WEBVTT 00:00:00.110 --> 00:00:02.320 The following content is provided under a Creative 00:00:02.320 --> 00:00:03.150 Commons license. 00:00:03.150 --> 00:00:05.930 Your support will help MIT OpenCourseWare 00:00:05.930 --> 00:00:09.960 continue to offer high quality educational resources for free. 00:00:09.960 --> 00:00:12.630 To make a donation, or to view additional materials 00:00:12.630 --> 00:00:15.910 from hundreds of MIT courses visit MIT OpenCourseWare 00:00:15.910 --> 00:00:22.240 at ocw.mit.edu. 00:00:22.240 --> 00:00:25.050 Professor: I am Haynes Miller, I am substituting 00:00:25.050 --> 00:00:26.510 for David Jerison today. 00:00:26.510 --> 00:00:41.880 So you have a substitute teacher today. 00:00:41.880 --> 00:00:44.400 So I haven't been here in this class with you 00:00:44.400 --> 00:00:47.580 so I'm not completely sure where you are. 00:00:47.580 --> 00:00:52.170 I think you've just been talking about differentiation 00:00:52.170 --> 00:00:56.840 and you've got some examples of differentiation like these 00:00:56.840 --> 00:00:59.860 basic examples: the derivative of x^n is nx^(n-1). 00:01:03.197 --> 00:01:05.280 But I think maybe you've spent some time computing 00:01:05.280 --> 00:01:11.230 the derivative of the sine function as well, recently. 00:01:11.230 --> 00:01:16.290 And I think you have some rules for extending 00:01:16.290 --> 00:01:18.950 these calculations as well. 00:01:18.950 --> 00:01:24.080 For instance, I think you know that if you differentiate 00:01:24.080 --> 00:01:27.770 a constant times a function, what do you get? 00:01:27.770 --> 00:01:32.590 Student: [INAUDIBLE]. 00:01:32.590 --> 00:01:36.670 Professor: The constant comes outside like this. 00:01:36.670 --> 00:01:40.030 Or I could write (cu)' = cu'. 00:01:42.550 --> 00:01:45.350 That's this rule, multiplying by a constant, 00:01:45.350 --> 00:01:58.870 and I think you also know about differentiating a sum. 00:01:58.870 --> 00:02:03.650 Or I could write this as (u + v)' = u' + v'. 00:02:06.870 --> 00:02:09.850 So I'm going to be using those but today I'll 00:02:09.850 --> 00:02:12.290 talk about a collection of other rules 00:02:12.290 --> 00:02:15.080 about how to deal with a product of functions, 00:02:15.080 --> 00:02:18.180 a quotient of functions, and, best of all, 00:02:18.180 --> 00:02:20.024 composition of functions. 00:02:20.024 --> 00:02:21.690 And then at the end, I'll have something 00:02:21.690 --> 00:02:23.480 to say about higher derivatives. 00:02:23.480 --> 00:02:26.670 So that's the story for today. 00:02:26.670 --> 00:02:29.120 That's the program. 00:02:29.120 --> 00:02:43.360 So let's begin by talking about the product rule. 00:02:43.360 --> 00:02:44.750 So the product rule tells you how 00:02:44.750 --> 00:02:46.760 to differentiate a product of functions, 00:02:46.760 --> 00:02:49.270 and I'll just give you the rule, first of all. 00:02:49.270 --> 00:02:51.600 The rule is it's u'v + uv'. 00:02:57.280 --> 00:02:58.500 It's a little bit funny. 00:02:58.500 --> 00:03:02.320 Differentiating a product gives you a sum. 00:03:02.320 --> 00:03:06.730 But let's see how that works out in a particular example. 00:03:06.730 --> 00:03:08.270 For example, suppose that I wanted 00:03:08.270 --> 00:03:11.280 to differentiate the product. 00:03:11.280 --> 00:03:13.890 Well, the product of these two basic 00:03:13.890 --> 00:03:15.584 examples that we just talked about. 00:03:15.584 --> 00:03:17.000 I'm going to use the same variable 00:03:17.000 --> 00:03:20.790 in both cases instead of different ones like I did here. 00:03:20.790 --> 00:03:23.230 So the derivative of x^n times sin x. 00:03:28.430 --> 00:03:30.300 So this is a new thing. 00:03:30.300 --> 00:03:36.120 We couldn't do this without using the product rule. 00:03:36.120 --> 00:03:39.670 So the first function is x^n and the second one is sin x. 00:03:39.670 --> 00:03:41.740 And we're going to apply this rule. 00:03:41.740 --> 00:03:49.450 So u is x^n. u' is, according to the rule, nx^(n-1). 00:03:49.450 --> 00:03:56.029 And then I take v and write it down the way it is, sine of x. 00:03:56.029 --> 00:03:57.320 And then I do it the other way. 00:03:57.320 --> 00:04:00.690 I take u the way it is, that's x^n, 00:04:00.690 --> 00:04:05.260 and multiply it by the derivative of v, v'. 00:04:05.260 --> 00:04:09.234 We just saw v' is cosine of x. 00:04:09.234 --> 00:04:11.520 So that's it. 00:04:11.520 --> 00:04:14.980 Obviously, you can differentiate longer products, 00:04:14.980 --> 00:04:20.560 products of more things by doing it one at a time. 00:04:20.560 --> 00:04:22.700 Let's see why this is true. 00:04:22.700 --> 00:04:25.730 I want to try to show you why the product rule holds. 00:04:25.730 --> 00:04:31.490 So you have a standard way of trying to understand this, 00:04:31.490 --> 00:04:34.590 and it involves looking at the change in the function 00:04:34.590 --> 00:04:37.350 that you're interested in differentiating. 00:04:37.350 --> 00:04:41.390 So I should look at how much the product uv changes 00:04:41.390 --> 00:04:44.630 when x changes a little bit. 00:04:44.630 --> 00:04:47.120 Well, so how do compute the change? 00:04:47.120 --> 00:04:49.330 Well, I write down the value of the function 00:04:49.330 --> 00:04:56.160 at some new value of x, x + delta x. 00:04:56.160 --> 00:04:58.410 Well, I better write down the whole new value 00:04:58.410 --> 00:05:01.480 of the function, and the function is uv. 00:05:01.480 --> 00:05:05.230 So the whole new value looks like this. 00:05:05.230 --> 00:05:09.450 It's u(x + delta x) times v(x + delta x). 00:05:09.450 --> 00:05:10.960 That's the new value. 00:05:10.960 --> 00:05:13.200 But what's the change in the product? 00:05:13.200 --> 00:05:15.410 Well, I better subtract off what the old value 00:05:15.410 --> 00:05:20.920 was, which is u(x) v(x). 00:05:20.920 --> 00:05:24.950 Okay, according to the rule we're trying to prove, 00:05:24.950 --> 00:05:27.750 I have to get u' involved. 00:05:27.750 --> 00:05:31.420 So I want to involve the change in u alone, by itself. 00:05:31.420 --> 00:05:32.990 Let's just try that. 00:05:32.990 --> 00:05:36.620 I see part of the formula for the change in u right there. 00:05:36.620 --> 00:05:40.260 Let's see if we can get the rest of it in place. 00:05:40.260 --> 00:05:46.330 So the change in x is (u(x + delta x) - u(x). 00:05:46.330 --> 00:05:50.080 That's the change in x [Correction: change in u]. 00:05:50.080 --> 00:05:54.760 This part of it occurs up here, multiplied by v(x + delta x), 00:05:54.760 --> 00:05:57.890 so let's put that in too. 00:05:57.890 --> 00:06:00.440 Now this equality sign isn't very good right now. 00:06:00.440 --> 00:06:04.770 I've got this product here so far, 00:06:04.770 --> 00:06:06.690 but I've introduced something I don't like. 00:06:06.690 --> 00:06:09.800 I've introduced u times v(x delta x), right? 00:06:09.800 --> 00:06:12.010 Minus that. 00:06:12.010 --> 00:06:17.120 So the next thing I'm gonna do is correct that little defect 00:06:17.120 --> 00:06:24.620 by adding in u(x) v(x + delta x). 00:06:24.620 --> 00:06:28.834 Okay, now I cancelled off what was wrong with this line. 00:06:28.834 --> 00:06:30.500 But I'm still not quite there, because I 00:06:30.500 --> 00:06:32.670 haven't put this in yet. 00:06:32.670 --> 00:06:38.420 So I better subtract off uv, and then I'll be home. 00:06:38.420 --> 00:06:40.500 But I'm going to do that in a clever way, 00:06:40.500 --> 00:06:43.900 because I noticed that I already have a u here. 00:06:43.900 --> 00:06:46.620 So I'm gonna take this factor of u 00:06:46.620 --> 00:06:48.560 and make it the same as this factor. 00:06:48.560 --> 00:06:52.510 So I get u(x) times this, minus u(x) times that. 00:06:52.510 --> 00:06:57.180 That's the same thing as u times the difference. 00:06:57.180 --> 00:06:59.406 So that was a little bit strange, 00:06:59.406 --> 00:07:01.030 but when you stand back and look at it, 00:07:01.030 --> 00:07:04.280 you can see multiplied out, the middle terms cancel. 00:07:04.280 --> 00:07:07.340 And you get the right answer. 00:07:07.340 --> 00:07:10.420 Well I like that because it's involved the change in u 00:07:10.420 --> 00:07:16.810 and the change in v. So this is equal to delta u times v(x + 00:07:16.810 --> 00:07:26.320 delta x) minus u(x) times the change in v. Well, 00:07:26.320 --> 00:07:27.510 I'm almost there. 00:07:27.510 --> 00:07:30.050 The next step in computing the derivative is 00:07:30.050 --> 00:07:42.910 take difference quotient, divide this by delta x. 00:07:42.910 --> 00:07:50.690 So, (delta (uv)) / (delta x) is well, 00:07:50.690 --> 00:08:03.150 I'll say (delta u / delta x) times v(x + delta x). 00:08:03.150 --> 00:08:10.000 Have I made a mistake here? 00:08:10.000 --> 00:08:13.340 This plus magically became a minus on the way down here, 00:08:13.340 --> 00:08:18.810 so I better fix that. 00:08:18.810 --> 00:08:23.260 Plus u times (delta v) / (delta x). 00:08:23.260 --> 00:08:27.950 This u is this u over here. 00:08:27.950 --> 00:08:30.310 So I've just divided this formula by delta x, 00:08:30.310 --> 00:08:34.520 and now I can take the limit as x goes to 0, so this 00:08:34.520 --> 00:08:42.070 is as delta x goes to 0. 00:08:42.070 --> 00:08:46.480 This becomes the definition of the derivative, 00:08:46.480 --> 00:08:51.790 and on this side, I get du/dx times... 00:08:51.790 --> 00:08:57.400 now what happens to this quantity when delta x goes 00:08:57.400 --> 00:09:02.750 to 0? 00:09:02.750 --> 00:09:05.710 So this quantity is getting closer and closer to x. 00:09:05.710 --> 00:09:09.000 So what happens to the value of v? 00:09:09.000 --> 00:09:14.030 It becomes equal to x of v. That uses continuity of v. So, 00:09:14.030 --> 00:09:22.590 v(x + delta x) goes to v(x) by continuity. 00:09:22.590 --> 00:09:26.586 So this gives me times v, and then I have u times, 00:09:26.586 --> 00:09:30.680 and delta v / delta x gives me dv/dx. 00:09:30.680 --> 00:09:31.819 And that's the formula. 00:09:31.819 --> 00:09:33.360 That's the formula as I wrote it down 00:09:33.360 --> 00:09:35.660 at the beginning over here. 00:09:35.660 --> 00:09:39.170 The derivative of a product is given by this sum. 00:09:39.170 --> 00:09:46.223 Yeah? 00:09:46.223 --> 00:09:48.056 Student: How did you get from the first line 00:09:48.056 --> 00:09:49.514 to the second of the long equation? 00:09:49.514 --> 00:09:51.390 Professor: From here to here? 00:09:51.390 --> 00:09:53.460 Student: Yes. 00:09:53.460 --> 00:09:56.380 Professor: So maybe it's easiest to work backwards and verify 00:09:56.380 --> 00:09:59.800 that what I wrote down is correct here. 00:09:59.800 --> 00:10:05.130 So, if you look there's a u times v(x + delta x) there. 00:10:05.130 --> 00:10:07.840 And there's also one here. 00:10:07.840 --> 00:10:09.940 And they occur with opposite signs. 00:10:09.940 --> 00:10:11.490 So they cancel. 00:10:11.490 --> 00:10:20.530 What's left is u(x + delta x) v(x + delta x) - uv. 00:10:20.530 --> 00:10:29.120 And that's just what I started with. 00:10:29.120 --> 00:10:33.920 Student: [INAUDIBLE] They cancel right? 00:10:33.920 --> 00:10:37.430 Professor: I cancelled out this term and this term, 00:10:37.430 --> 00:10:39.700 and what's left is the ends. 00:10:39.700 --> 00:10:41.490 Any other questions? 00:10:41.490 --> 00:10:49.660 Student: [INAUDIBLE]. 00:10:49.660 --> 00:10:55.640 Professor: Well, I just calculated what delta uv is, 00:10:55.640 --> 00:10:57.980 and now I'm gonna divide that by delta x on my way 00:10:57.980 --> 00:11:00.250 to computing the derivative. 00:11:00.250 --> 00:11:07.760 And so I copied down the right hand side and divided delta x. 00:11:07.760 --> 00:11:11.550 I just decided to divide the delta u by delta x and delta v 00:11:11.550 --> 00:11:16.230 by delta x. 00:11:16.230 --> 00:11:16.990 Good. 00:11:16.990 --> 00:11:22.490 Anything else? 00:11:22.490 --> 00:11:24.260 So we have the product rule here. 00:11:24.260 --> 00:11:26.980 The rule for differentiating a product of two functions. 00:11:26.980 --> 00:11:28.325 This is making us stronger. 00:11:28.325 --> 00:11:29.700 There are many more functions you 00:11:29.700 --> 00:11:31.420 can find derivatives of now. 00:11:31.420 --> 00:11:33.580 How about quotients? 00:11:33.580 --> 00:11:35.370 Let's find out how to differentiate 00:11:35.370 --> 00:11:47.669 a quotient of two functions. 00:11:47.669 --> 00:11:50.210 Well again, I'll write down what the answer is and then we'll 00:11:50.210 --> 00:11:52.370 try to verify it. 00:11:52.370 --> 00:11:55.192 So there's a quotient. 00:11:55.192 --> 00:11:56.150 Let me write this down. 00:11:56.150 --> 00:11:58.970 There's a quotient of two functions. 00:11:58.970 --> 00:12:00.340 And here's the rule for it. 00:12:00.340 --> 00:12:02.910 I always have to think about this and hope that I get it 00:12:02.910 --> 00:12:09.140 right. (u'v - uv') / v^2. 00:12:09.140 --> 00:12:11.900 This may be the craziest rule you'll see in this course, 00:12:11.900 --> 00:12:14.330 but there it is. 00:12:14.330 --> 00:12:18.014 And I'll try to show you why that's true and see an example. 00:12:18.014 --> 00:12:18.930 Yeah there was a hand? 00:12:18.930 --> 00:12:27.300 Student: [INAUDIBLE] 00:12:27.300 --> 00:12:33.040 Professor: What letters look the same? u and v look the same? 00:12:33.040 --> 00:12:37.040 I'll try to make them look more different. 00:12:37.040 --> 00:12:38.520 The v's have points on the bottom. 00:12:38.520 --> 00:12:41.160 u's have little round things on the bottom. 00:12:41.160 --> 00:12:44.980 What's the new value of u? 00:12:44.980 --> 00:12:55.442 The value of u at x + delta x is u + delta u, right? 00:12:55.442 --> 00:12:56.400 That's what delta u is. 00:12:56.400 --> 00:13:01.327 It's the change in u when x gets replaced by delta x 00:13:01.327 --> 00:13:02.410 [Correction: x + delta x]. 00:13:02.410 --> 00:13:09.700 And the change in v, the new value v, is v + delta v. 00:13:09.700 --> 00:13:13.130 So this is the new value of u divided by the new value of v. 00:13:13.130 --> 00:13:16.130 That's the beginning. 00:13:16.130 --> 00:13:18.705 And then I subtract off the old values, which 00:13:18.705 --> 00:13:22.910 are u minus v. This'll be easier to work out 00:13:22.910 --> 00:13:26.084 when I write it out this way. 00:13:26.084 --> 00:13:27.750 So now, we'll cross multiply, as I said. 00:13:27.750 --> 00:13:38.890 So I get uv + (delta u)v minus, now I cross multiply this way, 00:13:38.890 --> 00:13:46.330 you get uv - u(delta v). 00:13:46.330 --> 00:13:49.980 And I divide all this by (v + delta v)u. 00:13:52.840 --> 00:13:55.970 Okay, now the reason I like to do it 00:13:55.970 --> 00:13:59.490 this way is that you see the cancellation happening here. uv 00:13:59.490 --> 00:14:02.187 and uv occur twice and so I can cancel them. 00:14:02.187 --> 00:14:04.520 And I will, and I'll answer these questions in a minute. 00:14:04.520 --> 00:14:06.260 Audience: [INAUDIBLE]. 00:14:06.260 --> 00:14:14.030 Professor: Ooh, that's a v. All right. 00:14:14.030 --> 00:14:15.570 Good, anything else? 00:14:15.570 --> 00:14:16.750 That's what all hands were. 00:14:16.750 --> 00:14:17.880 Good. 00:14:17.880 --> 00:14:20.870 All right, so I cancel these and what I'm left with then 00:14:20.870 --> 00:14:24.900 is delta u times v minus u times delta v 00:14:24.900 --> 00:14:32.250 and all this is over v + delta v times v. Okay, 00:14:32.250 --> 00:14:33.360 there's the difference. 00:14:33.360 --> 00:14:36.600 There's the change in the quotient. 00:14:36.600 --> 00:14:39.580 The change in this function is given by this formula. 00:14:39.580 --> 00:14:41.570 And now to compute the derivative, 00:14:41.570 --> 00:14:45.000 I want to divide by delta x, and take the limit. 00:14:45.000 --> 00:14:53.560 So let's write that down, delta(u/v)/delta x is this 00:14:53.560 --> 00:14:56.820 formula here divided by delta x. 00:14:56.820 --> 00:15:00.986 And again, I'm going to put the delta x under these delta u 00:15:00.986 --> 00:15:02.780 and delta v. Okay? 00:15:02.780 --> 00:15:05.080 I'm gonna put delta x in the denominator, 00:15:05.080 --> 00:15:07.150 but I can think of that as dividing 00:15:07.150 --> 00:15:09.920 into this factor and this factor. 00:15:09.920 --> 00:15:16.980 So this is (delta u/ delta x)v - u(delta v/delta x). 00:15:21.130 --> 00:15:23.130 And all that is divided by the same denominator, 00:15:23.130 --> 00:15:28.970 (v + delta v)v. Right? 00:15:28.970 --> 00:15:33.010 Put the delta x up in the numerator there. 00:15:33.010 --> 00:15:37.830 Next up, take the limit as delta x goes to 0. 00:15:37.830 --> 00:15:43.470 I get, by definition, the derivative of (u/v). 00:15:43.470 --> 00:15:46.210 And on the right hand side, well, this 00:15:46.210 --> 00:15:51.300 is the derivative du/dx right? 00:15:51.300 --> 00:15:55.570 Times v. See and then u times, and here it's 00:15:55.570 --> 00:15:56.480 the derivative dv/dx. 00:16:00.420 --> 00:16:04.250 Now what about the denominator? 00:16:04.250 --> 00:16:10.220 So when delta x goes to 0, v stays the same, v stays 00:16:10.220 --> 00:16:10.720 the same. 00:16:10.720 --> 00:16:13.480 What happens to this delta v? 00:16:13.480 --> 00:16:17.970 It goes to 0, again, because v is continuous. 00:16:17.970 --> 00:16:23.330 So again, delta v goes to 0 with delta x 00:16:23.330 --> 00:16:28.180 because they're continuous and you just get v times v. 00:16:28.180 --> 00:16:30.867 I think that's the formula I wrote down over there. 00:16:30.867 --> 00:16:31.700 (du/dx)v - u(dv/dx). 00:16:35.510 --> 00:16:40.770 And all divided by the square of the old denominator. 00:16:40.770 --> 00:16:42.160 Well, that's it. 00:16:42.160 --> 00:16:43.540 That's the quotient rule. 00:16:43.540 --> 00:16:44.520 Weird formula. 00:16:44.520 --> 00:16:46.160 Let's see an application. 00:16:46.160 --> 00:16:51.070 Let's see an example. 00:16:51.070 --> 00:16:54.680 So the example I'm going to give is pretty simple. 00:16:54.680 --> 00:16:58.100 I'm going to take the numerator to be just 1. 00:16:58.100 --> 00:17:02.790 So I'm gonna take u = 1. 00:17:02.790 --> 00:17:07.580 So now I'm differentiating 1 / v, 00:17:07.580 --> 00:17:14.430 the reciprocal of a function; 1 over a function. 00:17:14.430 --> 00:17:16.880 Here's a copy of my rule. 00:17:16.880 --> 00:17:22.790 What's du/ dx in that case? u is a constant, 00:17:22.790 --> 00:17:27.050 so that term is 0 in this rule. 00:17:27.050 --> 00:17:28.700 I don't have to worry about this. 00:17:28.700 --> 00:17:31.650 I get a minus. 00:17:31.650 --> 00:17:36.800 And then u is 1, and dv/dx. 00:17:36.800 --> 00:17:38.820 Well, v is whatever v is. 00:17:38.820 --> 00:17:40.790 I'll write dv/dx as v'. 00:17:43.854 --> 00:17:45.520 And then I get a v^2 in the denominator. 00:17:45.520 --> 00:17:50.070 So that's the rule. 00:17:50.070 --> 00:17:51.380 I could write it as v^(-2) v'. 00:17:56.840 --> 00:17:59.300 Minus v' divided by v^2. 00:17:59.300 --> 00:18:03.730 That's the derivative of 1 / v. 00:18:03.730 --> 00:18:12.110 How about sub-example of that? 00:18:12.110 --> 00:18:15.840 I'm going to take the special case where u = 1 again. 00:18:15.840 --> 00:18:16.770 And v is a power of x. 00:18:21.000 --> 00:18:25.630 And I'm gonna use the rule that we developed earlier about 00:18:25.630 --> 00:18:29.080 the derivative of x^n. 00:18:29.080 --> 00:18:33.230 So what do I get here? 00:18:33.230 --> 00:18:42.645 d/dx (1/x^n) is, I'm plugging into this formula here with v = 00:18:42.645 --> 00:18:45.260 x^n. 00:18:45.260 --> 00:18:51.580 So I get minus, uh, v^-2. 00:18:51.580 --> 00:18:57.250 If v = x^n, v^-2 is, by the rule of exponents, x^(-2n). 00:19:01.430 --> 00:19:05.550 And then v' is the derivative of x^n, which is nx^(n-1). 00:19:10.150 --> 00:19:12.010 Okay, so let's put these together. 00:19:12.010 --> 00:19:13.550 There's several powers of x here. 00:19:13.550 --> 00:19:14.940 I can put them together. 00:19:14.940 --> 00:19:22.330 I get -n x to the -2n + n - 1. 00:19:22.330 --> 00:19:23.936 One of these n's cancels. 00:19:23.936 --> 00:19:25.310 And what I'm left with is -n - 1. 00:19:29.260 --> 00:19:32.550 So we've computed the derivative of 1 / x^n, 00:19:32.550 --> 00:19:39.210 which I could also write as x^-n, right? 00:19:39.210 --> 00:19:42.640 So I've computed the derivative of negative powers of x. 00:19:42.640 --> 00:19:46.560 And this is the formula that I get. 00:19:46.560 --> 00:19:51.990 If you think of this -n as a unit, as a thing to itself, 00:19:51.990 --> 00:19:54.310 it occurs here in the exponent. 00:19:54.310 --> 00:19:59.890 It occurs here, and it occurs here. 00:19:59.890 --> 00:20:01.820 So how does that compare with the formula 00:20:01.820 --> 00:20:04.120 that we had up here? 00:20:04.120 --> 00:20:06.860 The derivative of a power of x is 00:20:06.860 --> 00:20:12.300 that power times x to one less than that power. 00:20:12.300 --> 00:20:16.010 That's exactly the same as the rule that I wrote down here. 00:20:16.010 --> 00:20:19.050 But the power here happens to be a negative number, 00:20:19.050 --> 00:20:22.360 and the same negative number shows up as a coefficient 00:20:22.360 --> 00:20:23.780 and there in the exponent. 00:20:23.780 --> 00:20:24.280 Yeah? 00:20:24.280 --> 00:20:30.440 Student: [INAUDIBLE]. 00:20:30.440 --> 00:20:34.930 Professor: How did I do this? 00:20:34.930 --> 00:20:49.150 Student: [INAUDIBLE]. 00:20:49.150 --> 00:20:55.990 Professor: Where did that x^(-2n) come from? 00:20:55.990 --> 00:20:59.900 So I'm applying this rule. 00:20:59.900 --> 00:21:04.440 So the denominator in the quotient rule is v^2. 00:21:04.440 --> 00:21:11.109 And v was x^n, so the denominator is x^(2n). 00:21:11.109 --> 00:21:12.650 And I decided to write it as x^(-2n). 00:21:19.010 --> 00:21:22.080 So the green comments there... 00:21:22.080 --> 00:21:26.270 What they say is that I can enlarge this rule. 00:21:26.270 --> 00:21:31.230 This exact same rule is true for negative values of n, 00:21:31.230 --> 00:21:36.310 as well as positive values of n. 00:21:36.310 --> 00:21:40.300 So there's something new in your list 00:21:40.300 --> 00:21:46.670 of rules that you can apply, of values of the derivative. 00:21:46.670 --> 00:21:49.550 That standard rule is true for negative as well as positive 00:21:49.550 --> 00:21:51.120 exponents. 00:21:51.120 --> 00:21:57.290 And that comes out of a quotient rule. 00:21:57.290 --> 00:21:59.020 Okay, so we've done two rules. 00:21:59.020 --> 00:22:04.650 I've talked about the product rule and the quotient rule. 00:22:04.650 --> 00:22:05.670 What's next? 00:22:05.670 --> 00:22:07.150 Let's see the chain rule. 00:22:07.150 --> 00:22:22.220 So this is a composition rule. 00:22:22.220 --> 00:22:24.890 So the kind of thing that I have in mind, composition 00:22:24.890 --> 00:22:28.210 of functions is about substitution. 00:22:28.210 --> 00:22:31.106 So the kind of function that I have in mind is, for instance, 00:22:31.106 --> 00:22:31.730 y = (sin t)^10. 00:22:39.700 --> 00:22:42.695 That's a new one. 00:22:42.695 --> 00:22:44.820 We haven't seen how to differentiate that before, I 00:22:44.820 --> 00:22:46.590 think. 00:22:46.590 --> 00:22:50.600 This kind of power of a trig function happens very often. 00:22:50.600 --> 00:22:53.540 You've seen them happen, as well, I'm sure, already. 00:22:53.540 --> 00:22:58.020 And there's a little notational switch that people use. 00:22:58.020 --> 00:22:59.320 They'll write sin^10(t). 00:23:02.910 --> 00:23:05.100 But remember that when you write sin^10(t), 00:23:05.100 --> 00:23:08.030 what you mean is take the sine of t, 00:23:08.030 --> 00:23:10.440 and then take the 10th power of that. 00:23:10.440 --> 00:23:13.590 It's the meaning of sin^10(t). 00:23:13.590 --> 00:23:20.950 So the method of dealing with this kind of composition 00:23:20.950 --> 00:23:33.190 of functions is to use new variable names. 00:23:33.190 --> 00:23:36.830 What I mean is, I can think of this (sin t)^10. 00:23:39.710 --> 00:23:42.070 I can think of it it as a two step process. 00:23:42.070 --> 00:23:44.160 First of all, I compute the sine of t. 00:23:44.160 --> 00:23:47.450 And let's call the result x. 00:23:47.450 --> 00:23:50.150 There's the new variable name. 00:23:50.150 --> 00:23:53.340 And then, I express y in terms of x. 00:23:53.340 --> 00:23:58.070 So y says take this and raise it to the tenth power. 00:23:58.070 --> 00:23:59.360 In other words, y = x^10. 00:24:03.400 --> 00:24:06.420 And then you plug x = sin(t) into that, 00:24:06.420 --> 00:24:10.590 and you get the formula for what y is in terms of t. 00:24:10.590 --> 00:24:14.550 So it's good practice to introduce new letters when 00:24:14.550 --> 00:24:17.060 they're convenient, and this is one place where 00:24:17.060 --> 00:24:21.820 it's very convenient. 00:24:21.820 --> 00:24:24.260 So let's find a rule for differentiating 00:24:24.260 --> 00:24:25.860 a composition, a function that can 00:24:25.860 --> 00:24:27.770 be expressed by doing one function 00:24:27.770 --> 00:24:30.270 and then applying another function. 00:24:30.270 --> 00:24:32.880 And here's the rule. 00:24:32.880 --> 00:24:34.930 Well, maybe I'll actually derive this rule first, 00:24:34.930 --> 00:24:37.420 and then you'll see what it is. 00:24:37.420 --> 00:24:40.600 In fact, the rule is very simple to derive. 00:24:40.600 --> 00:24:43.890 So this is a proof first, and then we'll write down the rule. 00:24:43.890 --> 00:24:51.950 I'm interested in delta y / delta t. y is a function of x. 00:24:51.950 --> 00:24:53.760 x is a function of t. 00:24:53.760 --> 00:24:56.850 And I'm interested in how y changes with respect 00:24:56.850 --> 00:25:00.850 to t, with respect to the original variable t. 00:25:00.850 --> 00:25:05.160 Well, because of that intermediate variable, 00:25:05.160 --> 00:25:12.670 I can write this as (delta y / delta x) (delta x / delta t). 00:25:12.670 --> 00:25:15.330 It cancels, right? 00:25:15.330 --> 00:25:17.600 The delta x cancels. 00:25:17.600 --> 00:25:23.100 The change in that immediate variable cancels out. 00:25:23.100 --> 00:25:26.120 This is just basic algebra. 00:25:26.120 --> 00:25:29.930 But what happens when I let delta t get small? 00:25:29.930 --> 00:25:31.410 Well this give me dy/dt. 00:25:34.370 --> 00:25:42.220 On the right-hand side, I get (dy/dx) (dx/dt). 00:25:42.220 --> 00:25:44.430 So students will often remember this rule. 00:25:44.430 --> 00:25:47.130 This is the rule, by saying that you can cancel out 00:25:47.130 --> 00:25:49.080 for the dx's. 00:25:49.080 --> 00:25:51.860 And that's not so far from the truth. 00:25:51.860 --> 00:25:55.160 That's a good way to think of it. 00:25:55.160 --> 00:26:01.410 In other words, this is the so-called chain rule. 00:26:01.410 --> 00:26:26.690 And it says that differentiation of a composition is a product. 00:26:26.690 --> 00:26:34.910 It's just the product of the two derivatives. 00:26:34.910 --> 00:26:39.570 So that's how you differentiate a composite of two functions. 00:26:39.570 --> 00:26:42.070 And let's just do an example. 00:26:42.070 --> 00:26:44.690 Let's do this example. 00:26:44.690 --> 00:26:48.820 Let's see how that comes out. 00:26:48.820 --> 00:26:55.250 So let's differentiate, what did I say? 00:26:55.250 --> 00:26:56.530 (sin t)^10. 00:26:59.400 --> 00:27:03.130 Okay, there's an inside function and an outside function. 00:27:03.130 --> 00:27:07.910 The inside function is x as a function of t. 00:27:07.910 --> 00:27:19.170 This is the inside function, and this is the outside function. 00:27:19.170 --> 00:27:22.590 So the rule says, first of all let's differentiate 00:27:22.590 --> 00:27:23.550 the outside function. 00:27:23.550 --> 00:27:25.370 Take dy/dx. 00:27:25.370 --> 00:27:29.200 Differentiate it with respect to that variable x. 00:27:29.200 --> 00:27:31.020 The outside function is the 10th power. 00:27:31.020 --> 00:27:34.640 What's its derivative? 00:27:34.640 --> 00:27:37.530 So I get 10x^9. 00:27:42.440 --> 00:27:51.090 In this account, I'm using this newly introduced variable 00:27:51.090 --> 00:27:53.990 named x. 00:27:53.990 --> 00:27:58.150 So the derivative of the outside function is 10x^9. 00:27:58.150 --> 00:28:00.360 And then here's the inside function, 00:28:00.360 --> 00:28:03.130 and the next thing I want to do is differentiate it. 00:28:03.130 --> 00:28:07.730 So what's dx/dt, d/dt (sin t), the derivative of sine t? 00:28:07.730 --> 00:28:11.619 All right, that's cosine t. 00:28:11.619 --> 00:28:13.160 That's what the chain rule gives you. 00:28:13.160 --> 00:28:17.490 This is correct, but since we were the ones 00:28:17.490 --> 00:28:20.730 to introduce this notation x here, 00:28:20.730 --> 00:28:24.560 that wasn't given to us in the original problem here. 00:28:24.560 --> 00:28:26.400 The last step in this process should 00:28:26.400 --> 00:28:28.980 be to put back, to substitute back 00:28:28.980 --> 00:28:32.440 in what x is in terms of t. 00:28:32.440 --> 00:28:35.320 So x = sin t. 00:28:35.320 --> 00:28:45.980 So that tells me that I get 10(sin(t))^9, that's x^9, 00:28:45.980 --> 00:28:47.860 times the cos(t). 00:28:47.860 --> 00:28:50.860 Or the same thing is sin^9(t)cos(t). 00:28:56.040 --> 00:28:59.540 So there's an application of the chain rule. 00:28:59.540 --> 00:29:02.504 You know, people often wonder where the name chain 00:29:02.504 --> 00:29:03.170 rule comes from. 00:29:03.170 --> 00:29:06.340 I was just wondering about that myself. 00:29:06.340 --> 00:29:15.230 So is it because it chains you down? 00:29:15.230 --> 00:29:18.070 Is it like a chain fence? 00:29:18.070 --> 00:29:19.590 I decided what it is. 00:29:19.590 --> 00:29:21.910 It's because by using it, you burst 00:29:21.910 --> 00:29:25.880 the chains of differentiation, and you can differentiate 00:29:25.880 --> 00:29:28.040 many more functions using it. 00:29:28.040 --> 00:29:31.553 So when you want to think of the chain rule, 00:29:31.553 --> 00:29:35.640 just think of that chain there. 00:29:35.640 --> 00:29:47.960 It lets you burst free. 00:29:47.960 --> 00:30:04.830 Let me give you another application of the chain rule. 00:30:04.830 --> 00:30:16.220 Ready for this one? 00:30:16.220 --> 00:30:17.970 So I'd like to differentiate the sin(10t). 00:30:25.524 --> 00:30:27.440 Again, this is the composite of two functions. 00:30:27.440 --> 00:30:30.220 What's the inside function? 00:30:30.220 --> 00:30:35.640 Okay, so I think I'll introduce this new notation. x = 10t, 00:30:35.640 --> 00:30:38.260 and the outside function is the sine. 00:30:38.260 --> 00:30:41.320 So y = sin x. 00:30:41.320 --> 00:30:46.660 So now the chain rule says dy/dt is... 00:30:46.660 --> 00:30:47.920 Okay, let's see. 00:30:47.920 --> 00:30:50.710 I take the derivative of the outside function, 00:30:50.710 --> 00:30:54.240 and what's that? 00:30:54.240 --> 00:30:56.470 Sine prime and we can substitute because we 00:30:56.470 --> 00:30:58.520 know what sine prime is. 00:30:58.520 --> 00:31:06.470 So I get cosine of whatever, x, and then times what? 00:31:06.470 --> 00:31:11.400 Now I differentiate the inside function, which is just 10. 00:31:11.400 --> 00:31:16.380 So I could write this as 10cos of what? 00:31:16.380 --> 00:31:17.360 10t, x is 10t. 00:31:20.260 --> 00:31:26.170 Now, once you get used to this, this middle variable, 00:31:26.170 --> 00:31:33.190 you don't have to give a name for it. 00:31:33.190 --> 00:31:35.150 You can just to think about it in your mind 00:31:35.150 --> 00:31:44.890 without actually writing it down, d/dt (sin(10t)). 00:31:47.980 --> 00:31:49.860 I'll just do it again without introducing 00:31:49.860 --> 00:31:52.240 this middle variable explicitly. 00:31:52.240 --> 00:31:54.530 Think about it. 00:31:54.530 --> 00:31:58.100 I first of all differentiate the outside function, 00:31:58.100 --> 00:31:59.740 and I get cosine. 00:31:59.740 --> 00:32:03.170 But I don't change the thing that I'm plugging into it. 00:32:03.170 --> 00:32:08.560 It's still x that I'm plugging into it. x is 10t. 00:32:08.560 --> 00:32:11.470 So let's just write 10t and not worry about the name 00:32:11.470 --> 00:32:12.720 of that extra variable. 00:32:12.720 --> 00:32:15.510 If it confuses you, introduce the new variable. 00:32:15.510 --> 00:32:18.180 And do it carefully and slowly like this. 00:32:18.180 --> 00:32:19.970 But, quite quickly, I think you'll 00:32:19.970 --> 00:32:23.202 get to be able to keep that step in your mind. 00:32:23.202 --> 00:32:24.160 I'm not quite done yet. 00:32:24.160 --> 00:32:26.900 I haven't differentiated the inside function, 00:32:26.900 --> 00:32:29.190 the derivative of 10t = 10. 00:32:29.190 --> 00:32:33.250 So you get, again, the same result. 00:32:33.250 --> 00:32:36.420 A little shortcut that you'll get used to. 00:32:36.420 --> 00:32:38.680 Really and truly, once you have the chain rule, 00:32:38.680 --> 00:32:41.110 the world is yours to conquer. 00:32:41.110 --> 00:32:46.730 It puts you in a very, very powerful position. 00:32:46.730 --> 00:32:50.210 Okay, well let's see. 00:32:50.210 --> 00:32:51.310 What have I covered today? 00:32:51.310 --> 00:32:57.370 I've talked about product rule, quotient rule, composition. 00:32:57.370 --> 00:32:59.580 I should tell you something about higher derivatives, 00:32:59.580 --> 00:33:00.670 as well. 00:33:00.670 --> 00:33:10.440 So let's do that. 00:33:10.440 --> 00:33:12.150 This is a simple story. 00:33:12.150 --> 00:33:14.950 Higher is kind of a strange word. 00:33:14.950 --> 00:33:32.950 It just means differentiate over and over again. 00:33:32.950 --> 00:33:34.600 All right, so let's see. 00:33:34.600 --> 00:33:38.510 If we have a function u or u(x), please 00:33:38.510 --> 00:33:45.010 allow me to just write it as briefly as u. 00:33:45.010 --> 00:33:49.330 Well, this is a sort of notational thing. 00:33:49.330 --> 00:33:51.780 I can differentiate it and get u'. 00:33:54.790 --> 00:33:55.900 That's a new function. 00:33:55.900 --> 00:33:57.680 Like if you started with the sine, that's 00:33:57.680 --> 00:34:00.760 gonna be the cosine. 00:34:00.760 --> 00:34:03.570 A new function, so I can differentiate it again. 00:34:03.570 --> 00:34:05.780 And the notation for the differentiating of it again, 00:34:05.780 --> 00:34:07.470 is u prime prime. 00:34:07.470 --> 00:34:12.930 So u'' is just u' differentiated again. 00:34:12.930 --> 00:34:21.380 For example, if u is the sine of x, so u' is the cosine of x. 00:34:21.380 --> 00:34:24.150 Has Professor Jerison talked about what 00:34:24.150 --> 00:34:26.580 the derivative of cosine is? 00:34:26.580 --> 00:34:28.220 What is it? 00:34:28.220 --> 00:34:33.020 Ha, okay so u'' is -sin x. 00:34:36.810 --> 00:34:38.930 Let me go on. 00:34:38.930 --> 00:34:42.970 What do you suppose u''' means? 00:34:42.970 --> 00:34:46.420 I guess it's the derivative of u''. 00:34:46.420 --> 00:34:53.050 It's called the third derivative. 00:34:53.050 --> 00:34:56.210 And u'' is called the second derivative. 00:34:56.210 --> 00:34:59.000 And it's u'' differentiated again. 00:34:59.000 --> 00:35:03.680 So to compute u''' in this example, what do I do? 00:35:03.680 --> 00:35:05.340 I differentiate that again. 00:35:05.340 --> 00:35:08.460 There's a constant term, -1, constant factor. 00:35:08.460 --> 00:35:09.950 That comes out. 00:35:09.950 --> 00:35:13.500 The derivative of sine is what? 00:35:13.500 --> 00:35:17.930 Okay, so u''' = -cos x. 00:35:17.930 --> 00:35:18.690 Let's do it again. 00:35:18.690 --> 00:35:21.890 Now after a while, you get tired of writing these things. 00:35:21.890 --> 00:35:24.650 And so maybe I'll use the notation u^(4). 00:35:24.650 --> 00:35:27.290 That's the fourth derivative. 00:35:27.290 --> 00:35:29.490 That's u''''. 00:35:29.490 --> 00:35:33.440 Or it's (u''') differentiated again, the fourth derivative. 00:35:33.440 --> 00:35:37.970 And what is that in this example? 00:35:37.970 --> 00:35:41.290 Okay, the cosine has derivative minus the sine, 00:35:41.290 --> 00:35:42.010 like you told me. 00:35:42.010 --> 00:35:44.430 And that minus sign cancels with that sign, 00:35:44.430 --> 00:35:47.640 and all together, I get sin x. 00:35:47.640 --> 00:35:48.940 That's pretty bizarre. 00:35:48.940 --> 00:35:51.720 When I differentiate the function sine of x four times, 00:35:51.720 --> 00:35:56.920 I get back to the sine of x again. 00:35:56.920 --> 00:36:00.290 That's the way it is. 00:36:00.290 --> 00:36:03.491 Now this notation, prime prime prime prime, and things 00:36:03.491 --> 00:36:03.990 like that. 00:36:03.990 --> 00:36:13.650 There are different variants of that notation. 00:36:13.650 --> 00:36:24.070 For example, that's another notation. 00:36:24.070 --> 00:36:29.320 Well, you've used the notation du/dx before. u' 00:36:29.320 --> 00:36:30.630 could also be denoted du/dx. 00:36:35.730 --> 00:36:38.460 I think we've already here, today, 00:36:38.460 --> 00:36:43.230 used this way of rewriting du/dx. 00:36:43.230 --> 00:36:48.150 I think when I was talking about d/dt(uv) and so on, I pulled 00:36:48.150 --> 00:36:52.360 that d/dt outside and put whatever function 00:36:52.360 --> 00:36:55.010 you're differentiating over to the right. 00:36:55.010 --> 00:36:57.430 So that's just a notational switch. 00:36:57.430 --> 00:36:58.110 It looks good. 00:36:58.110 --> 00:37:06.260 It looks like good algebra doesn't it? 00:37:06.260 --> 00:37:12.410 But what it's doing is regarding this notation as an operator. 00:37:12.410 --> 00:37:16.920 It's something you apply to a function to get a new function. 00:37:16.920 --> 00:37:20.680 I apply it to the sine function, and I get the cosine function. 00:37:20.680 --> 00:37:24.220 I apply it to x^2, and I get 2x. 00:37:24.220 --> 00:37:31.140 This thing here, that symbol, represents an operator, 00:37:31.140 --> 00:37:40.340 which you apply to a function. 00:37:40.340 --> 00:37:44.860 And the operator says, take the function and differentiate it. 00:37:44.860 --> 00:37:47.330 So further notation that people often use, 00:37:47.330 --> 00:37:49.460 is they give a different name to that operator. 00:37:49.460 --> 00:37:52.270 And they'll write capital D for it. 00:37:52.270 --> 00:38:02.980 So this is just using capital D for the symbol d/dx. 00:38:02.980 --> 00:38:05.050 So in terms of that notation, let's see. 00:38:05.050 --> 00:38:20.440 Let's write down what higher derivatives look like. 00:38:20.440 --> 00:38:21.870 So let's see. 00:38:21.870 --> 00:38:23.090 That's what u' is. 00:38:23.090 --> 00:38:24.360 How about u''? 00:38:24.360 --> 00:38:28.890 Let's write that in terms of the d/dx notation. 00:38:28.890 --> 00:38:31.710 Well I'm supposed to differentiate u' right? 00:38:31.710 --> 00:38:35.590 So that's d/dx applied to the function du/dx. 00:38:40.920 --> 00:38:43.030 Differentiate the derivative. 00:38:43.030 --> 00:38:47.240 That's what I've done. 00:38:47.240 --> 00:38:54.350 Or I could write that as d/dx applied to d/dx applied to u. 00:38:54.350 --> 00:38:57.850 Just pulling that u outside. 00:38:57.850 --> 00:38:59.570 So I'm doing d/dx twice. 00:38:59.570 --> 00:39:01.590 I'm doing that operator twice. 00:39:01.590 --> 00:39:08.030 I could write that as (d/dx)^2 applied to u. 00:39:08.030 --> 00:39:15.170 Differentiate twice, and do it to the function u. 00:39:15.170 --> 00:39:23.130 Or, I can write it as, now this is a strange one. 00:39:23.130 --> 00:39:33.330 I could also write as-- like that. 00:39:33.330 --> 00:39:36.630 It's getting stranger and stranger, isn't it? 00:39:36.630 --> 00:39:40.770 This is definitely just a kind of abuse of notation. 00:39:40.770 --> 00:39:45.030 But people will go even further and write d squared 00:39:45.030 --> 00:39:46.030 u divided by dx squared. 00:39:50.500 --> 00:39:52.190 So this is the strangest one. 00:39:52.190 --> 00:39:56.190 This identity quality is the strangest one, 00:39:56.190 --> 00:40:00.130 because you may think that you're taking d of the quantity 00:40:00.130 --> 00:40:01.330 x^2. 00:40:01.330 --> 00:40:03.930 But that's not what's intended. 00:40:03.930 --> 00:40:08.240 This is not d(x^2). 00:40:08.240 --> 00:40:12.750 What's intended is the quantity dx squared. 00:40:12.750 --> 00:40:14.630 In this notation, which is very common, 00:40:14.630 --> 00:40:16.410 what's intended by the denominator 00:40:16.410 --> 00:40:18.250 is the quantity dx squared. 00:40:18.250 --> 00:40:23.630 It's part of this second differentiation operator. 00:40:23.630 --> 00:40:26.240 So I've written a bunch of equalities down here, 00:40:26.240 --> 00:40:28.570 and the only content to them is that these 00:40:28.570 --> 00:40:32.320 are all different notations for the same thing. 00:40:32.320 --> 00:40:34.940 You'll see this notation very commonly. 00:40:34.940 --> 00:40:37.050 So for instance the third derivative 00:40:37.050 --> 00:40:47.330 is d cubed u divided by dx cubed, and so on. 00:40:47.330 --> 00:40:47.830 Sorry? 00:40:47.830 --> 00:40:58.755 Student: [INAUDIBLE]. 00:40:58.755 --> 00:40:59.880 Professor: Yes, absolutely. 00:40:59.880 --> 00:41:04.417 Or an equally good notation is to write the operator capital 00:41:04.417 --> 00:41:05.500 D, done three times, to u. 00:41:09.400 --> 00:41:11.502 Absolutely. 00:41:11.502 --> 00:41:13.960 So I guess I should also write over here when I was talking 00:41:13.960 --> 00:41:16.180 about d^2, the second derivative, 00:41:16.180 --> 00:41:20.820 another notation is do the operator capital D twice. 00:41:20.820 --> 00:41:22.820 Let's see an example of how this can be applied. 00:41:22.820 --> 00:41:23.903 I'll answer this question. 00:41:23.903 --> 00:41:32.582 Student: [INAUDIBLE]. 00:41:32.582 --> 00:41:34.040 Professor: Yeah, so the question is 00:41:34.040 --> 00:41:36.230 whether the fourth derivative always gives you 00:41:36.230 --> 00:41:38.880 the original function back, like what happened here. 00:41:38.880 --> 00:41:39.580 No. 00:41:39.580 --> 00:41:43.470 That's very, very special to sines and cosines. 00:41:43.470 --> 00:41:45.200 All right? 00:41:45.200 --> 00:41:47.850 And, in fact, let's see an example of that. 00:41:47.850 --> 00:41:50.920 I'll do a calculation. 00:41:50.920 --> 00:42:06.130 Let's calculate the nth derivative of x^n. 00:42:06.130 --> 00:42:13.190 Okay, n is a number, like 1, 2, 3, 4. 00:42:13.190 --> 00:42:13.720 Here we go. 00:42:13.720 --> 00:42:15.360 Let's do this. 00:42:15.360 --> 00:42:17.650 So, let's do this bit by bit. 00:42:17.650 --> 00:42:22.500 What's the first derivative of x^n? 00:42:22.500 --> 00:42:24.090 So everybody knows this. 00:42:24.090 --> 00:42:27.830 I'm just using a new notation, this capital D notation. 00:42:27.830 --> 00:42:30.520 So it's nx^(n-1). 00:42:30.520 --> 00:42:34.380 Now know, by the way, n could be a negative number for that, 00:42:34.380 --> 00:42:37.250 but for now, for this application, 00:42:37.250 --> 00:42:41.280 I wanna take n to be 1, 2, 3, and so on; 00:42:41.280 --> 00:42:43.070 one of those numbers. 00:42:43.070 --> 00:42:44.550 Okay, we did one derivative. 00:42:44.550 --> 00:42:49.530 Let's compute the second derivative of x^n. 00:42:49.530 --> 00:42:52.070 Well there's this n constant that comes out, 00:42:52.070 --> 00:42:59.980 and then the exponent comes down, and it gets reduced by 1. 00:42:59.980 --> 00:43:01.190 All right? 00:43:01.190 --> 00:43:03.780 Should I do one more? 00:43:03.780 --> 00:43:07.600 D^3 (x^n) is n(n-1). 00:43:07.600 --> 00:43:09.410 That's the constant from here. 00:43:09.410 --> 00:43:13.220 Times that exponent, n - 2, times 1 less, n 00:43:13.220 --> 00:43:15.740 - 3 is the new exponent. 00:43:15.740 --> 00:43:26.430 Well, I keep on going until I come to a new blackboard. 00:43:26.430 --> 00:43:28.100 Now, I think I'm going to stop when 00:43:28.100 --> 00:43:29.980 I get to the n minus first derivative, 00:43:29.980 --> 00:43:35.370 so we can see what's likely to happen. 00:43:35.370 --> 00:43:38.970 So when I took the third derivative, 00:43:38.970 --> 00:43:42.957 I had the n minus third power of x. 00:43:42.957 --> 00:43:44.540 And when I took the second derivative, 00:43:44.540 --> 00:43:45.760 I had the second power of x. 00:43:45.760 --> 00:43:48.310 So, I think what'll happen when I 00:43:48.310 --> 00:43:49.730 have the n minus first derivative 00:43:49.730 --> 00:43:53.510 is I'll have the first power of x left over. 00:43:53.510 --> 00:43:55.390 The powers of x keep coming down. 00:43:55.390 --> 00:43:59.350 And what I've done it n - 1 times, I get the first power. 00:43:59.350 --> 00:44:04.230 And then I get a big constant out in front here times more 00:44:04.230 --> 00:44:07.450 and more and more of these smaller and smaller integers 00:44:07.450 --> 00:44:08.500 that come down. 00:44:08.500 --> 00:44:12.310 What's the last integer that came down before I got x^1 00:44:12.310 --> 00:44:17.460 here? 00:44:17.460 --> 00:44:19.390 Well, let's see. 00:44:19.390 --> 00:44:23.320 It's just 2, because this x^1 occurred as the derivative 00:44:23.320 --> 00:44:24.340 of x^2. 00:44:24.340 --> 00:44:27.800 And the coefficient in front of that is 2. 00:44:27.800 --> 00:44:29.730 So that's what you get. 00:44:29.730 --> 00:44:35.140 The numbers n, n-1, and so on down to 2, times x^1. 00:44:35.140 --> 00:44:41.560 And now we can differentiate one more time and calculate what 00:44:41.560 --> 00:44:42.770 D^n x^n is. 00:44:42.770 --> 00:44:48.070 So I get the same number, n times n-1 and so on and so on, 00:44:48.070 --> 00:44:49.680 times 2. 00:44:49.680 --> 00:44:52.500 And then I guess I'll say times 1. 00:44:52.500 --> 00:44:58.640 Times, what's the derivative of x^1= 1, so times 1. 00:44:58.640 --> 00:45:01.260 Time 1, times 1. 00:45:01.260 --> 00:45:10.490 Where this one means the constant function 1. 00:45:10.490 --> 00:45:14.070 Does anyone know what this number is called? 00:45:14.070 --> 00:45:15.110 That has a name. 00:45:15.110 --> 00:45:19.720 It's called n factorial. 00:45:19.720 --> 00:45:21.400 And it's written n exclamation point. 00:45:24.240 --> 00:45:28.830 And we just used an example of mathematical induction. 00:45:28.830 --> 00:45:37.750 So the end result is D^n x^n is n!, constant. 00:45:37.750 --> 00:45:42.460 Okay that's a neat fact. 00:45:42.460 --> 00:45:47.570 Final question for the lecture is what's D^(n + 1) applied 00:45:47.570 --> 00:45:49.730 to x^n? 00:45:49.730 --> 00:45:50.850 Ha. 00:45:50.850 --> 00:45:54.340 Excellent. 00:45:54.340 --> 00:45:56.620 It's the derivative of a constant. 00:45:56.620 --> 00:45:57.940 So it's 0. 00:45:57.940 --> 00:45:58.440 Okay. 00:45:58.440 --> 00:45:59.980 Thank you.