"Just need to change the path to go through x to s or through y to s,"},{"Start":"02:59.245 ","End":"03:02.210","Text":"and that gives us u with respect to x,"},{"Start":"03:02.210 ","End":"03:06.260","Text":"x with respect to s plus u with respect to y,"},{"Start":"03:06.260 ","End":"03:09.320","Text":"y with respect to s, u_x."},{"Start":"03:09.320 ","End":"03:12.310","Text":"Now, x with respect to s,"},{"Start":"03:12.310 ","End":"03:15.915","Text":"r is the constant, so it stays."},{"Start":"03:15.915 ","End":"03:20.190","Text":"The derivative of cosh is sinh,"},{"Start":"03:20.190 ","End":"03:24.440","Text":"there\u0027s no minus like with the trigonometric functions,"},{"Start":"03:24.440 ","End":"03:27.860","Text":"sinh of s. I better put brackets, sometimes I will."},{"Start":"03:27.860 ","End":"03:32.160","Text":"Maybe I really should be using brackets."},{"Start":"03:33.380 ","End":"03:37.610","Text":"Plus u_y, and similarly,"},{"Start":"03:37.610 ","End":"03:39.410","Text":"y with respect to s,"},{"Start":"03:39.410 ","End":"03:49.125","Text":"derivative of sinh is cosh and the r is the constant, so cosh."},{"Start":"03:49.125 ","End":"03:52.205","Text":"Each 1 is derivative of the other."},{"Start":"03:52.205 ","End":"03:58.900","Text":"Cosh of s. Now, let\u0027s see."},{"Start":"03:59.240 ","End":"04:04.120","Text":"If I take u_r squared,"},{"Start":"04:04.790 ","End":"04:08.640","Text":"I\u0027ll put them in brackets because they did, here,"},{"Start":"04:08.640 ","End":"04:13.490","Text":"this will equal, taking it off here,"},{"Start":"04:13.490 ","End":"04:18.800","Text":"I get u_x squared cosh squared of"},{"Start":"04:18.800 ","End":"04:27.690","Text":"s plus u_y squared sinh squared of s,"},{"Start":"04:29.170 ","End":"04:32.675","Text":"and the middle term."},{"Start":"04:32.675 ","End":"04:34.985","Text":"I wasn\u0027t careful there for a moment."},{"Start":"04:34.985 ","End":"04:44.120","Text":"I\u0027m using the formula a plus b squared is a squared plus 2ab plus b squared."},{"Start":"04:44.120 ","End":"04:46.380","Text":"Then I almost forgot the 2ab,"},{"Start":"04:46.450 ","End":"04:56.569","Text":"plus 2 u_x u_y,"},{"Start":"04:56.569 ","End":"05:03.350","Text":"I need more room, cosh sinh,"},{"Start":"05:03.350 ","End":"05:13.290","Text":"cosh of s, sinh of s. Just squeezed it in there."},{"Start":"05:13.290 ","End":"05:15.689","Text":"That\u0027s u_r squared."},{"Start":"05:15.689 ","End":"05:20.700","Text":"Now u_s squared,"},{"Start":"05:24.770 ","End":"05:27.945","Text":"and that will give me from here,"},{"Start":"05:27.945 ","End":"05:32.320","Text":"again using the binomial expansion,"},{"Start":"05:35.780 ","End":"05:46.680","Text":"u_x squared times r squared times sinh squared of"},{"Start":"05:46.680 ","End":"05:53.130","Text":"s plus twice u_x"},{"Start":"05:53.130 ","End":"06:01.335","Text":"u_y r squared,"},{"Start":"06:01.335 ","End":"06:04.530","Text":"I\u0027m picking the order I could do the multiplication,"},{"Start":"06:04.530 ","End":"06:07.350","Text":"and sinh and cosh,"},{"Start":"06:07.350 ","End":"06:14.100","Text":"sinh of s cosh of s and finally,"},{"Start":"06:14.100 ","End":"06:15.540","Text":"the b squared part,"},{"Start":"06:15.540 ","End":"06:20.745","Text":"u_y squared r squared"},{"Start":"06:20.745 ","End":"06:28.460","Text":"cosh squared of s. I\u0027m just building up to the right-hand side."},{"Start":"06:28.460 ","End":"06:31.540","Text":"I\u0027ve got this thing squared, I got this thing squared."},{"Start":"06:31.540 ","End":"06:33.955","Text":"Let\u0027s do 1 over r squared of this."},{"Start":"06:33.955 ","End":"06:40.265","Text":"1 over r squared of u_s squared equals,"},{"Start":"06:40.265 ","End":"06:43.530","Text":"notice I have r squared here,"},{"Start":"06:43.530 ","End":"06:47.364","Text":"r squared here, and r squared here."},{"Start":"06:47.364 ","End":"06:49.750","Text":"So if I divide by r squared,"},{"Start":"06:49.750 ","End":"06:56.800","Text":"I just get u_x squared sinh"},{"Start":"06:56.800 ","End":"07:02.270","Text":"squared s plus 2u_x u_y,"},{"Start":"07:05.190 ","End":"07:08.480","Text":"the r squared\u0027s gone."},{"Start":"07:09.680 ","End":"07:16.570","Text":"Let me just change the order on these 2 because I\u0027m already looking ahead"},{"Start":"07:16.570 ","End":"07:19.630","Text":"and see that here I have cosh sinh and here sinh"},{"Start":"07:19.630 ","End":"07:22.960","Text":"cosh and I\u0027d like them to be in the same order, obviously multiplication."},{"Start":"07:22.960 ","End":"07:26.780","Text":"I\u0027ll write it as cosh of s,"},{"Start":"07:26.900 ","End":"07:34.035","Text":"sinh of s plus u_y squared"},{"Start":"07:34.035 ","End":"07:41.310","Text":"cosh squared of s. Now what I want to do is subtract this minus this."},{"Start":"07:41.310 ","End":"07:46.280","Text":"In other words, I\u0027m taking this and subtracting this."},{"Start":"07:46.280 ","End":"07:50.010","Text":"Let me see. I\u0027ll just mark them."},{"Start":"07:50.590 ","End":"07:56.700","Text":"These 2, this 1 minus this 1."},{"Start":"07:56.780 ","End":"08:03.520","Text":"U_r squared minus 1 over r squared,"},{"Start":"08:04.550 ","End":"08:13.275","Text":"the brackets here, u_s squared equals."},{"Start":"08:13.275 ","End":"08:16.975","Text":"Now, I have 3 terms here."},{"Start":"08:16.975 ","End":"08:18.680","Text":"I\u0027m going to subtract them."},{"Start":"08:18.680 ","End":"08:22.585","Text":"This minus this, this minus this, this minus this."},{"Start":"08:22.585 ","End":"08:24.140","Text":"From here and here,"},{"Start":"08:24.140 ","End":"08:26.060","Text":"I\u0027ve got a u_x squared in common."},{"Start":"08:26.060 ","End":"08:31.030","Text":"It\u0027s u_x squared cosh squared minus sinh squared."},{"Start":"08:31.030 ","End":"08:38.095","Text":"Cosh squared s minus sinh squared s."},{"Start":"08:38.095 ","End":"08:47.495","Text":"This minus this just disappears because this is equal to this."},{"Start":"08:47.495 ","End":"08:50.400","Text":"It doesn\u0027t appear in the difference."},{"Start":"08:50.410 ","End":"08:54.200","Text":"Then I get this minus this."},{"Start":"08:54.200 ","End":"08:57.180","Text":"U_y squared is in common,"},{"Start":"08:57.190 ","End":"09:03.540","Text":"and I get sinh squared s minus"},{"Start":"09:03.540 ","End":"09:13.090","Text":"cosh squared s. Now,"},{"Start":"09:13.090 ","End":"09:19.070","Text":"I\u0027m looking all the time ahead to what I want to prove,"},{"Start":"09:19.070 ","End":"09:28.790","Text":"which is this, and there\u0027s 2 things that encourage me."},{"Start":"09:28.790 ","End":"09:35.195","Text":"First of all, there\u0027s the identity from hyperbolic functions"},{"Start":"09:35.195 ","End":"09:41.520","Text":"that in general cosh squared of an angle,"},{"Start":"09:41.520 ","End":"09:44.310","Text":"let\u0027s leave it as s,"},{"Start":"09:44.310 ","End":"09:50.960","Text":"minus sinh squared of the same thing is equal to 1."},{"Start":"09:50.960 ","End":"09:53.720","Text":"That\u0027s an identity for every s, this is true."},{"Start":"09:53.720 ","End":"09:57.350","Text":"Just like in trigonometry cosine squared plus sine squared is 1,"},{"Start":"09:57.350 ","End":"09:59.300","Text":"here it works a bit different."},{"Start":"09:59.300 ","End":"10:04.785","Text":"In other words, this thing here will be 1,"},{"Start":"10:04.785 ","End":"10:09.360","Text":"but this thing here is the reverse subtraction."},{"Start":"10:09.360 ","End":"10:12.770","Text":"When you subtract something in the reverse order,"},{"Start":"10:12.770 ","End":"10:17.640","Text":"you just get negative of the same thing and this is going to be minus 1."},{"Start":"10:17.930 ","End":"10:21.670","Text":"If I just wrap it up there,"},{"Start":"10:21.980 ","End":"10:28.365","Text":"what I get is this 1 is just u_x squared,"},{"Start":"10:28.365 ","End":"10:34.500","Text":"and this is the minus u_y squared."},{"Start":"10:34.500 ","End":"10:38.930","Text":"I hope that was okay. When you subtract in the opposite order,"},{"Start":"10:38.930 ","End":"10:44.060","Text":"just like minus of a minus b is b minus a."},{"Start":"10:44.060 ","End":"10:47.645","Text":"We\u0027ve done this before, that\u0027s where we get the minus."},{"Start":"10:47.645 ","End":"10:52.070","Text":"This is exactly what we had to prove."},{"Start":"10:52.070 ","End":"10:54.710","Text":"This is the right-hand side."},{"Start":"10:54.710 ","End":"10:58.710","Text":"So we\u0027ve actually shown this."},{"Start":"10:59.120 ","End":"11:02.760","Text":"It\u0027s hard to get them both on screen,"},{"Start":"11:02.760 ","End":"11:07.780","Text":"but roughly there. That\u0027s it."}],"ID":8983}],"Thumbnail":null,"ID":4964}]

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