1 00:00:01,200 --> 00:00:04,600 [...] sometimes ... Let's ... 2 00:00:05,900 --> 00:00:07,300 The chain rule. 3 00:00:07,400 --> 00:00:09,850 Right? So what's the chain rule about? 4 00:00:09,950 --> 00:00:13,000 It's about thinking of ... 5 00:00:14,200 --> 00:00:17,400 ... a function f of independent variable x. 6 00:00:17,500 --> 00:00:20,900 And then all of a sudden you think of x ... 7 00:00:21,400 --> 00:00:23,700 ... as a function of something else. 8 00:00:23,800 --> 00:00:28,900 And then you ask well if you know the rate of change of f with respect to x ... 9 00:00:29,000 --> 00:00:31,900 ... and you also know the rate of change of with respect to t. 10 00:00:32,000 --> 00:00:36,300 Can you possibly find the rate of change of f with respect to t? 11 00:00:36,900 --> 00:00:40,200 Because now you can consider f as a function of t. 12 00:00:40,350 --> 00:00:43,900 Because now you can substitute that x into f, right? 13 00:00:44,000 --> 00:00:48,750 So being as a function of t you can ask for derivative. 14 00:00:48,850 --> 00:00:51,300 And that means ... 15 00:00:52,000 --> 00:00:54,100 So what is it that you can do? 16 00:00:54,200 --> 00:01:00,800 You can say df is f prime dx. 17 00:01:01,200 --> 00:01:05,300 Right? This is what you know from f being a function of x. 18 00:01:05,400 --> 00:01:09,400 And this f prime is f prime with respect to x. 19 00:01:09,500 --> 00:01:12,400 So let's put x there. 20 00:01:12,900 --> 00:01:16,400 And then from this formula you can say ... 21 00:01:16,500 --> 00:01:21,100 ... dx is x prime ... 22 00:01:22,400 --> 00:01:24,500 Now it is a function of t. 23 00:01:24,800 --> 00:01:25,800 dt. 24 00:01:26,750 --> 00:01:31,700 Because in this setup you treat x as dependent on t. 25 00:01:31,800 --> 00:01:34,850 So dx is exactly that. 26 00:01:34,950 --> 00:01:39,150 And now ... dx was independent in this setting. 27 00:01:39,250 --> 00:01:44,200 But we switched to thinking of x being dependent. 28 00:01:44,300 --> 00:01:47,800 So you just substitute that dx into this ... 29 00:01:48,100 --> 00:01:53,600 ... formula and you get df equals f prime ... 30 00:01:54,450 --> 00:01:57,700 ... derivative with respect to x times ... 31 00:01:57,900 --> 00:01:59,500 ... x prime. 32 00:02:00,300 --> 00:02:02,400 Derivative with respect to t. 33 00:02:02,900 --> 00:02:04,600 Times dt. 34 00:02:06,000 --> 00:02:08,000 And that's exactly the chain rule. 35 00:02:08,800 --> 00:02:11,200 Right? If you divide by dt ... 36 00:02:14,000 --> 00:02:16,800 ... and then operating like this ... 37 00:02:17,750 --> 00:02:21,700 Those being quantities that I can divide. 38 00:02:22,200 --> 00:02:25,000 But essentially that derivative is ... 39 00:02:25,100 --> 00:02:27,500 ... f prime of x times ... 40 00:02:28,200 --> 00:02:29,900 ... x prime of t. 41 00:02:30,800 --> 00:02:33,500 So the chain rule follows from this basic idea ... 42 00:02:33,600 --> 00:02:36,000 ... of what differentials are. 43 00:02:36,100 --> 00:02:40,600 And we will follow the same idea in function of two variables. 44 00:02:40,700 --> 00:02:43,850 Because the chain rule for function of many variables is ... 45 00:02:43,950 --> 00:02:46,500 ... not one formula. It's much more complicated. 46 00:02:46,600 --> 00:02:49,700 But it will based on this very, very simple idea. 47 00:02:50,000 --> 00:02:53,400 [...] you operate at differentials and substitute. 48 00:02:54,700 --> 00:02:58,100 And essentially the chain rule ... 49 00:02:58,550 --> 00:03:04,000 Well in my mind the chain rule has no place in calculus because it doesn't belong to calculus. 50 00:03:04,750 --> 00:03:07,700 Because the chain rule basically tells you ... 51 00:03:08,200 --> 00:03:13,800 ... something about combination of linear functions. 52 00:03:13,900 --> 00:03:15,900 So what is it about? 53 00:03:16,700 --> 00:03:19,000 What is it that we are saying here? 54 00:03:19,250 --> 00:03:23,600 We are saying that if you linearize this dependence ... 55 00:03:24,300 --> 00:03:26,600 And this is what we do here, right? 56 00:03:26,700 --> 00:03:30,800 We pretend dependence of f on x is linear. 57 00:03:30,900 --> 00:03:33,000 Which is simply multiplication by a number. 58 00:03:33,100 --> 00:03:36,700 So if you linearize that this is what you get. 59 00:03:37,000 --> 00:03:40,100 If you linearize this dependence this is what you get. 60 00:03:40,200 --> 00:03:43,200 And then you ask -- if you have ... 61 00:03:43,450 --> 00:03:45,400 ... two linear dependencies ... 62 00:03:45,500 --> 00:03:49,200 f linearly depends on x and x linearly depends on t. 63 00:03:49,300 --> 00:03:52,000 How does f linearly depend on t? 64 00:03:53,000 --> 00:03:55,800 Well you just have to combine these two linear ... 65 00:03:55,900 --> 00:04:00,000 ... dependencies so you have to multiply those two numbers. 66 00:04:00,550 --> 00:04:04,100 Right? So the idea is fundamentally very simple. 67 00:04:04,300 --> 00:04:08,400 If x is twice as fast as t, f is three times as fast as x ... 68 00:04:08,500 --> 00:04:11,200 ... then f is six times as fast as t. 69 00:04:11,300 --> 00:04:13,100 So that's all it is about it. 70 00:04:13,200 --> 00:04:18,300 So that idea is about linear algebra. 71 00:04:18,600 --> 00:04:20,700 And for many variables ... 72 00:04:22,350 --> 00:04:24,500 ... what we will have is ... 73 00:04:26,100 --> 00:04:29,450 ... f being multivariable object. 74 00:04:29,550 --> 00:04:32,300 x being multivariable object. 75 00:04:32,400 --> 00:04:36,800 And how do you describe linear dependence of one object on the other? 76 00:04:36,900 --> 00:04:38,600 Well you use a matrix ... 77 00:04:38,700 --> 00:04:40,100 ... instead of a number. 78 00:04:40,200 --> 00:04:44,200 So you multiply a vector by a matrix to get another vector. 79 00:04:44,300 --> 00:04:47,200 And then similar thing here. 80 00:04:47,300 --> 00:04:52,600 Multivariable vector, multivariable vector, multiplication by a matrix. 81 00:04:52,700 --> 00:04:57,950 And then you ask 'well, how do you do you combine these two linear dependences?' 82 00:04:58,050 --> 00:05:01,550 Well that's exactly the topic of linear algebra saying you ... 83 00:05:01,650 --> 00:05:04,300 You have to multiply these two matrices. 84 00:05:06,200 --> 00:05:09,550 Right? So the chain rule is saying effectively that ... 85 00:05:09,650 --> 00:05:14,800 ... all you have to do is you have to multiply those two multivariable objects. 86 00:05:14,900 --> 00:05:19,400 And actually the concept of multiplication of matrices ... 87 00:05:19,500 --> 00:05:22,700 ... is designed by looking at this rule. 88 00:05:23,200 --> 00:05:25,000 With the way you ... 89 00:05:25,750 --> 00:05:26,750 ... [...] did it. 90 00:05:26,850 --> 00:05:29,100 The way you multiply matrices together ... 91 00:05:29,200 --> 00:05:32,200 Well it something like an artificial rule ... well somebody tells you ... 92 00:05:32,300 --> 00:05:35,600 Well you have to take this, multiply by that plus take that, multiply by this. 93 00:05:35,700 --> 00:05:43,200 So essentially that rule is designed to make this simple rule work. 94 00:05:43,550 --> 00:05:46,000 If you want this to happen ... 95 00:05:46,100 --> 00:05:51,300 ... that combination is determined by multiplication of these coefficients. 96 00:05:51,400 --> 00:05:57,000 As you make those coefficients matrices you have to design multiplication and ... 97 00:05:57,300 --> 00:05:59,500 ... linear algebra tells you what it is. 98 00:06:02,900 --> 00:06:07,000 So unfortunately as linear algebra is not [...] ... 99 00:06:07,100 --> 00:06:10,300 ... this chain rule is not something very easy for you ... 100 00:06:10,400 --> 00:06:15,100 ... that is well understood at the level of linear dependences. 101 00:06:15,200 --> 00:06:17,100 So I will go to that quickly. 102 00:06:17,200 --> 00:06:23,000 But there is nothing mysterious and it's really not a part of calculus from my point of view. 103 00:06:24,050 --> 00:06:27,200 It's something that happens after we linearize. 104 00:06:27,300 --> 00:06:31,500 And calculus to me is about this linearization. 105 00:06:33,500 --> 00:06:36,400 All right. So that's kind of preview ... 106 00:06:37,200 --> 00:06:39,500 ... of multivariable stuff. 107 00:06:39,700 --> 00:06:42,600 So any questions about differentials? 108 00:06:44,650 --> 00:06:47,250 Because that's basically the ... 109 00:06:47,950 --> 00:06:53,400 ... the different way of wording what Taylor series says after degree one. 110 00:06:53,500 --> 00:06:55,600 Nothing more, nothing less. 111 00:06:55,700 --> 00:06:57,800 Just different notation for the same stuff. |

Video Lectures > Calculus III (2011 Summer) > Lecture 2011.06.30 Differential. Circular Motion. Curvature > Part 02 Chain Rule Explained Using Differentials >