WEBVTT 00:00:00.110 --> 00:00:02.320 The following content was created under a Creative 00:00:02.320 --> 00:00:03.660 Commons License. 00:00:03.660 --> 00:00:05.930 Your support will help MIT OpenCourseWare 00:00:05.930 --> 00:00:09.970 continue to offer high quality educational resources for free. 00:00:09.970 --> 00:00:12.560 To make a donation or to view additional materials 00:00:12.560 --> 00:00:16.150 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.150 --> 00:00:19.350 at ocw.mit.edu. 00:00:19.350 --> 00:00:24.720 Professor: So, we're ready to begin the fifth lecture. 00:00:24.720 --> 00:00:25.790 I'm glad to be back. 00:00:25.790 --> 00:00:33.240 Thank you for entertaining my colleague, Haynes Miller. 00:00:33.240 --> 00:00:35.600 So, today we're going to continue 00:00:35.600 --> 00:00:40.740 where he started, namely what he talked about was the chain 00:00:40.740 --> 00:00:43.690 rule, which is probably the most powerful technique 00:00:43.690 --> 00:00:45.860 for extending the kinds of functions 00:00:45.860 --> 00:00:47.650 that you can differentiate. 00:00:47.650 --> 00:00:50.920 And we're going to use the chain rule in some rather clever 00:00:50.920 --> 00:00:54.410 algebraic ways today. 00:00:54.410 --> 00:00:57.420 So the topic for today is what's known 00:00:57.420 --> 00:01:10.670 as implicit differentiation. 00:01:10.670 --> 00:01:15.040 So implicit differentiation is a technique 00:01:15.040 --> 00:01:17.690 that allows you to differentiate a lot of functions you didn't 00:01:17.690 --> 00:01:20.970 even know how to find before. 00:01:20.970 --> 00:01:24.940 And it's a technique - let's wait for a few people 00:01:24.940 --> 00:01:28.260 to sit down here. 00:01:28.260 --> 00:01:29.210 Physics, huh? 00:01:29.210 --> 00:01:36.419 Okay, more Physics. 00:01:36.419 --> 00:01:37.210 Let's take a break. 00:01:37.210 --> 00:01:40.980 You can get those after class. 00:01:40.980 --> 00:01:46.220 All right, so we're talking about implicit differentiation, 00:01:46.220 --> 00:01:53.760 and I'm going to illustrate it by several examples. 00:01:53.760 --> 00:01:57.610 So this is one of the most important and basic formulas 00:01:57.610 --> 00:01:59.770 that we've already covered part way. 00:01:59.770 --> 00:02:06.950 Namely, the derivative of x to a power is ax^(a-1). 00:02:06.950 --> 00:02:15.780 Now, what we've got so far is the exponents, 0, plus or minus 00:02:15.780 --> 00:02:19.010 1, plus or minus 2, etc. 00:02:19.010 --> 00:02:24.440 You did the positive integer powers in the first lecture, 00:02:24.440 --> 00:02:30.710 and then yesterday Professor Miller 00:02:30.710 --> 00:02:32.470 told you about the negative powers. 00:02:32.470 --> 00:02:35.470 So what we're going to do right now, 00:02:35.470 --> 00:02:39.500 today, is we're going to consider 00:02:39.500 --> 00:02:44.420 the exponents which are rational numbers, ratios of integers. 00:02:44.420 --> 00:02:46.540 So a is m/n. 00:02:46.540 --> 00:02:53.380 m and n are integers. 00:02:53.380 --> 00:02:55.414 All right, so that's our goal for right now, 00:02:55.414 --> 00:02:56.830 and we're going to use this method 00:02:56.830 --> 00:02:58.150 of implicit differentiation. 00:02:58.150 --> 00:03:01.010 In particular, it's important to realize that this 00:03:01.010 --> 00:03:03.260 covers the case m = 1. 00:03:03.260 --> 00:03:04.890 And those are the nth roots. 00:03:04.890 --> 00:03:07.370 So when we take the one over n power, 00:03:07.370 --> 00:03:09.680 we're going to cover that right now, 00:03:09.680 --> 00:03:13.110 along with many other examples. 00:03:13.110 --> 00:03:16.360 So this is our first example. 00:03:16.360 --> 00:03:17.700 So how do we get started? 00:03:17.700 --> 00:03:20.510 Well we just write down a formula for the function. 00:03:20.510 --> 00:03:24.620 The function is y = x^(m/n). 00:03:24.620 --> 00:03:26.620 That's what we're trying to deal with. 00:03:26.620 --> 00:03:30.610 And now there's really only two steps. 00:03:30.610 --> 00:03:38.120 The first step is to take this equation to the nth power, 00:03:38.120 --> 00:03:42.900 so write it y^n = x^m. 00:03:42.900 --> 00:03:46.170 Alright, so that's just the same equation re-written. 00:03:46.170 --> 00:03:50.180 And now, what we're going to do is 00:03:50.180 --> 00:03:52.170 we're going to differentiate. 00:03:52.170 --> 00:04:01.600 So we're going to apply d/dx to the equation. 00:04:01.600 --> 00:04:05.730 Now why is it that we can apply it to the second equation, not 00:04:05.730 --> 00:04:06.670 the first equation? 00:04:06.670 --> 00:04:10.220 So maybe I should call these equation 1 and equation 2. 00:04:10.220 --> 00:04:13.150 So, the point is, we can apply it to equation 2. 00:04:13.150 --> 00:04:17.860 Now, the reason is that we don't know how to differentiate 00:04:17.860 --> 00:04:18.860 x^(m/n). 00:04:18.860 --> 00:04:21.320 That's something we just don't know yet. 00:04:21.320 --> 00:04:24.630 But we do know how to differentiate integer powers. 00:04:24.630 --> 00:04:29.080 Those are the things that we took care of before. 00:04:29.080 --> 00:04:32.980 So now we're in shape to be able to do the differentiation. 00:04:32.980 --> 00:04:34.900 So I'm going to write it out explicitly 00:04:34.900 --> 00:04:37.930 over here, without carrying it out just yet. 00:04:37.930 --> 00:04:46.460 That's d/dx of y^n = d/dx of x^m. 00:04:46.460 --> 00:04:51.920 And now you see this expression here 00:04:51.920 --> 00:04:55.020 requires us to do something we couldn't do before yesterday. 00:04:55.020 --> 00:04:58.780 Namely, this y is a function of x. 00:04:58.780 --> 00:05:01.610 So we have to apply the chain rule here. 00:05:01.610 --> 00:05:06.710 So this is the same as - this is by the chain rule now - 00:05:06.710 --> 00:05:13.332 d/dy of y^n times dy/dx. 00:05:13.332 --> 00:05:15.790 And then, on the right hand side, we can just carry it out. 00:05:15.790 --> 00:05:17.280 We know the formula. 00:05:17.280 --> 00:05:18.420 It's mx^(m-1). 00:05:21.550 --> 00:05:24.960 Right, now this is our scheme. 00:05:24.960 --> 00:05:29.360 And you'll see in a minute why we win with this. 00:05:29.360 --> 00:05:32.390 So, first of all, there are two factors here. 00:05:32.390 --> 00:05:33.870 One of them is unknown. 00:05:33.870 --> 00:05:35.870 In fact, it's what we're looking for. 00:05:35.870 --> 00:05:38.570 But the other one is going to be a known quantity, 00:05:38.570 --> 00:05:40.507 because we know how to differentiate y 00:05:40.507 --> 00:05:42.507 to the n with respect to y. 00:05:42.507 --> 00:05:44.340 That's the same formula, although the letter 00:05:44.340 --> 00:05:46.330 has been changed. 00:05:46.330 --> 00:05:53.070 And so this is the same as - I'll write it underneath here - 00:05:53.070 --> 00:06:07.040 n y^(n-1) dy/dx = m x^(m-1). 00:06:07.040 --> 00:06:14.565 Okay, now comes, if you like, the non-calculus part 00:06:14.565 --> 00:06:15.190 of the problem. 00:06:15.190 --> 00:06:17.140 Remember the non-calculus part of the problem 00:06:17.140 --> 00:06:20.030 is always the messier part of the problem. 00:06:20.030 --> 00:06:22.210 So we want to figure out this formula. 00:06:22.210 --> 00:06:25.600 This formula, the answer over here, 00:06:25.600 --> 00:06:29.850 which maybe I'll put in a box now, 00:06:29.850 --> 00:06:33.420 has this expressed much more simply, only in terms of x. 00:06:33.420 --> 00:06:36.140 And what we have to do now is just solve for dy/dx 00:06:36.140 --> 00:06:39.730 using algebra, and then solve all the way in terms of x. 00:06:39.730 --> 00:06:41.700 So, first of all, we solve for dy/dx. 00:06:44.230 --> 00:06:47.930 So I do that by dividing the factor on the left-hand side. 00:06:47.930 --> 00:06:52.960 So I get here mx^(m-1) divided by ny^(n-1). 00:06:56.030 --> 00:07:02.020 And now I'm going to plug in-- so I'll write this as m/n. 00:07:02.020 --> 00:07:04.540 This is x^(m-1). 00:07:04.540 --> 00:07:10.540 Now over here I'm going to put in for y, x^(m/n) times n-1. 00:07:15.690 --> 00:07:18.700 So now we're almost done, but unfortunately we 00:07:18.700 --> 00:07:22.070 have this mess of exponents that we have to work out. 00:07:22.070 --> 00:07:25.060 I'm going to write it one more time. 00:07:25.060 --> 00:07:28.584 So I already recognize the factor a out front. 00:07:28.584 --> 00:07:30.250 That's not going to be a problem for me, 00:07:30.250 --> 00:07:31.980 and that's what I'm aiming for here. 00:07:31.980 --> 00:07:34.700 But now I have to encode all of these powers, 00:07:34.700 --> 00:07:36.390 so let's just write it. 00:07:36.390 --> 00:07:41.790 It's m-1, and then it's minus the quantity (n-1) m/n. 00:07:46.270 --> 00:07:50.250 All right, so that's the law of exponents applied to this ratio 00:07:50.250 --> 00:07:50.750 here. 00:07:50.750 --> 00:07:58.410 And then we'll do the arithmetic over here on the next board. 00:07:58.410 --> 00:08:08.700 So we have here m - 1 - (n-1) m/n = m - 1. 00:08:08.700 --> 00:08:12.460 And if I multiply n by this, I get -m. 00:08:12.460 --> 00:08:15.560 And if the second factor is minus minus, that's a plus. 00:08:15.560 --> 00:08:18.220 And that's +m/n. 00:08:18.220 --> 00:08:21.080 Altogether the two m's cancel. 00:08:21.080 --> 00:08:23.740 I have here -1 + m/n. 00:08:23.740 --> 00:08:27.492 And lo and behold that's the same thing as a - 1, 00:08:27.492 --> 00:08:29.510 just what we wanted. 00:08:29.510 --> 00:08:31.900 All right, so this equals a x^(n-1). 00:08:31.900 --> 00:08:39.560 Okay, again just a bunch of arithmetic. 00:08:39.560 --> 00:08:42.530 From this point forward, from this substitution 00:08:42.530 --> 00:08:51.180 on, it's just the arithmetic of exponents. 00:08:51.180 --> 00:08:58.000 All right, so we've done our first example here. 00:08:58.000 --> 00:09:00.590 I want to give you a couple more examples, 00:09:00.590 --> 00:09:04.030 so let's just continue. 00:09:04.030 --> 00:09:08.720 The next example I'll keep relatively simple. 00:09:08.720 --> 00:09:15.590 So we have example two, which is going to be the function x^2 + 00:09:15.590 --> 00:09:18.060 y^2 = 1. 00:09:18.060 --> 00:09:21.200 Well, that's not really a function. 00:09:21.200 --> 00:09:29.180 It's a way of defining y as a function of x implicitly. 00:09:29.180 --> 00:09:34.700 There's the idea that I could solve for y if I wanted to. 00:09:34.700 --> 00:09:36.490 And indeed let's do that. 00:09:36.490 --> 00:09:42.490 So if you solve for y here, what happens is you get y^2 = 1 - 00:09:42.490 --> 00:09:47.380 x^2, and y is equal to plus or minus the square root of 1 - 00:09:47.380 --> 00:09:52.940 x^2. 00:09:52.940 --> 00:09:58.200 So this, if you like, is the implicit definition. 00:09:58.200 --> 00:10:00.910 And here is the explicit function y, 00:10:00.910 --> 00:10:04.140 which is a function of x. 00:10:04.140 --> 00:10:06.360 And now just for my own convenience, 00:10:06.360 --> 00:10:09.310 I'm just going to take the positive branch. 00:10:09.310 --> 00:10:13.450 This is the function. 00:10:13.450 --> 00:10:15.900 It's just really a circle in disguise. 00:10:15.900 --> 00:10:19.600 And I'm just going to take the top part of the circle, 00:10:19.600 --> 00:10:24.920 so we'll take that top hump here. 00:10:24.920 --> 00:10:27.900 All right, so that means I'm erasing this minus sign. 00:10:27.900 --> 00:10:35.660 I'm just taking the positive branch, just 00:10:35.660 --> 00:10:36.470 for my convenience. 00:10:36.470 --> 00:10:40.980 I could do it just as well with the negative branch. 00:10:40.980 --> 00:10:46.000 Alright, so now I've taken the solution, 00:10:46.000 --> 00:10:49.540 and I can differentiate with this. 00:10:49.540 --> 00:10:53.070 So rather than using the dy/dx notation over here, 00:10:53.070 --> 00:10:55.510 I'm going to switch notations over here, 00:10:55.510 --> 00:10:56.620 because it's less writing. 00:10:56.620 --> 00:10:59.670 I'm going to write y' and change notations. 00:10:59.670 --> 00:11:04.450 Okay, so I want to take the derivative of this. 00:11:04.450 --> 00:11:11.570 Well this is a somewhat complicated function here. 00:11:11.570 --> 00:11:15.770 It's the square root of 1 - x^2, and the right way always 00:11:15.770 --> 00:11:21.720 to look at functions like this is to rewrite them using 00:11:21.720 --> 00:11:26.210 the fractional power notation. 00:11:26.210 --> 00:11:28.910 That's the first step in computing 00:11:28.910 --> 00:11:32.600 a derivative of a square root. 00:11:32.600 --> 00:11:38.030 And then the second step here is what? 00:11:38.030 --> 00:11:40.740 Does somebody want to tell me? 00:11:40.740 --> 00:11:43.650 Chain rule, right. 00:11:43.650 --> 00:11:44.300 That's it. 00:11:44.300 --> 00:11:45.320 So we have two things. 00:11:45.320 --> 00:11:47.720 We start with one, and then we do something else to it. 00:11:47.720 --> 00:11:50.090 So whenever we do two things to something, 00:11:50.090 --> 00:11:52.020 we need to apply the chain rule. 00:11:52.020 --> 00:11:55.240 So 1 - x^2, square root. 00:11:55.240 --> 00:11:57.180 All right, so how do we do that? 00:11:57.180 --> 00:11:58.830 Well, the first factor I claim is 00:11:58.830 --> 00:12:01.220 the derivative of this thing. 00:12:01.220 --> 00:12:06.500 So this is 1/2 blah to the -1/2. 00:12:06.500 --> 00:12:09.760 So I'm doing this kind of by the advanced method 00:12:09.760 --> 00:12:11.390 now, because we've already graduated. 00:12:11.390 --> 00:12:14.450 You already did the chain rule last time. 00:12:14.450 --> 00:12:15.920 So what does this mean? 00:12:15.920 --> 00:12:20.940 This is an abbreviation for the derivative with respect 00:12:20.940 --> 00:12:27.697 to blah of blah ^ 1/2, whatever it is. 00:12:27.697 --> 00:12:30.280 All right, so that's the first factor that we're going to use. 00:12:30.280 --> 00:12:34.480 Rather than actually write out a variable for it 00:12:34.480 --> 00:12:36.890 and pass through as I did previously 00:12:36.890 --> 00:12:39.160 with this y and x variable here, I'm 00:12:39.160 --> 00:12:41.840 just going to skip that step and let 00:12:41.840 --> 00:12:45.370 you imagine it as being a placeholder for that variable 00:12:45.370 --> 00:12:45.870 here. 00:12:45.870 --> 00:12:48.960 So this variable is now parenthesis. 00:12:48.960 --> 00:12:52.370 And then I have to multiply that by the rate of change of what's 00:12:52.370 --> 00:12:55.050 inside with respect to x. 00:12:55.050 --> 00:12:58.580 And that is going to be -2x. 00:12:58.580 --> 00:13:02.830 The derivative of 1 - x^2 is -2x. 00:13:02.830 --> 00:13:09.030 And now again, we couldn't have done this example two 00:13:09.030 --> 00:13:11.730 before example one, because we needed 00:13:11.730 --> 00:13:17.470 to know that the power rule worked not just for a integer 00:13:17.470 --> 00:13:19.690 but also for a = 1/2. 00:13:19.690 --> 00:13:22.740 We're using the case a = 1/2 right here. 00:13:22.740 --> 00:13:29.540 It's 1/2 times, and this -1/2 here is a-1. - 00:13:29.540 --> 00:13:33.430 So this is the case a = 1/2. 00:13:33.430 --> 00:13:39.790 a-1 happens to be -1/2. 00:13:39.790 --> 00:13:41.940 Okay, so I'm putting all those things together. 00:13:41.940 --> 00:13:44.380 And you know within a week you have 00:13:44.380 --> 00:13:45.970 to be doing this very automatically. 00:13:45.970 --> 00:13:47.900 So we're going to do it at this speed now. 00:13:47.900 --> 00:13:49.780 You want to do it even faster, ultimately. 00:13:49.780 --> 00:13:50.280 Yes? 00:13:50.280 --> 00:13:53.630 Student: [INAUDIBLE] 00:13:53.630 --> 00:13:56.110 Professor: The question is could I have done it implicitly 00:13:56.110 --> 00:13:58.060 without the square roots. 00:13:58.060 --> 00:13:59.440 And the answer is yes. 00:13:59.440 --> 00:14:02.040 That's what I'm about to do. 00:14:02.040 --> 00:14:04.570 So this is an illustration of what's 00:14:04.570 --> 00:14:07.430 called the explicit solution. 00:14:07.430 --> 00:14:13.800 So this guy is what's called explicit. 00:14:13.800 --> 00:14:17.024 And I want to contrast it with the method 00:14:17.024 --> 00:14:18.440 that we're going to now use today. 00:14:18.440 --> 00:14:20.410 So it involves a lot of complications. 00:14:20.410 --> 00:14:21.701 It involves the chain rule. 00:14:21.701 --> 00:14:23.700 And as we'll see it can get messier and messier. 00:14:23.700 --> 00:14:27.260 And then there's the implicit method, 00:14:27.260 --> 00:14:29.830 which I claim is easier. 00:14:29.830 --> 00:14:36.170 So let's see what happens if you do it implicitly 00:14:36.170 --> 00:14:41.010 The implicit method involves, instead of writing 00:14:41.010 --> 00:14:43.610 the function in this relatively complicated way, 00:14:43.610 --> 00:14:47.380 with the square root, it involves leaving it alone. 00:14:47.380 --> 00:14:50.050 Don't do anything to it. 00:14:50.050 --> 00:14:52.820 In this previous case, we were left with something which was 00:14:52.820 --> 00:14:56.820 complicated, say x^(1/3) or x^(1/2) or something 00:14:56.820 --> 00:14:57.370 complicated. 00:14:57.370 --> 00:14:59.326 We had to simplify it. 00:14:59.326 --> 00:15:01.450 We had an equation one, which was more complicated. 00:15:01.450 --> 00:15:03.970 We simplified it then differentiated it. 00:15:03.970 --> 00:15:05.590 And so that was a simpler case. 00:15:05.590 --> 00:15:09.590 Well here, the simplest thing us to differentiate 00:15:09.590 --> 00:15:13.470 is the one we started with, because squares are practically 00:15:13.470 --> 00:15:16.725 the easiest thing after first powers, or maybe zeroth powers 00:15:16.725 --> 00:15:18.830 to differentiate. 00:15:18.830 --> 00:15:19.940 So we're leaving it alone. 00:15:19.940 --> 00:15:21.689 This is the simplest possible form for it, 00:15:21.689 --> 00:15:23.640 and now we're going to differentiate. 00:15:23.640 --> 00:15:24.570 So what happens? 00:15:24.570 --> 00:15:26.640 So again what's the method? 00:15:26.640 --> 00:15:27.810 Let me remind you. 00:15:27.810 --> 00:15:30.210 You're applying d/dx to the equation. 00:15:30.210 --> 00:15:33.640 So you have to differentiate the left side of the equation, 00:15:33.640 --> 00:15:35.810 and differentiate the right side of the equation. 00:15:35.810 --> 00:15:51.100 So it's this, and what you get is 2x + 2yy' is equal to what? 00:15:51.100 --> 00:15:52.760 0. 00:15:52.760 --> 00:15:56.390 The derivative of 1 0. 00:15:56.390 --> 00:15:58.630 So this is the chain rule again. 00:15:58.630 --> 00:16:00.390 I did it a different way. 00:16:00.390 --> 00:16:02.690 I'm trying to get you used to many different notations 00:16:02.690 --> 00:16:04.420 at once. 00:16:04.420 --> 00:16:05.350 Well really just two. 00:16:05.350 --> 00:16:10.600 Just the prime notation and the dy/dx notation. 00:16:10.600 --> 00:16:14.370 And this is what I get. 00:16:14.370 --> 00:16:19.670 So now all I have to do is solve for y'. 00:16:19.670 --> 00:16:24.270 So that y', if I put the 2x on the other side, is -2x, 00:16:24.270 --> 00:16:27.760 and then divide by 2y, which is -x/y. 00:16:30.630 --> 00:16:34.600 So let's compare our solutions, and I'll apologize, 00:16:34.600 --> 00:16:39.080 I'm going to have to erase something to do that. 00:16:39.080 --> 00:16:44.480 So let's compare our two solutions. 00:16:44.480 --> 00:16:46.460 I'm going to put this underneath and simplify. 00:16:46.460 --> 00:16:48.880 So what was our solution over here? 00:16:48.880 --> 00:16:51.500 It was 1/2(1-x^2)^(-1/2) (-2x). 00:16:56.720 --> 00:17:02.170 That was what we got over here. 00:17:02.170 --> 00:17:06.992 And that is the same thing, if I cancel the 2's, and I change it 00:17:06.992 --> 00:17:08.450 back to looking like a square root, 00:17:08.450 --> 00:17:11.536 that's the same thing as -x divided by square root of 1 - 00:17:11.536 --> 00:17:13.960 x^2. 00:17:13.960 --> 00:17:18.380 So this is the formula for the derivative 00:17:18.380 --> 00:17:21.340 when I do it the explicit way. 00:17:21.340 --> 00:17:29.550 And I'll just compare them, these two expressions here. 00:17:29.550 --> 00:17:32.630 And notice they are the same. 00:17:32.630 --> 00:17:37.860 They're the same, because y is equal to square root of 1 - 00:17:37.860 --> 00:17:40.230 x^2. 00:17:40.230 --> 00:17:40.730 Yeah? 00:17:40.730 --> 00:17:41.230 Question? 00:17:41.230 --> 00:17:46.140 Student: [INAUDIBLE] 00:17:46.140 --> 00:17:48.530 Professor: The question is why did the implicit method 00:17:48.530 --> 00:17:50.930 not give the bottom half of the circle? 00:17:50.930 --> 00:17:53.200 Very good question. 00:17:53.200 --> 00:17:57.000 The answer to that is that it did. 00:17:57.000 --> 00:17:59.520 I just didn't mention it. 00:17:59.520 --> 00:18:00.890 Wait, I'll explain. 00:18:00.890 --> 00:18:05.300 So suppose I stuck in a minus sign here. 00:18:05.300 --> 00:18:08.040 I would have gotten this with the difference, so 00:18:08.040 --> 00:18:10.080 with an extra minus sign. 00:18:10.080 --> 00:18:12.480 But then when I compared it to what was over there, 00:18:12.480 --> 00:18:15.620 I would have had to have another different minus sign over here. 00:18:15.620 --> 00:18:19.310 So actually both places would get an extra minus sign. 00:18:19.310 --> 00:18:20.630 And they would still coincide. 00:18:20.630 --> 00:18:22.760 So actually the implicit method is a little better. 00:18:22.760 --> 00:18:23.940 It doesn't even notice the difference 00:18:23.940 --> 00:18:24.940 between the branches. 00:18:24.940 --> 00:18:28.930 It does the job on both the top and bottom half. 00:18:28.930 --> 00:18:31.390 Another way of saying that is that you're 00:18:31.390 --> 00:18:33.200 calculating the slopes here. 00:18:33.200 --> 00:18:35.170 So let's look at this picture. 00:18:35.170 --> 00:18:36.740 Here's a slope. 00:18:36.740 --> 00:18:39.090 Let's just take a look at a positive value 00:18:39.090 --> 00:18:42.770 of x and just check the sign to see what's happening. 00:18:42.770 --> 00:18:46.982 If you take a positive value of x over here, x is positive. 00:18:46.982 --> 00:18:48.190 This denominator is positive. 00:18:48.190 --> 00:18:49.106 The slope is negative. 00:18:49.106 --> 00:18:52.620 You can see that it's tilting down. 00:18:52.620 --> 00:18:53.960 So it's okay. 00:18:53.960 --> 00:18:59.520 Now on the bottom side, it's going to be tilting up. 00:18:59.520 --> 00:19:01.690 And similarly what's happening up here 00:19:01.690 --> 00:19:05.275 is that both x and y are positive, and this x and this y 00:19:05.275 --> 00:19:06.167 are positive. 00:19:06.167 --> 00:19:07.250 And the slope is negative. 00:19:07.250 --> 00:19:10.430 On the other hand, on the bottom side, x is still positive, 00:19:10.430 --> 00:19:11.780 but y is negative. 00:19:11.780 --> 00:19:15.168 And it's tilting up because the denominator is negative. 00:19:15.168 --> 00:19:17.084 The numerator is positive, and this minus sign 00:19:17.084 --> 00:19:19.050 has a positive slope. 00:19:19.050 --> 00:19:23.330 So it matches perfectly in every category. 00:19:23.330 --> 00:19:26.920 This complicated, however, and it's easier 00:19:26.920 --> 00:19:30.000 just to keep track of one branch at a time, 00:19:30.000 --> 00:19:32.850 even in advanced math. 00:19:32.850 --> 00:19:37.590 Okay, so we only do it one branch at a time. 00:19:37.590 --> 00:19:43.970 Other questions? 00:19:43.970 --> 00:19:47.360 Okay, so now I want to give you a slightly more 00:19:47.360 --> 00:19:49.210 complicated example here. 00:19:49.210 --> 00:19:52.690 And indeed some of the-- so here's 00:19:52.690 --> 00:19:54.490 a little more complicated example. 00:19:54.490 --> 00:19:56.900 It's not going to be the most complicated example, 00:19:56.900 --> 00:20:17.980 but you know it'll be a little tricky. 00:20:17.980 --> 00:20:22.520 So this example, I'm going to give you a fourth order 00:20:22.520 --> 00:20:23.020 equation. 00:20:23.020 --> 00:20:31.980 So y^4 + xy^2 - 2 = 0. 00:20:31.980 --> 00:20:35.320 Now it just so happens that there's 00:20:35.320 --> 00:20:38.700 a trick to solving this equation, 00:20:38.700 --> 00:20:41.420 so actually you can do both the explicit method 00:20:41.420 --> 00:20:46.210 and the non-explicit method. 00:20:46.210 --> 00:20:50.379 So the explicit method would say okay well, 00:20:50.379 --> 00:20:51.420 I want to solve for this. 00:20:51.420 --> 00:20:55.760 So I'm going to use the quadratic formula, but on y^2. 00:20:55.760 --> 00:20:59.230 This is quadratic in y^2, because there's a fourth power 00:20:59.230 --> 00:21:02.891 and a second power, and the first and third powers are 00:21:02.891 --> 00:21:03.390 missing. 00:21:03.390 --> 00:21:09.940 So this is y^2 is equal to -x plus or minus the square root 00:21:09.940 --> 00:21:19.570 of x^2 - 4(-2) divided by 2. 00:21:19.570 --> 00:21:22.790 And so this x is the b. 00:21:22.790 --> 00:21:29.750 This -2 is the c, and a = 1 in the quadratic formula. 00:21:29.750 --> 00:21:37.200 And so the formula for y is plus or minus the square root of -x 00:21:37.200 --> 00:21:45.070 plus or minus the square root x^2 + 8 divided by 2. 00:21:45.070 --> 00:21:47.880 So now you can see this problem of branches, 00:21:47.880 --> 00:21:50.540 this happens actually in a lot of cases, 00:21:50.540 --> 00:21:53.066 coming up in an elaborate way. 00:21:53.066 --> 00:21:54.690 You have two choices for the sign here. 00:21:54.690 --> 00:21:56.610 You have two choices for the sign here. 00:21:56.610 --> 00:21:59.410 Conceivably as many as four roots for this equation, 00:21:59.410 --> 00:22:02.031 because it's a fourth degree equation. 00:22:02.031 --> 00:22:02.780 It's quite a mess. 00:22:02.780 --> 00:22:06.000 You should have to check each branch separately. 00:22:06.000 --> 00:22:09.180 And this really is that level of complexity, 00:22:09.180 --> 00:22:11.750 and in general it's very difficult 00:22:11.750 --> 00:22:17.840 to figure out the formulas for quartic equations. 00:22:17.840 --> 00:22:21.630 But fortunately we're never going to use them. 00:22:21.630 --> 00:22:24.830 That is, we're never going to need those formulas. 00:22:24.830 --> 00:22:31.850 So the implicit method is far easier. 00:22:31.850 --> 00:22:35.230 The implicit method just says okay I'll 00:22:35.230 --> 00:22:38.980 leave the equation in its simplest form. 00:22:38.980 --> 00:22:40.520 And now differentiate. 00:22:40.520 --> 00:22:47.300 So when I differentiate, I get 4y^3 y' plus - 00:22:47.300 --> 00:22:50.920 now here I have to apply the product rule. 00:22:50.920 --> 00:22:56.090 So I differentiate the x and the y^2 separately. 00:22:56.090 --> 00:22:59.720 First I differentiate with respect to x, so I get y^2. 00:22:59.720 --> 00:23:03.220 Then I differentiate with respect to the other factor, 00:23:03.220 --> 00:23:04.410 the y^2 factor. 00:23:04.410 --> 00:23:08.950 And I get x(2 y y'). 00:23:08.950 --> 00:23:10.440 And then the 0 gives me 0. 00:23:10.440 --> 00:23:16.100 So minus 0 equals 0. 00:23:16.100 --> 00:23:21.970 So there's the implicit differentiation step. 00:23:21.970 --> 00:23:26.260 And now I just want to solve for y'. 00:23:26.260 --> 00:23:32.570 So I'm going to factor out 4y^3 + 2xy. 00:23:32.570 --> 00:23:35.740 That's the factor on y'. 00:23:35.740 --> 00:23:39.780 And I'm going to put the y^2 on the other side. 00:23:39.780 --> 00:23:43.400 -y^2 over here. 00:23:43.400 --> 00:23:55.110 And so the formula for y' is -y^2 divided by 4y^3 + 2xy. 00:23:55.110 --> 00:24:01.420 So that's the formula for the solution. 00:24:01.420 --> 00:24:06.947 For the slope. 00:24:06.947 --> 00:24:07.780 You have a question? 00:24:07.780 --> 00:24:16.340 Student: [INAUDIBLE] 00:24:16.340 --> 00:24:18.350 Professor: So the question is for the y would 00:24:18.350 --> 00:24:22.140 we have to put in what solved for in the explicit equation. 00:24:22.140 --> 00:24:24.120 And the answer is absolutely yes. 00:24:24.120 --> 00:24:25.280 That's exactly the point. 00:24:25.280 --> 00:24:30.950 So this is not a complete solution to a problem. 00:24:30.950 --> 00:24:32.710 We started with an implicit equation. 00:24:32.710 --> 00:24:33.990 We differentiated. 00:24:33.990 --> 00:24:36.660 And we got in the end, also an implicit equation. 00:24:36.660 --> 00:24:39.540 It doesn't tell us what y is as a function of x. 00:24:39.540 --> 00:24:43.040 You have to go back to this formula 00:24:43.040 --> 00:24:45.520 to get the formula for x. 00:24:45.520 --> 00:24:49.310 So for example, let me give you an example here. 00:24:49.310 --> 00:24:54.660 So this hides a degree of complexity of the problem. 00:24:54.660 --> 00:24:58.550 But it's a degree of complexity that we must live with. 00:24:58.550 --> 00:25:10.460 So for example, at x = 1, you can see that y = 1 solves. 00:25:10.460 --> 00:25:16.750 That happens to be-- solves y^4 + xy^2 - 2 = 0. 00:25:16.750 --> 00:25:18.400 That's why I picked the 2 actually, 00:25:18.400 --> 00:25:21.000 so it would be 1 + 1 - 2 = 0. 00:25:21.000 --> 00:25:23.060 I just wanted to have a convenient solution there 00:25:23.060 --> 00:25:25.630 to pull out of my hat at this point. 00:25:25.630 --> 00:25:26.670 So I did that. 00:25:26.670 --> 00:25:30.250 And so we now know that when x = 1, y = 1. 00:25:30.250 --> 00:25:41.740 So at (1, 1) along the curve, the slope is equal to what? 00:25:41.740 --> 00:25:52.200 Well, I have to plug in here, -1^2 / (4*1^3 + 2*1*1). 00:25:52.200 --> 00:25:54.290 That's just plugging in that formula over there, 00:25:54.290 --> 00:25:59.170 which turns out to be -1/6. 00:25:59.170 --> 00:26:00.670 So I can get it. 00:26:00.670 --> 00:26:13.940 On the other hand, at say x = 2, we're 00:26:13.940 --> 00:26:32.890 stuck using this formula star here to find y. 00:26:32.890 --> 00:26:37.170 Now, so let me just make two points 00:26:37.170 --> 00:26:40.020 about this, which are just philosophical points for you 00:26:40.020 --> 00:26:42.420 right now. 00:26:42.420 --> 00:26:45.427 The first is, when I promised you 00:26:45.427 --> 00:26:47.010 at the beginning of this class that we 00:26:47.010 --> 00:26:48.840 were going to be able to differentiate 00:26:48.840 --> 00:26:53.230 any function you know, I meant it very literally. 00:26:53.230 --> 00:26:56.021 What I meant is if you know the function, 00:26:56.021 --> 00:26:58.020 we'll be able give a formula for the derivative. 00:26:58.020 --> 00:27:00.210 If you don't know how to find a function, 00:27:00.210 --> 00:27:02.490 you'll have a lot of trouble finding the derivative. 00:27:02.490 --> 00:27:05.370 So we didn't make any promises that if you 00:27:05.370 --> 00:27:06.830 can't find the function you will be 00:27:06.830 --> 00:27:09.300 able to find the derivative by some magic. 00:27:09.300 --> 00:27:10.450 That will never happen. 00:27:10.450 --> 00:27:12.900 And however complex the function is, 00:27:12.900 --> 00:27:16.130 a root of a fourth degree polynomial 00:27:16.130 --> 00:27:20.300 can be pretty complicated function of the coefficients, 00:27:20.300 --> 00:27:23.990 we're stuck with this degree of complexity in the problem. 00:27:23.990 --> 00:27:27.220 But the big advantage of his method, notice, 00:27:27.220 --> 00:27:29.340 is that although we've had to find star, 00:27:29.340 --> 00:27:31.180 we had to find this formula star, 00:27:31.180 --> 00:27:34.100 and there are many other ways of doing these things numerically, 00:27:34.100 --> 00:27:36.250 by the way, which we'll learn later, 00:27:36.250 --> 00:27:39.940 so there's a good method for doing it numerically. 00:27:39.940 --> 00:27:43.190 Although we had to find star, we never had to differentiate it. 00:27:43.190 --> 00:27:46.080 We had a fast way of getting the slope. 00:27:46.080 --> 00:27:48.280 So we had to know what x and y were. 00:27:48.280 --> 00:27:50.980 But y' we got by an algebraic formula, 00:27:50.980 --> 00:27:54.450 in terms of the values here. 00:27:54.450 --> 00:27:57.140 So this is very fast, forgetting the slope, 00:27:57.140 --> 00:28:02.382 once you know the point. yes? 00:28:02.382 --> 00:28:03.840 Student: What's in the parentheses? 00:28:03.840 --> 00:28:06.790 Professor: Sorry, this is-- Well let's see if I can manage this. 00:28:06.790 --> 00:28:16.522 Is this the parentheses you're talking about? 00:28:16.522 --> 00:28:17.022 Ah, "say". 00:28:17.022 --> 00:28:17.310 That says "say". 00:28:17.310 --> 00:28:19.185 Well, so maybe I should put commas around it. 00:28:19.185 --> 00:28:24.560 But it was S A Y, comma comma, okay? 00:28:24.560 --> 00:28:28.900 Well here was at x = 1. 00:28:28.900 --> 00:28:33.100 I'm just throwing out a value. 00:28:33.100 --> 00:28:34.200 Any other value. 00:28:34.200 --> 00:28:36.360 Actually there is one value, my favorite value. 00:28:36.360 --> 00:28:39.700 Well this is easy to evaluate right? x = 0, 00:28:39.700 --> 00:28:42.610 I can do it there. 00:28:42.610 --> 00:28:45.400 That's maybe the only one. 00:28:45.400 --> 00:28:55.820 The others are a nuisance. 00:28:55.820 --> 00:29:03.750 All right, other questions? 00:29:03.750 --> 00:29:06.060 Now we have to do something more here. 00:29:06.060 --> 00:29:10.470 So I claimed to you that we could differentiate 00:29:10.470 --> 00:29:11.580 all the functions we know. 00:29:11.580 --> 00:29:13.220 But really we can learn a tremendous 00:29:13.220 --> 00:29:17.910 about functions which are really hard to get at. 00:29:17.910 --> 00:29:20.330 So this implicit differentiation method 00:29:20.330 --> 00:29:30.750 has one very, very important application 00:29:30.750 --> 00:29:38.010 to finding inverse functions, or finding derivatives 00:29:38.010 --> 00:29:40.650 of inverse functions. 00:29:40.650 --> 00:29:51.790 So let's talk about that next. 00:29:51.790 --> 00:29:55.700 So first, maybe we'll just illustrate by an example. 00:29:55.700 --> 00:29:59.310 If you have the function y is equal to square root x, 00:29:59.310 --> 00:30:04.600 for x positive, then of course this idea 00:30:04.600 --> 00:30:07.170 is that we should simplify this equation 00:30:07.170 --> 00:30:10.350 and we should square it so we get this somewhat simpler 00:30:10.350 --> 00:30:11.860 equation here. 00:30:11.860 --> 00:30:14.080 And then we have a notation for this. 00:30:14.080 --> 00:30:21.690 If we call f(x) equal to square root of x, and g(y) = x, 00:30:21.690 --> 00:30:25.340 this is the reversal of this. 00:30:25.340 --> 00:30:33.150 Then the formula for g(y) is that it should be y^2. 00:30:33.150 --> 00:30:48.310 And in general, if we start with any old y = f(x), 00:30:48.310 --> 00:30:52.700 and we just write down, this is the defining relationship 00:30:52.700 --> 00:30:57.050 for a function g, the property that we're saying is that 00:30:57.050 --> 00:31:01.040 g(f(x)) has got to bring us back to x. 00:31:01.040 --> 00:31:04.620 And we write that in a couple of different ways. 00:31:04.620 --> 00:31:08.260 We call g the inverse of f. 00:31:08.260 --> 00:31:13.400 And also we call f the inverse of g, 00:31:13.400 --> 00:31:15.960 although I'm going to be silent about which variable 00:31:15.960 --> 00:31:19.300 I want to use, because people mix them up a little bit, 00:31:19.300 --> 00:31:31.540 as we'll be doing when we draw some pictures of this. 00:31:31.540 --> 00:31:32.620 So let's see. 00:31:32.620 --> 00:31:42.310 Let's draw pictures of both f and f inverse 00:31:42.310 --> 00:31:50.260 on the same graph. 00:31:50.260 --> 00:32:02.130 So first of all, I'm going to draw the graph of f(x) 00:32:02.130 --> 00:32:06.470 = square root of x. 00:32:06.470 --> 00:32:11.260 That's some shape like this. 00:32:11.260 --> 00:32:16.390 And now, in order to understand what g(y) is, 00:32:16.390 --> 00:32:20.030 so let's do the analysis in general, 00:32:20.030 --> 00:32:23.420 but then we'll draw it in this particular case. 00:32:23.420 --> 00:32:31.780 If you have g(y) = x, that's really 00:32:31.780 --> 00:32:34.460 just the same equation right? 00:32:34.460 --> 00:32:37.440 This is the equation g(y) = x, that's y^2 = x. 00:32:37.440 --> 00:32:40.650 This is y = square root of x, those are the same equations, 00:32:40.650 --> 00:32:43.330 it's the same curve. 00:32:43.330 --> 00:32:49.807 But suppose now that we wanted to write down what g(x) is. 00:32:49.807 --> 00:32:51.890 In other words, we wanted to switch the variables, 00:32:51.890 --> 00:32:55.650 so draw them as I said on the same graph with the same x, 00:32:55.650 --> 00:32:59.800 and the same y axes. 00:32:59.800 --> 00:33:04.340 Then that would be, in effect, trading the roles of x and y. 00:33:04.340 --> 00:33:07.670 We have to rename every point on the graph which 00:33:07.670 --> 00:33:12.290 is the ordered pair (x, y), and trade it for the opposite one. 00:33:12.290 --> 00:33:15.250 And when you exchange x and y, so 00:33:15.250 --> 00:33:23.800 to do this, exchange x and y, and when 00:33:23.800 --> 00:33:27.030 you do that, graphically what that looks 00:33:27.030 --> 00:33:30.440 like is the following: suppose you have a place here, 00:33:30.440 --> 00:33:33.580 and this is the x and this is the y, 00:33:33.580 --> 00:33:35.030 then you want to trade them. 00:33:35.030 --> 00:33:39.620 So you want the y here right? 00:33:39.620 --> 00:33:41.360 And the x up there. 00:33:41.360 --> 00:33:44.400 It's sort of the opposite place over there. 00:33:44.400 --> 00:33:51.130 And that is the place which is directly opposite this point 00:33:51.130 --> 00:33:55.790 across the diagonal line x = y. 00:33:55.790 --> 00:33:58.410 So you reflect across this or you flip across that. 00:33:58.410 --> 00:34:01.120 You get this other shape that looks like that. 00:34:01.120 --> 00:34:10.780 Maybe I'll draw it with a colored piece of chalk here. 00:34:10.780 --> 00:34:24.090 So this guy here is y = f^(-1)(x). 00:34:24.090 --> 00:34:26.224 And indeed, if you look at these graphs, 00:34:26.224 --> 00:34:27.390 this one is the square root. 00:34:27.390 --> 00:34:34.787 This one happens to be y = x^2. 00:34:34.787 --> 00:34:36.370 If you take this one, and you turn it, 00:34:36.370 --> 00:34:39.890 you reverse the roles of the x axis and the y axis, 00:34:39.890 --> 00:34:43.620 and tilt it on its side. 00:34:43.620 --> 00:34:51.150 So that's the picture of what an inverse function is, and now I 00:34:51.150 --> 00:34:56.070 want to show you that the method of implicit differentiation 00:34:56.070 --> 00:34:59.890 allows us to compute the derivatives 00:34:59.890 --> 00:35:03.250 of inverse functions. 00:35:03.250 --> 00:35:05.100 So let me just say it in general, 00:35:05.100 --> 00:35:07.310 and then I'll carry it out in particular. 00:35:07.310 --> 00:35:16.390 So implicit differentiation allows 00:35:16.390 --> 00:35:32.880 us to find the derivative of any inverse function, 00:35:32.880 --> 00:35:53.380 provided we know the derivative of the function. 00:35:53.380 --> 00:35:58.360 So let's do that for what is an example, which 00:35:58.360 --> 00:36:02.510 is truly complicated and a little subtle here. 00:36:02.510 --> 00:36:04.770 It has a very pretty answer. 00:36:04.770 --> 00:36:09.700 So we'll carry out an example here, 00:36:09.700 --> 00:36:19.660 which is the function y is equal to the inverse tangent. 00:36:19.660 --> 00:36:25.900 So again, for the inverse tangent 00:36:25.900 --> 00:36:30.020 all of the things that we're going to do 00:36:30.020 --> 00:36:32.360 are going to be based on simplifying 00:36:32.360 --> 00:36:36.250 this equation by taking the tangent of both sides. 00:36:36.250 --> 00:36:38.260 So, us let me remind you by the way, 00:36:38.260 --> 00:36:41.780 the inverse tangent is what's also known as arctangent. 00:36:41.780 --> 00:36:45.210 That's just another notation for the same thing. 00:36:45.210 --> 00:36:49.770 And what we're going to use to describe 00:36:49.770 --> 00:36:55.565 this function is the equation tan y = x. 00:36:55.565 --> 00:36:56.940 That's what happens when you take 00:36:56.940 --> 00:36:59.110 the tangent of this function. 00:36:59.110 --> 00:37:01.690 This is how we're going to figure out 00:37:01.690 --> 00:37:19.650 what the function looks like. 00:37:19.650 --> 00:37:23.370 So first of all, I want to draw it, 00:37:23.370 --> 00:37:26.610 and then we'll do the computation. 00:37:26.610 --> 00:37:32.120 So let's make the diagram first. 00:37:32.120 --> 00:37:33.630 So I want to do something which is 00:37:33.630 --> 00:37:35.879 analogous to what I did over here with the square root 00:37:35.879 --> 00:37:38.060 function. 00:37:38.060 --> 00:37:43.740 So first of all, I remind you that the tangent function 00:37:43.740 --> 00:37:52.850 is defined between two values here, which are pi/2 and -pi/2. 00:37:52.850 --> 00:37:55.010 And it starts out at minus infinity 00:37:55.010 --> 00:37:58.560 and curves up like this. 00:37:58.560 --> 00:38:08.050 So that's the function tan x. 00:38:08.050 --> 00:38:11.350 And so the one that we have to sketch 00:38:11.350 --> 00:38:14.870 is this one which we get by reflecting this 00:38:14.870 --> 00:38:21.780 across the axis. 00:38:21.780 --> 00:38:25.150 Well not the axis, the diagonal. 00:38:25.150 --> 00:38:33.080 This slope by the way, should be less - a little lower here so 00:38:33.080 --> 00:38:37.580 that we can have it going down and up. 00:38:37.580 --> 00:38:42.000 So let me show you what it looks like. 00:38:42.000 --> 00:38:44.830 On the front, it's going to look a lot like this one. 00:38:44.830 --> 00:38:50.620 So this one had curved down, and so the reflection 00:38:50.620 --> 00:38:52.480 across the diagonal curved up. 00:38:52.480 --> 00:38:54.320 Here this is curving up, so the reflection 00:38:54.320 --> 00:38:56.070 is going to curve down. 00:38:56.070 --> 00:38:58.760 It's going to look like this. 00:38:58.760 --> 00:39:02.360 Maybe I should, sorry, let's use a different color, 00:39:02.360 --> 00:39:04.450 because it's reversed from before. 00:39:04.450 --> 00:39:10.420 I'll just call it green. 00:39:10.420 --> 00:39:15.630 Now, the original curve in the first quadrant 00:39:15.630 --> 00:39:17.990 eventually had an asymptote which was straight up. 00:39:17.990 --> 00:39:24.090 So this one is going to have an asymptote which is horizontal. 00:39:24.090 --> 00:39:27.260 And that level is what? 00:39:27.260 --> 00:39:29.950 What's the highest? 00:39:29.950 --> 00:39:30.780 It is just pi/2. 00:39:33.470 --> 00:39:35.910 Now similarly, the other way, we're 00:39:35.910 --> 00:39:40.040 going to do this: and this bottom level 00:39:40.040 --> 00:39:42.940 is going to be -pi/2. 00:39:42.940 --> 00:39:47.480 So there's the picture of this function. 00:39:47.480 --> 00:39:50.300 It's defined for all x. 00:39:50.300 --> 00:39:57.530 So this green guy is y = arctan x. 00:39:57.530 --> 00:39:59.530 And it's defined all the way from minus infinity 00:39:59.530 --> 00:40:05.290 to infinity. 00:40:05.290 --> 00:40:11.410 And to use a notation that we had from limit notation 00:40:11.410 --> 00:40:21.290 as x goes to infinity, let's say, x is equal to pi/2. 00:40:21.290 --> 00:40:24.823 That's an example of one value that's of interest in addition 00:40:24.823 --> 00:40:28.250 to the finite values. 00:40:28.250 --> 00:40:31.430 Okay, so now the first ingredient 00:40:31.430 --> 00:40:34.580 that we're going to need, is we're 00:40:34.580 --> 00:40:37.190 going to need the derivative of the tangent function. 00:40:37.190 --> 00:40:40.060 So I'm going to recall for you, and maybe you 00:40:40.060 --> 00:40:43.180 haven't worked this out yet, but I hope that many of you have, 00:40:43.180 --> 00:40:48.790 that if you take the derivative with respect to y of tan y. 00:40:48.790 --> 00:40:55.500 So this you do by the quotient rule. 00:40:55.500 --> 00:40:59.150 So this is of the form u/v, right? 00:40:59.150 --> 00:41:00.630 You use the quotient rule. 00:41:00.630 --> 00:41:06.090 So I'm going to get this. 00:41:06.090 --> 00:41:09.560 But what you get in the end is some marvelous simplification 00:41:09.560 --> 00:41:12.580 that comes out to cos^2 y. 00:41:12.580 --> 00:41:14.740 1 over cosine squared. 00:41:14.740 --> 00:41:17.460 You can recognize the cosine squared from the fact that you 00:41:17.460 --> 00:41:19.240 should get v^2 in the denominator, 00:41:19.240 --> 00:41:26.630 and somehow the numerators all cancel and simplifies to 1. 00:41:26.630 --> 00:41:32.560 This is also known as secant squared y. 00:41:32.560 --> 00:41:38.010 So that something that if you haven't done yet, 00:41:38.010 --> 00:41:48.450 you're going to have to do this as an exercise. 00:41:48.450 --> 00:41:50.220 So we need that ingredient, and now we're 00:41:50.220 --> 00:41:59.180 just going to differentiate our equation. 00:41:59.180 --> 00:42:00.540 And what do we get? 00:42:00.540 --> 00:42:15.460 We get, again, (d/dy tan y) times dy/dx is equal to 1. 00:42:15.460 --> 00:42:22.930 Or, if you like, 1 / cos^2 y times, in the other notation, 00:42:22.930 --> 00:42:30.090 y', is equal to 1. 00:42:30.090 --> 00:42:35.740 So I've just used the formulas that I just wrote down there. 00:42:35.740 --> 00:42:37.860 Now all I have to do is solve for y'. 00:42:37.860 --> 00:42:44.060 It's cos^2 y. 00:42:44.060 --> 00:42:47.040 Unfortunately, this is not the form 00:42:47.040 --> 00:42:49.610 that we ever want to leave these things in. 00:42:49.610 --> 00:42:52.180 This is the same problem we had with that ugly square root 00:42:52.180 --> 00:42:54.500 expression, or with any of the others. 00:42:54.500 --> 00:42:58.070 We want to rewrite in terms of x. 00:42:58.070 --> 00:43:05.280 Our original question was what is d/dx of arctan x. 00:43:05.280 --> 00:43:08.260 Now so far we have the following answer to that question: 00:43:08.260 --> 00:43:15.210 it's cos^2 (arctan x). 00:43:15.210 --> 00:43:31.780 Now this is a correct answer, but way too complicated. 00:43:31.780 --> 00:43:33.180 Now that doesn't mean that if you 00:43:33.180 --> 00:43:35.100 took a random collection of functions, 00:43:35.100 --> 00:43:37.560 you wouldn't end up with something this complicated. 00:43:37.560 --> 00:43:41.000 But these particular functions, these beautiful circular 00:43:41.000 --> 00:43:42.900 functions involved with trigonometry all 00:43:42.900 --> 00:43:45.610 have very nice formulas associated with them. 00:43:45.610 --> 00:43:48.740 And this simplifies tremendously. 00:43:48.740 --> 00:43:50.360 So one of the skills that you need 00:43:50.360 --> 00:43:54.660 to develop when you're dealing with trig functions 00:43:54.660 --> 00:43:56.690 is to simplify this. 00:43:56.690 --> 00:44:03.620 And so let's see now that expressions like this all 00:44:03.620 --> 00:44:07.950 simplify. 00:44:07.950 --> 00:44:10.400 So here we go. 00:44:10.400 --> 00:44:12.264 There's only one formula, one ingredient 00:44:12.264 --> 00:44:14.180 that we need to use to do this, and then we're 00:44:14.180 --> 00:44:15.460 going to draw a diagram. 00:44:15.460 --> 00:44:18.530 So the ingredient again, is the original defining relationship 00:44:18.530 --> 00:44:22.150 that tan y = x. 00:44:22.150 --> 00:44:27.780 So tan y = x can be encoded in a right triangle 00:44:27.780 --> 00:44:34.520 in the following way: here's the right triangle and tan 00:44:34.520 --> 00:44:38.590 y means that y should be represented as an angle. 00:44:38.590 --> 00:44:40.820 And then, its tangent is the ratio 00:44:40.820 --> 00:44:43.500 of this vertical to this horizontal side. 00:44:43.500 --> 00:44:46.250 So I'm just going to pick two values that work, 00:44:46.250 --> 00:44:48.990 namely x and 1. 00:44:48.990 --> 00:44:51.930 Those are the simplest ones. 00:44:51.930 --> 00:44:57.670 So I've encoded this equation in this picture. 00:44:57.670 --> 00:45:01.560 And now all I have to do is figure out what the cosine of y 00:45:01.560 --> 00:45:03.540 is in this right triangle here. 00:45:03.540 --> 00:45:06.081 In order to do that, I need to figure out what the hypotenuse 00:45:06.081 --> 00:45:13.280 is, but that's just square root of 1 + x^2. 00:45:13.280 --> 00:45:18.580 And now I can read off what the cosine of y is. 00:45:18.580 --> 00:45:23.560 So the cosine of y is 1 divided by the hypotenuse. 00:45:23.560 --> 00:45:32.480 So it's 1 over square root, whoops, yeah, 1 + x^2. 00:45:32.480 --> 00:45:39.900 And so cosine squared is just 1 / 1 + x^2. 00:45:39.900 --> 00:45:43.050 And so our answer over here, the preferred answer which is way 00:45:43.050 --> 00:45:45.670 simpler than what I wrote up there, 00:45:45.670 --> 00:46:04.690 is that d/dx of tan inverse x is equal to 1 over 1 + x^2. 00:46:04.690 --> 00:46:06.770 Maybe I'll stop here for one more question. 00:46:06.770 --> 00:46:10.240 I have one more calculation which I can do even 00:46:10.240 --> 00:46:11.500 in less than a minute. 00:46:11.500 --> 00:46:16.360 So we have a whole minute for questions. 00:46:16.360 --> 00:46:20.480 Yeah? 00:46:20.480 --> 00:46:26.620 Student: [INAUDIBLE] 00:46:26.620 --> 00:46:34.210 Professor: What happens to the inverse tangent? 00:46:34.210 --> 00:46:41.530 The inverse tangent-- Okay, this inverse tangent 00:46:41.530 --> 00:46:44.030 is the same as this y here. 00:46:44.030 --> 00:46:46.140 Those are the same thing. 00:46:46.140 --> 00:46:50.480 So what I did was I skipped this step here entirely. 00:46:50.480 --> 00:46:52.070 I never wrote that down. 00:46:52.070 --> 00:46:54.560 But the inverse tangent was that y. 00:46:54.560 --> 00:46:56.690 The issue was what's a good formula 00:46:56.690 --> 00:47:01.080 for cos y in terms of x? 00:47:01.080 --> 00:47:04.510 So I am evaluating that, but I'm doing it using the letter y. 00:47:04.510 --> 00:47:06.600 So in other words, what happened to the inverse 00:47:06.600 --> 00:47:10.350 tangent is that I called it y, which 00:47:10.350 --> 00:47:15.340 is what it's been all along. 00:47:15.340 --> 00:47:17.450 Okay, so now I'm going to do the case 00:47:17.450 --> 00:47:20.400 of the sine, the inverse sine. 00:47:20.400 --> 00:47:22.890 And I'll show you how easy this is 00:47:22.890 --> 00:47:27.420 if I don't fuss with-- because this one has an easy trig 00:47:27.420 --> 00:47:29.760 identity associated with it. 00:47:29.760 --> 00:47:37.870 So if y = sin^(-1) x, and sin y = x, 00:47:37.870 --> 00:47:40.970 and now watch how simple it is when I do the differentiation. 00:47:40.970 --> 00:47:42.440 I just differentiate. 00:47:42.440 --> 00:47:50.780 I get (cos y) y' = 1. 00:47:50.780 --> 00:48:00.580 And then, y', so that implies that = 1 / cos y, 00:48:00.580 --> 00:48:03.080 and now to rewrite that in terms of x, 00:48:03.080 --> 00:48:10.550 I have to just recognize that this is the same as this, 00:48:10.550 --> 00:48:14.910 which is the same as 1 / square root of 1 - x^2. 00:48:14.910 --> 00:48:19.600 So all told, the derivative with respect to x of the arcsine 00:48:19.600 --> 00:48:30.420 function is 1 / square root of 1 - x^2. 00:48:30.420 --> 00:48:32.540 So these implicit differentiations 00:48:32.540 --> 00:48:34.700 are very convenient. 00:48:34.700 --> 00:48:38.190 However, I warn you that you do have 00:48:38.190 --> 00:48:42.670 to be careful about the range of applicability of these things. 00:48:42.670 --> 00:48:44.550 You have to draw a picture like this one 00:48:44.550 --> 00:48:47.210 to make sure you know where this makes sense. 00:48:47.210 --> 00:48:50.380 In other words, you have to pick a branch for the sine function 00:48:50.380 --> 00:48:52.050 to work that out, and there's something 00:48:52.050 --> 00:48:53.330 like that on your problem set. 00:48:53.330 --> 00:48:56.030 And it's also discussed in your text. 00:48:56.030 --> 00:48:58.070 So we'll stop here.