WEBVTT 00:00:00.040 --> 00:00:02.320 The following content is provided under a Creative 00:00:02.320 --> 00:00:03.600 Commons License. 00:00:03.600 --> 00:00:05.950 Your support will help MIT OpenCourseWare 00:00:05.950 --> 00:00:09.940 continue to offer high quality educational resources for free. 00:00:09.940 --> 00:00:12.520 To make a donation, or to view additional materials 00:00:12.520 --> 00:00:16.120 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.120 --> 00:00:21.900 at ocw.mit.edu. 00:00:21.900 --> 00:00:27.060 PROFESSOR: All right, so let's begin lecture six. 00:00:27.060 --> 00:00:44.630 We're talking today about exponentials and logarithms. 00:00:44.630 --> 00:00:49.390 And these are the last functions that I need to introduce, 00:00:49.390 --> 00:00:50.950 the last standard functions that we 00:00:50.950 --> 00:00:55.110 need to connect with calculus, that you've learned about. 00:00:55.110 --> 00:00:58.230 And they're certainly as fundamental, if not more so, 00:00:58.230 --> 00:01:00.830 than trigonometric functions. 00:01:00.830 --> 00:01:04.960 So first of all, we'll start out with a number, 00:01:04.960 --> 00:01:09.030 a, which is positive, which is usually called a base. 00:01:09.030 --> 00:01:12.790 And then we have these properties that 00:01:12.790 --> 00:01:14.870 a to the power 0 is always 1. 00:01:14.870 --> 00:01:17.180 That's how we get started. 00:01:17.180 --> 00:01:21.270 And a^1 is a. 00:01:21.270 --> 00:01:24.030 And of course a^2, not surprisingly, 00:01:24.030 --> 00:01:26.200 is a times a, etc. 00:01:26.200 --> 00:01:35.400 And the general rule is that a^(x_1 + x_2) is a^(x_1) times 00:01:35.400 --> 00:01:36.970 a^(x_2). 00:01:36.970 --> 00:01:41.440 So this is the basic rule of exponents, and with these two 00:01:41.440 --> 00:01:49.070 initial properties, that defines the exponential function. 00:01:49.070 --> 00:01:52.220 And then there's an additional property, 00:01:52.220 --> 00:01:54.360 which is deduced from these, which 00:01:54.360 --> 00:01:59.020 is the composition of exponential functions, which 00:01:59.020 --> 00:02:03.330 is that you take a to the x_1 power, to the x_2 power. 00:02:03.330 --> 00:02:08.390 Then that turns out to be a to the x_1 times x_2. 00:02:08.390 --> 00:02:11.540 So that's an additional property that we'll take for granted, 00:02:11.540 --> 00:02:14.140 which you learned in high school. 00:02:14.140 --> 00:02:22.650 Now, in order to understand what all the values of a^x are, 00:02:22.650 --> 00:02:28.620 we need to first remember that if you're taking a rational 00:02:28.620 --> 00:02:34.350 power that it's the ratio of two integers power of a. 00:02:34.350 --> 00:02:37.190 That's going to be a^m, and then we're going to have to take 00:02:37.190 --> 00:02:39.440 the nth root of that. 00:02:39.440 --> 00:02:40.840 So that's the definition. 00:02:40.840 --> 00:02:45.050 And then, when you're defining a^x, 00:02:45.050 --> 00:03:00.370 so a^x is defined for all x by filling in. 00:03:00.370 --> 00:03:03.480 So I'm gonna use that expression in quotation marks, 00:03:03.480 --> 00:03:09.930 "filling in" by continuity. 00:03:09.930 --> 00:03:11.540 This is really what your calculator 00:03:11.540 --> 00:03:13.960 does when it gives you a to the power x, 00:03:13.960 --> 00:03:17.540 because you can't even punch in the square root of x. 00:03:17.540 --> 00:03:19.590 It doesn't really exist on your calculator. 00:03:19.590 --> 00:03:21.230 There's some decimal expansion. 00:03:21.230 --> 00:03:24.400 So it takes the decimal expansion to a certain length 00:03:24.400 --> 00:03:26.080 and spits out a number which is pretty 00:03:26.080 --> 00:03:28.120 close to the correct answer. 00:03:28.120 --> 00:03:31.550 But indeed, in theory, there is an a to the power 00:03:31.550 --> 00:03:33.710 square root of 2, even though the square root of 2 00:03:33.710 --> 00:03:34.740 is irrational. 00:03:34.740 --> 00:03:37.540 And there's a to the pi and so forth. 00:03:37.540 --> 00:03:41.000 All right, so that's the exponential function, 00:03:41.000 --> 00:03:46.830 and let's draw a picture of one. 00:03:46.830 --> 00:03:52.230 So we'll try, say y = 2^x here. 00:03:52.230 --> 00:03:55.180 And I'm not going to draw such a careful graph, 00:03:55.180 --> 00:03:58.630 but let's just plot the most important point, which 00:03:58.630 --> 00:04:01.360 is the point (0,1). 00:04:01.360 --> 00:04:04.510 That's 2^0, which is 1. 00:04:04.510 --> 00:04:08.940 And then maybe we'll go back up here to -1 here. 00:04:08.940 --> 00:04:13.860 And 2 to the -1 is this point here. 00:04:13.860 --> 00:04:18.990 This is (-1, 1/2), the reciprocal. 00:04:18.990 --> 00:04:23.460 And over here, we have 1, and so that goes all the way up to 2. 00:04:23.460 --> 00:04:26.870 And then exponentials are remarkably fast. 00:04:26.870 --> 00:04:30.680 So it's off the board what happens next out at 2. 00:04:30.680 --> 00:04:34.360 It's already above my range here, 00:04:34.360 --> 00:04:37.970 but the graph looks something like this. 00:04:37.970 --> 00:04:38.610 All right. 00:04:38.610 --> 00:04:41.200 Now I've just visually, at least, graphically 00:04:41.200 --> 00:04:43.240 filled in all the rest of the points. 00:04:43.240 --> 00:04:46.730 You have to imagine all these rational numbers, and so forth. 00:04:46.730 --> 00:04:51.670 So this point here would have been (1, 2). 00:04:51.670 --> 00:04:53.330 And so forth. 00:04:53.330 --> 00:04:54.610 All right? 00:04:54.610 --> 00:05:00.992 So that's not too far along. 00:05:00.992 --> 00:05:01.950 So now what's our goal? 00:05:01.950 --> 00:05:04.020 Well, obviously we want to do calculus here. 00:05:04.020 --> 00:05:08.050 So our goal, here, for now - and it's gonna take a while. 00:05:08.050 --> 00:05:10.220 We have to think about it pretty hard. 00:05:10.220 --> 00:05:22.250 We have to calculate what this derivative is. 00:05:22.250 --> 00:05:26.020 All right, so we'll get started. 00:05:26.020 --> 00:05:28.520 And the way we get started is simply 00:05:28.520 --> 00:05:31.190 by plugging in the definition of the derivative. 00:05:31.190 --> 00:05:34.760 The derivative is the limit as delta 00:05:34.760 --> 00:05:41.690 x goes to 0 of a to the x plus delta x, minus a to the x, 00:05:41.690 --> 00:05:45.080 divided by delta x. 00:05:45.080 --> 00:05:50.320 So that's what it is. 00:05:50.320 --> 00:05:56.140 And now, the only step that we can really perform here 00:05:56.140 --> 00:05:58.790 to make this is into something a little bit simpler 00:05:58.790 --> 00:06:03.200 is to use this very first rule that we have here. 00:06:03.200 --> 00:06:06.930 That the exponential of the sum is the product 00:06:06.930 --> 00:06:08.070 of the exponentials. 00:06:08.070 --> 00:06:10.340 So we have here, a^x . 00:06:10.340 --> 00:06:15.750 So what I want to use is just the property that a^(x + delta 00:06:15.750 --> 00:06:22.460 x) = a^x a^(delta x). 00:06:22.460 --> 00:06:26.880 And if I do that, I see that I can factor out a common factor 00:06:26.880 --> 00:06:29.760 in the numerator, which is a^x. 00:06:29.760 --> 00:06:35.120 So we'll write this as the limit as delta x goes to 0, 00:06:35.120 --> 00:06:41.300 of a to the x times this ratio, now a to the delta x, minus 1, 00:06:41.300 --> 00:06:49.280 divided by delta x. 00:06:49.280 --> 00:06:50.160 So far, so good? 00:06:50.160 --> 00:06:53.640 We're actually almost to some serious progress here. 00:06:53.640 --> 00:06:58.460 So there's one other important conceptual step 00:06:58.460 --> 00:07:00.100 which we need to understand. 00:07:00.100 --> 00:07:03.010 And this is a relatively simple one. 00:07:03.010 --> 00:07:05.340 We actually did this before, by the way. 00:07:05.340 --> 00:07:08.162 We did this with sines and cosines. 00:07:08.162 --> 00:07:09.870 The next thing I want to point out to you 00:07:09.870 --> 00:07:15.680 is that you're used to thinking of x as being the variable. 00:07:15.680 --> 00:07:18.140 And indeed, already we were discussing 00:07:18.140 --> 00:07:20.130 x as being the variable and a as being fixed. 00:07:20.130 --> 00:07:22.220 But for the purposes of this limit, 00:07:22.220 --> 00:07:26.290 there's a different variable that's moving. x is fixed 00:07:26.290 --> 00:07:29.340 and delta x is the thing that's moving. 00:07:29.340 --> 00:07:33.440 So that means that this factor here, which is a common factor, 00:07:33.440 --> 00:07:34.799 is constant. 00:07:34.799 --> 00:07:36.590 And we can just factor it out of the limit. 00:07:36.590 --> 00:07:39.450 It doesn't affect the limit at all. 00:07:39.450 --> 00:07:41.130 A constant times a limit is the same 00:07:41.130 --> 00:07:44.580 as whether we multiply before or after we take the limit. 00:07:44.580 --> 00:07:46.830 So I'm just going to factor that out. 00:07:46.830 --> 00:07:49.110 So that's my next step here. 00:07:49.110 --> 00:07:53.450 a^x, and then I have the limit delta x goes to 0 00:07:53.450 --> 00:07:59.830 of a to the delta x minus 1, divided by delta x. 00:07:59.830 --> 00:08:02.180 All right? 00:08:02.180 --> 00:08:03.910 And so what I have here, so this is 00:08:03.910 --> 00:08:05.220 by definition the derivative. 00:08:05.220 --> 00:08:12.730 So here is d/dx of a^x, and it's equal to this expression here. 00:08:12.730 --> 00:08:17.340 Now, I want to stare at this expression, 00:08:17.340 --> 00:08:22.880 and see what it's telling us, because it's telling us 00:08:22.880 --> 00:08:27.680 as much as we can get so far, without some-- 00:08:27.680 --> 00:08:34.810 So first let's just look at what this says. 00:08:34.810 --> 00:08:40.260 So what it's saying is that the derivative of a^x is a^x times 00:08:40.260 --> 00:08:42.820 something that we don't yet know. 00:08:42.820 --> 00:08:46.060 And I'm going to call this something, this mystery number, 00:08:46.060 --> 00:08:47.130 M(a). 00:08:47.130 --> 00:08:52.990 So I'm gonna make the label, M(a) is equal to the limit 00:08:52.990 --> 00:08:56.640 as delta x goes to 0 of a to the delta x 00:08:56.640 --> 00:08:59.800 minus 1 divided by delta x. 00:08:59.800 --> 00:09:00.300 All right? 00:09:00.300 --> 00:09:08.870 So this is a definition. 00:09:08.870 --> 00:09:14.540 So this mystery number M(a) has a geometric interpretation, 00:09:14.540 --> 00:09:16.000 as well. 00:09:16.000 --> 00:09:17.554 So let me describe that. 00:09:17.554 --> 00:09:18.970 It has a geometric interpretation, 00:09:18.970 --> 00:09:20.678 and it's a very, very significant number. 00:09:20.678 --> 00:09:22.200 So let's work out what that is. 00:09:22.200 --> 00:09:25.690 So first of all, let's rewrite the expression in the box, 00:09:25.690 --> 00:09:28.470 using the shorthand for this number. 00:09:28.470 --> 00:09:33.840 So if I just rewrite it, it says d/dx of a^x is equal to this 00:09:33.840 --> 00:09:37.800 factor, which is M(a), times a^x. 00:09:37.800 --> 00:09:43.450 So the derivative of the exponential is this mystery 00:09:43.450 --> 00:09:44.790 number times a^x. 00:09:44.790 --> 00:09:48.870 So we've almost solved the problem of finding 00:09:48.870 --> 00:09:50.560 the derivative of a^x. 00:09:50.560 --> 00:09:53.270 We just have to figure out this one number, M(a), 00:09:53.270 --> 00:09:55.720 and we get the rest. 00:09:55.720 --> 00:10:01.780 So let me point out two more things about this number, M(a). 00:10:01.780 --> 00:10:09.350 So first of all, if I plug in x = 0, 00:10:09.350 --> 00:10:14.250 that's going to be d/dx of a^x , at x = 0. 00:10:14.250 --> 00:10:19.150 According to this formula, that's M(a) times a^0, 00:10:19.150 --> 00:10:21.370 which of course M(a). 00:10:21.370 --> 00:10:23.540 So what is M(a)? 00:10:23.540 --> 00:10:26.410 M(a) is the derivative of this function at 0. 00:10:26.410 --> 00:10:39.790 So M(a) is the slope of a^x at x = 0, of the graph. 00:10:39.790 --> 00:10:41.330 The graph of a^x at 0. 00:10:41.330 --> 00:10:46.170 So again over here, if you looked at the picture. 00:10:46.170 --> 00:10:48.260 I'll draw the one tangent line in here, 00:10:48.260 --> 00:10:50.640 which is this one here. 00:10:50.640 --> 00:11:00.050 And this thing has slope, what we're calling M(2). 00:11:00.050 --> 00:11:02.491 So, if I graph the function y = 2^x, 00:11:02.491 --> 00:11:03.740 I'll get a certain slope here. 00:11:03.740 --> 00:11:05.315 If I graph it with a different base, 00:11:05.315 --> 00:11:07.590 I might get another slope. 00:11:07.590 --> 00:11:12.820 And what we got so far is the following phenomenon: 00:11:12.820 --> 00:11:16.310 if we know this one number, if we know the slope at this one 00:11:16.310 --> 00:11:18.940 place, we will be able to figure out the formula for the slope 00:11:18.940 --> 00:11:23.320 everywhere else. 00:11:23.320 --> 00:11:25.690 Now, that's actually exactly the same thing 00:11:25.690 --> 00:11:28.040 that we did for sines and cosines. 00:11:28.040 --> 00:11:33.120 We knew the slope of the sine and the cosine function 00:11:33.120 --> 00:11:35.900 at x = 0. 00:11:35.900 --> 00:11:37.450 The sine function had slope 1. 00:11:37.450 --> 00:11:39.470 The cosine function had slope 0. 00:11:39.470 --> 00:11:41.560 And then from the sum formulas, well 00:11:41.560 --> 00:11:44.004 that's exactly this kind of thing here, 00:11:44.004 --> 00:11:44.920 from the sum formulas. 00:11:44.920 --> 00:11:47.110 This sum formula, in fact is easier than the ones 00:11:47.110 --> 00:11:49.320 for sines and cosines. 00:11:49.320 --> 00:11:50.960 From the sum formulas, we worked out 00:11:50.960 --> 00:11:53.620 what the slope was everywhere. 00:11:53.620 --> 00:11:57.610 So we're following the same procedure that we did before. 00:11:57.610 --> 00:12:00.960 But at this point we're stuck. 00:12:00.960 --> 00:12:04.660 We're stuck, because that time using radians, 00:12:04.660 --> 00:12:07.140 this very clever idea of radians in geometry, 00:12:07.140 --> 00:12:09.640 we were able to actually figure out what the slope is. 00:12:09.640 --> 00:12:14.920 Whereas here, we're not so sure, what M(2) is, for instance. 00:12:14.920 --> 00:12:17.200 We just don't know yet. 00:12:17.200 --> 00:12:22.790 So, the basic question that we have to deal with right now 00:12:22.790 --> 00:12:32.060 is what is M(a)? 00:12:32.060 --> 00:12:34.680 That's what we're left with. 00:12:34.680 --> 00:12:42.990 And, the curious fact is that the clever thing to do 00:12:42.990 --> 00:12:51.260 is to beg the question. 00:12:51.260 --> 00:12:54.730 So we're going to go through a very circular route here. 00:12:54.730 --> 00:12:56.580 That is circuitous, not circular. 00:12:56.580 --> 00:12:58.360 Circular is a bad word in math. 00:12:58.360 --> 00:13:00.380 That means that one thing depends on another, 00:13:00.380 --> 00:13:03.220 and that depends on it, and maybe both are wrong. 00:13:03.220 --> 00:13:07.460 Circuitous means, we're going to be taking a roundabout route. 00:13:07.460 --> 00:13:10.384 And we're going to discover that even though we refuse 00:13:10.384 --> 00:13:11.800 to answer this question right now, 00:13:11.800 --> 00:13:14.841 we'll succeed in answering it eventually. 00:13:14.841 --> 00:13:15.340 All right? 00:13:15.340 --> 00:13:18.340 So how are we going to beg the question? 00:13:18.340 --> 00:13:20.150 What we're going to say instead is 00:13:20.150 --> 00:13:30.650 we're going to define a mystery base, or number e, 00:13:30.650 --> 00:13:45.790 as the unique number, so that M(e) = 1. 00:13:45.790 --> 00:13:47.930 That's the trick that we're going to use. 00:13:47.930 --> 00:13:50.610 We don't yet know what e is, but we're just going 00:13:50.610 --> 00:13:53.900 to suppose that we have it. 00:13:53.900 --> 00:13:57.340 Now, I'm going to show you a bunch of consequences of this, 00:13:57.340 --> 00:14:00.540 and also I have to persuade you that it actually does exist. 00:14:00.540 --> 00:14:03.640 So first, let me explain what the first consequence is. 00:14:03.640 --> 00:14:06.670 First of all, if M(e) is 1, then if you 00:14:06.670 --> 00:14:09.540 look at this formula over here and you write it down for e, 00:14:09.540 --> 00:14:13.650 you have something which is a very usable formula. 00:14:13.650 --> 00:14:19.930 d/dx of e^x is just e^x. 00:14:19.930 --> 00:14:22.750 All right, so that's an incredibly important formula 00:14:22.750 --> 00:14:24.210 which is the fundamental one. 00:14:24.210 --> 00:14:26.710 It's the only one you have to remember from what we've done. 00:14:26.710 --> 00:14:28.251 So maybe I should have highlighted it 00:14:28.251 --> 00:14:34.760 in several colors here. 00:14:34.760 --> 00:14:37.800 That's a big deal. 00:14:37.800 --> 00:14:40.630 Very happy. 00:14:40.630 --> 00:14:42.770 And again, let me just emphasize, 00:14:42.770 --> 00:14:52.077 also that this is the one which at x = 0 has slope 1. 00:14:52.077 --> 00:14:53.660 That's the way we defined it, alright? 00:14:53.660 --> 00:15:00.640 So if you plug in x = 0 here on the right hand side, you got 1. 00:15:00.640 --> 00:15:03.540 Slope 1 at x = 0. 00:15:03.540 --> 00:15:05.580 So that's great. 00:15:05.580 --> 00:15:07.980 Except of course, since we don't know what e is, 00:15:07.980 --> 00:15:15.770 this is a little bit dicey. 00:15:15.770 --> 00:15:21.750 So, next even before explaining what e is... 00:15:21.750 --> 00:15:23.440 In fact, we won't get to what e really 00:15:23.440 --> 00:15:26.110 is until the very end of this lecture. 00:15:26.110 --> 00:15:34.530 But I have to persuade you why e exists. 00:15:34.530 --> 00:15:37.406 We have to have some explanation for why 00:15:37.406 --> 00:15:40.740 we know there is such a number. 00:15:40.740 --> 00:15:44.100 Okay, so first of all, let me start with the one that we 00:15:44.100 --> 00:15:46.970 supposedly know, which is the function 2^x. 00:15:46.970 --> 00:15:49.710 We'll call it f(x) is 2^x. 00:15:49.710 --> 00:15:50.460 All right? 00:15:50.460 --> 00:15:51.820 So that's the first thing. 00:15:51.820 --> 00:15:54.460 And remember, that the property that it had, 00:15:54.460 --> 00:15:58.170 was that f'(0) was M(2). 00:15:58.170 --> 00:16:04.430 That was the derivative of this function, the slope at x = 0 00:16:04.430 --> 00:16:06.610 of the graph. 00:16:06.610 --> 00:16:09.870 Of the tangent line, that is. 00:16:09.870 --> 00:16:12.980 So now, what we're going to consider 00:16:12.980 --> 00:16:16.880 is any kind of stretching. 00:16:16.880 --> 00:16:22.750 We're going to stretch this function by a factor k. 00:16:22.750 --> 00:16:23.610 Any number k. 00:16:23.610 --> 00:16:29.090 So what we're going to consider is f(kx). 00:16:29.090 --> 00:16:34.790 If you do that, that's the same as 2^(kx). 00:16:34.790 --> 00:16:37.420 Right? 00:16:37.420 --> 00:16:41.030 But now if I use the second law of exponents that I have over 00:16:41.030 --> 00:16:46.700 there, that's the same thing as 2 to the k to the power x, 00:16:46.700 --> 00:16:51.110 which is the same thing as some base b^x, 00:16:51.110 --> 00:16:55.420 where b is equal to-- Let's write that down over here. 00:16:55.420 --> 00:16:55.940 b is 2^k. 00:16:59.081 --> 00:16:59.580 Right. 00:16:59.580 --> 00:17:03.630 So whatever it is, if I have a different base which is 00:17:03.630 --> 00:17:07.740 expressed in terms of 2, of the form 2^k, 00:17:07.740 --> 00:17:14.110 then that new function is described by this function 00:17:14.110 --> 00:17:17.700 f(kx), the stretch. 00:17:17.700 --> 00:17:20.730 So what happens when you stretch a function? 00:17:20.730 --> 00:17:24.720 That's the same thing as shrinking the x axis. 00:17:24.720 --> 00:17:30.035 So when k gets larger, this corresponding point over here 00:17:30.035 --> 00:17:32.160 would be over here, and so this corresponding point 00:17:32.160 --> 00:17:32.990 would be over here. 00:17:32.990 --> 00:17:38.910 So you shrink this picture, and the slope here tilts up. 00:17:38.910 --> 00:17:43.140 So, as we increase k, the slope gets steeper and steeper. 00:17:43.140 --> 00:17:47.570 Let's see that explicitly, numerically, here. 00:17:47.570 --> 00:17:51.870 Explicitly, numerically, if I take the derivative here... 00:17:51.870 --> 00:17:56.580 So the derivative with respect to x of b^x, 00:17:56.580 --> 00:18:00.944 that's the chain rule, right? 00:18:00.944 --> 00:18:02.360 That's the derivative with respect 00:18:02.360 --> 00:18:08.260 to x of f(kx), which is what? 00:18:08.260 --> 00:18:11.780 It's k times f'(kx). 00:18:11.780 --> 00:18:20.850 And so if we do it at 0, we're just getting k times 00:18:20.850 --> 00:18:26.570 f'(0), which is k times this M(2). 00:18:26.570 --> 00:18:31.390 So how is it exactly that we cook up the right base b? 00:18:31.390 --> 00:18:40.260 So b = e when k = 1 over this number. 00:18:40.260 --> 00:18:44.279 In other words, we can pick all possible slopes that we want. 00:18:44.279 --> 00:18:46.320 This just has the effect of multiplying the slope 00:18:46.320 --> 00:18:47.550 by a factor. 00:18:47.550 --> 00:18:50.130 And we can shift the slope at 0 however we want, 00:18:50.130 --> 00:18:56.240 and we're going to do it so that the slope exactly matches 00:18:56.240 --> 00:18:58.150 1, the one that we want. 00:18:58.150 --> 00:18:59.580 We still don't know what k is. 00:18:59.580 --> 00:19:01.340 We still don't know what e is. 00:19:01.340 --> 00:19:04.940 But at least we know that it's there somewhere. 00:19:04.940 --> 00:19:05.640 Yes? 00:19:05.640 --> 00:19:08.299 Student: How do you know it's f(kx)? 00:19:08.299 --> 00:19:09.340 PROFESSOR: How do I know? 00:19:09.340 --> 00:19:13.440 Well, f(x) is 2^x. 00:19:13.440 --> 00:19:19.160 If f(x) is 2^x, then the formula for f(kx) is this. 00:19:19.160 --> 00:19:23.060 I've decided what f(x) is, so therefore there's 00:19:23.060 --> 00:19:25.120 a formula for f(kx). 00:19:25.120 --> 00:19:26.609 And furthermore, by the chain rule, 00:19:26.609 --> 00:19:28.150 there's a formula for the derivative. 00:19:28.150 --> 00:19:33.930 And it's k times the derivative of f. 00:19:33.930 --> 00:19:35.150 So again, scaling does this. 00:19:35.150 --> 00:19:37.890 By the way, we did exactly the same thing 00:19:37.890 --> 00:19:39.920 with the sine and cosine function. 00:19:39.920 --> 00:19:41.650 If you think of the sine function 00:19:41.650 --> 00:19:44.660 here, let me just remind you here, 00:19:44.660 --> 00:19:46.310 what happens with the chain rule, 00:19:46.310 --> 00:19:51.720 you get k times cosine k t here. 00:19:51.720 --> 00:19:55.340 So the fact that we set things up beautifully with radians 00:19:55.340 --> 00:19:58.650 that this thing is, but we could change the scale to anything, 00:19:58.650 --> 00:20:02.340 such as degrees, by the appropriate factor k. 00:20:02.340 --> 00:20:05.320 And then there would be this scale factor shift 00:20:05.320 --> 00:20:07.500 of the derivative formulas. 00:20:07.500 --> 00:20:09.500 Of course, the one with radians is the easy one, 00:20:09.500 --> 00:20:11.130 because the factor is 1. 00:20:11.130 --> 00:20:13.850 The one with degrees is horrible, 00:20:13.850 --> 00:20:20.085 because the factor is some crazy number like 180 over pi, 00:20:20.085 --> 00:20:22.410 or something like that. 00:20:22.410 --> 00:20:25.880 Okay, so there's something going on here 00:20:25.880 --> 00:20:30.420 which is exactly the same as that kind of re-scaling. 00:20:30.420 --> 00:20:37.040 So, so far we've got only one formula which is a keeper here. 00:20:37.040 --> 00:20:38.810 This one. 00:20:38.810 --> 00:20:40.870 We have a preliminary formula that we still 00:20:40.870 --> 00:20:42.510 haven't completely explained which 00:20:42.510 --> 00:20:45.820 has a little wavy line there. 00:20:45.820 --> 00:20:49.260 And we have to fit all these things together. 00:20:49.260 --> 00:20:52.260 Okay, so now to fit them together, 00:20:52.260 --> 00:21:11.450 I need to introduce the natural log. 00:21:11.450 --> 00:21:21.590 So the natural log is denoted this way, ln(x). 00:21:21.590 --> 00:21:24.570 So maybe I'll call it a new letter name, 00:21:24.570 --> 00:21:28.470 we'll call it w = ln x here. 00:21:28.470 --> 00:21:32.040 But if we were reversing things, if we started out with 00:21:32.040 --> 00:21:37.880 a function y = e^x , the property that it would have is 00:21:37.880 --> 00:21:41.030 that it's the inverse function of e^x. 00:21:41.030 --> 00:21:45.870 So it has the property that the log of y is equal to x. 00:21:45.870 --> 00:21:46.370 Right? 00:21:46.370 --> 00:21:58.910 So this defines the log. 00:21:58.910 --> 00:22:01.790 Now the logarithm has a bunch of properties 00:22:01.790 --> 00:22:04.270 and they come from the exponential properties 00:22:04.270 --> 00:22:04.940 in principle. 00:22:04.940 --> 00:22:07.500 You remember these. 00:22:07.500 --> 00:22:10.379 And I'm just going to remind you of them. 00:22:10.379 --> 00:22:12.420 So the main one that I just want to remind you of 00:22:12.420 --> 00:22:21.100 is that the logarithm of x_1 * x_2 00:22:21.100 --> 00:22:28.130 is equal to the logarithm of x_1 plus the logarithm of x_2. 00:22:28.130 --> 00:22:32.170 And maybe a few more are worth reminding you of. 00:22:32.170 --> 00:22:37.120 One is that the logarithm of 1 is 0. 00:22:37.120 --> 00:22:43.310 A second is that the logarithm of e is 1. 00:22:43.310 --> 00:22:43.840 All right? 00:22:43.840 --> 00:22:47.160 So these correspond to the inverse relationships here. 00:22:47.160 --> 00:22:51.170 If I plug in here, x = 0 and x = 1. 00:22:51.170 --> 00:22:56.650 If I plug in x = 0 and x = 1, I get the corresponding numbers 00:22:56.650 --> 00:23:04.030 here: y = 1 and y = e. 00:23:04.030 --> 00:23:10.430 And maybe it would be worth it to plot the picture once 00:23:10.430 --> 00:23:13.430 to reinforce this. 00:23:13.430 --> 00:23:16.620 So here I'll put them on the same chart. 00:23:16.620 --> 00:23:20.200 If you have here e^x over here. 00:23:20.200 --> 00:23:21.790 It looks like this. 00:23:21.790 --> 00:23:28.610 Then the logarithm which I'll maybe put in a different color. 00:23:28.610 --> 00:23:31.120 So this crosses at this all-important point 00:23:31.120 --> 00:23:32.700 here, (0,1). 00:23:32.700 --> 00:23:35.260 And now in order to figure out what the inverse function is, 00:23:35.260 --> 00:23:40.750 I have to take the flip across the diagonal x = y. 00:23:40.750 --> 00:23:44.600 So that's this shape here, going down like this. 00:23:44.600 --> 00:23:47.090 And here's the point (1, 0). 00:23:47.090 --> 00:23:50.700 So (1, 0) corresponds to this identity here. 00:23:50.700 --> 00:23:53.000 But the log of 1 is 0. 00:23:53.000 --> 00:24:00.120 And notice, so this is ln x, the graph of ln x. 00:24:00.120 --> 00:24:05.680 And notice it's only defined for x positive, 00:24:05.680 --> 00:24:09.570 which corresponds to the fact that e^x is always positive. 00:24:09.570 --> 00:24:15.130 So in other words, this white curve is only above this axis, 00:24:15.130 --> 00:24:19.210 and the orange one is to the right here. 00:24:19.210 --> 00:24:27.990 It's only defined for x positive. 00:24:27.990 --> 00:24:31.740 Oh, one other thing I should mention is the slope here is 1. 00:24:31.740 --> 00:24:35.380 And so the slope there is also going to be 1. 00:24:35.380 --> 00:24:41.180 Now, what we're allowed to do relatively easily, because we 00:24:41.180 --> 00:24:44.470 have the tools to do it, is to compute the derivative 00:24:44.470 --> 00:24:49.960 of the logarithm. 00:24:49.960 --> 00:24:59.090 So to find the derivative of a log, 00:24:59.090 --> 00:25:04.060 we're going to use implicit differentiation. 00:25:04.060 --> 00:25:08.250 This is how we find the derivative 00:25:08.250 --> 00:25:09.806 of any inverse function. 00:25:09.806 --> 00:25:11.180 So remember the way that works is 00:25:11.180 --> 00:25:12.971 if you know the derivative of the function, 00:25:12.971 --> 00:25:15.590 you can find the derivative of the inverse function. 00:25:15.590 --> 00:25:18.280 And the mechanism is the following: 00:25:18.280 --> 00:25:22.677 you write down here w = ln x. 00:25:22.677 --> 00:25:23.510 Here's the function. 00:25:23.510 --> 00:25:25.510 We're trying to find the derivative of w. 00:25:25.510 --> 00:25:28.750 But now we don't know how to differentiate this equation, 00:25:28.750 --> 00:25:38.200 but if we exponentiate it, so that's the same thing as e^w = 00:25:38.200 --> 00:25:42.660 x. 00:25:42.660 --> 00:25:46.420 Because let's just stick this in here. 00:25:46.420 --> 00:25:52.330 e^(ln x) = x. 00:25:52.330 --> 00:25:54.680 Now we can differentiate this. 00:25:54.680 --> 00:25:56.800 So let's do the differentiation here. 00:25:56.800 --> 00:26:04.010 We have d/dx e^w is equal to d/dx x, which is 1. 00:26:04.010 --> 00:26:06.030 And then this, by the chain rule, 00:26:06.030 --> 00:26:11.450 is d/dw of e^w times dw/dx. 00:26:11.450 --> 00:26:14.560 The product of these two factors. 00:26:14.560 --> 00:26:15.580 That's equal to 1. 00:26:15.580 --> 00:26:18.680 And now this guy, the one little guy 00:26:18.680 --> 00:26:27.980 that we actually know and can use, that's this guy here. 00:26:27.980 --> 00:26:33.800 So this is e^w times dw/dx, which is 1. 00:26:33.800 --> 00:26:44.730 And so finally, dw/dx = 1 / e^w . 00:26:44.730 --> 00:26:47.080 But what is that? 00:26:47.080 --> 00:26:48.250 It's x. 00:26:48.250 --> 00:26:50.740 So this is 1/x. 00:26:50.740 --> 00:26:53.140 So what we discovered is, and now I 00:26:53.140 --> 00:26:57.000 get to put another green guy around here, 00:26:57.000 --> 00:27:01.870 is that this is equal to 1/x. 00:27:01.870 --> 00:27:16.710 So alright, now we have two companion formulas here. 00:27:16.710 --> 00:27:20.210 The rate of change of ln x is 1/x. 00:27:20.210 --> 00:27:24.830 And the rate of change of e^x is itself, is e^x. 00:27:24.830 --> 00:27:30.729 And it's time to return to the problem 00:27:30.729 --> 00:27:32.770 that we were having a little bit of trouble with, 00:27:32.770 --> 00:27:37.690 which is somewhat not explicit, which is this M(a) times x. 00:27:37.690 --> 00:27:44.090 We want to now differentiate a^x in general, not just e^x . 00:27:44.090 --> 00:27:47.140 So let's work that out, and I want 00:27:47.140 --> 00:27:50.732 to explain it in a couple of ways, 00:27:50.732 --> 00:27:52.440 so you're going to have to remember this, 00:27:52.440 --> 00:27:55.530 because I'm going to erase it. 00:27:55.530 --> 00:28:01.780 But what I'd like you to do is, so now I 00:28:01.780 --> 00:28:03.570 want to teach you how to differentiate 00:28:03.570 --> 00:28:17.530 basically any exponential. 00:28:17.530 --> 00:28:31.580 So now to differentiate any exponential. 00:28:31.580 --> 00:28:37.941 There are two methods. 00:28:37.941 --> 00:28:39.440 They're practically the same method. 00:28:39.440 --> 00:28:41.480 They have the same amount of arithmetic. 00:28:41.480 --> 00:28:45.610 You'll see both of them, and they're equally valuable. 00:28:45.610 --> 00:28:48.150 So we're going to just describe them. 00:28:48.150 --> 00:28:55.940 Method one I'm going to illustrate on the function a^x. 00:28:55.940 --> 00:29:00.020 So we're interested in differentiating 00:29:00.020 --> 00:29:04.280 this thing, exactly this problem that I still didn't solve yet. 00:29:04.280 --> 00:29:05.080 Okay? 00:29:05.080 --> 00:29:06.950 So here it is. 00:29:06.950 --> 00:29:08.080 And here's the procedure. 00:29:08.080 --> 00:29:17.190 The procedure is to write, so the method is to use base e, 00:29:17.190 --> 00:29:20.350 or convert to base e. 00:29:20.350 --> 00:29:22.430 So how do you convert to base e? 00:29:22.430 --> 00:29:27.660 Well, you write a^x as e to some power. 00:29:27.660 --> 00:29:29.020 So what power is it? 00:29:29.020 --> 00:29:34.980 It's e to the power ln a, to the power x. 00:29:34.980 --> 00:29:40.730 And that is just e^(x ln a). 00:29:40.730 --> 00:29:44.870 So we've made our conversion now to base e. 00:29:44.870 --> 00:29:46.810 The exponential of something. 00:29:46.810 --> 00:29:50.410 So now I'm going to carry out the differentiation. 00:29:50.410 --> 00:29:59.270 So d/dx of a^x is equal to d/dx of e^(x ln a). 00:29:59.270 --> 00:30:05.970 And now, this is a step which causes great confusion when 00:30:05.970 --> 00:30:06.870 you first see it. 00:30:06.870 --> 00:30:10.920 And you must get used to it, because it's easy, not hard. 00:30:10.920 --> 00:30:13.450 Okay? 00:30:13.450 --> 00:30:18.660 The rate of change of this with respect to x is, 00:30:18.660 --> 00:30:23.040 let me do it by analogy here. 00:30:23.040 --> 00:30:27.520 Because say I had e^(3x) and I were differentiating it. 00:30:27.520 --> 00:30:31.600 The chain rule would say that this is just 3, 00:30:31.600 --> 00:30:36.330 the rate of change of 3x with respect to x times e^(3x). 00:30:36.330 --> 00:30:41.060 The rate of change of e to the u with respect to u. 00:30:41.060 --> 00:30:43.500 So this is the ordinary chain rule. 00:30:43.500 --> 00:30:47.610 And what we're doing up here is exactly the same thing, 00:30:47.610 --> 00:30:50.270 because ln a, as frightening as it 00:30:50.270 --> 00:30:54.690 looks, with all three letters there, is just a fixed number. 00:30:54.690 --> 00:30:55.860 It's not moving. 00:30:55.860 --> 00:30:57.170 It's a constant. 00:30:57.170 --> 00:31:01.080 So the constant just accelerates the rate of change 00:31:01.080 --> 00:31:04.980 by that factor, which is what the chain rule is doing. 00:31:04.980 --> 00:31:11.830 So this is equal to ln a times e^(x ln a). 00:31:11.830 --> 00:31:17.390 Same business here with ln a replacing 3. 00:31:17.390 --> 00:31:19.790 So this is something you've got to get used to in time 00:31:19.790 --> 00:31:21.789 for the exam, for instance, because you're going 00:31:21.789 --> 00:31:25.360 to be doing a million of these. 00:31:25.360 --> 00:31:27.810 So do get used to it. 00:31:27.810 --> 00:31:29.230 So here's the formula. 00:31:29.230 --> 00:31:33.770 On the other hand, this expression here was the same 00:31:33.770 --> 00:31:34.730 as a^x. 00:31:34.730 --> 00:31:39.200 So another way of writing this, and I'll put this into a box, 00:31:39.200 --> 00:31:41.790 but actually I never remember this particularly. 00:31:41.790 --> 00:31:48.650 I just re-derive it every time, is that the derivative of a^x 00:31:48.650 --> 00:31:51.100 is equal to (ln a) a^x . 00:31:51.100 --> 00:31:56.930 Now I'm going to get rid of what's underneath it. 00:31:56.930 --> 00:32:01.970 So this is another formula. 00:32:01.970 --> 00:32:05.500 So there's the formula I've essentially finished here. 00:32:05.500 --> 00:32:11.190 And notice, what is the magic number? 00:32:11.190 --> 00:32:16.089 The magic number is the natural log of a. 00:32:16.089 --> 00:32:16.880 That's what it was. 00:32:16.880 --> 00:32:18.679 We didn't know what it was in advance. 00:32:18.679 --> 00:32:19.470 This is what it is. 00:32:19.470 --> 00:32:21.450 It's the natural log of a. 00:32:21.450 --> 00:32:26.740 Let me emphasize to you again, something about what's 00:32:26.740 --> 00:32:34.510 going on here, which has to do with scale change. 00:32:34.510 --> 00:32:44.290 So, for example, the derivative with respect to x of 2^x is (ln 00:32:44.290 --> 00:32:47.290 2) 2^x. 00:32:47.290 --> 00:32:49.080 The derivative with respect to x, 00:32:49.080 --> 00:32:51.730 these are the two most obvious bases that you might want 00:32:51.730 --> 00:32:56.760 to use, is ln 10 times 10^x . 00:32:56.760 --> 00:32:59.470 So one of the things that's natural about the natural 00:32:59.470 --> 00:33:02.860 logarithm is that even if we insisted 00:33:02.860 --> 00:33:07.410 that we must use base 2, or that we must use base 10, 00:33:07.410 --> 00:33:11.360 we'd still be stuck with natural logarithms. 00:33:11.360 --> 00:33:12.540 They come up naturally. 00:33:12.540 --> 00:33:14.780 They're the ones which are independent 00:33:14.780 --> 00:33:20.079 of our human construct of base 2 and base 10. 00:33:20.079 --> 00:33:21.620 The natural logarithm is the one that 00:33:21.620 --> 00:33:25.360 comes up without reference. 00:33:25.360 --> 00:33:27.210 And we'll be mentioning a few other ways 00:33:27.210 --> 00:33:31.110 in which it's natural later. 00:33:31.110 --> 00:33:34.750 So I told you about this first method, 00:33:34.750 --> 00:33:41.730 now I want to tell you about a second method here. 00:33:41.730 --> 00:34:05.700 So the second is called logarithmic differentiation. 00:34:05.700 --> 00:34:07.770 So how does this work? 00:34:07.770 --> 00:34:10.930 Well, sometimes you're having trouble 00:34:10.930 --> 00:34:17.640 differentiating a function, and it's easier 00:34:17.640 --> 00:34:21.780 to differentiate its logarithm. 00:34:21.780 --> 00:34:23.830 That may seem peculiar, but actually we'll 00:34:23.830 --> 00:34:26.640 give several examples where this is clearly the case, 00:34:26.640 --> 00:34:28.560 that the logarithm is easier to differentiate 00:34:28.560 --> 00:34:30.830 than the function. 00:34:30.830 --> 00:34:34.090 So it could be that this is an easier quantity to understand. 00:34:34.090 --> 00:34:39.470 So we want to relate it back to the function u. 00:34:39.470 --> 00:34:44.170 So I'm going to write it a slightly different way. 00:34:44.170 --> 00:34:47.270 Let's write it in terms of primes here. 00:34:47.270 --> 00:34:51.060 So the basic identity is the chain rule again, 00:34:51.060 --> 00:34:52.730 and the derivative of the logarithm, 00:34:52.730 --> 00:34:54.920 well maybe I'll write it out this way first. 00:34:54.920 --> 00:35:01.720 So this would be d ln u / du, times d/dx u. 00:35:05.120 --> 00:35:10.110 These are the two factors. 00:35:10.110 --> 00:35:12.040 And that's the same thing, so remember 00:35:12.040 --> 00:35:14.140 what the derivative of the logarithm is. 00:35:14.140 --> 00:35:17.820 This is 1/u. 00:35:17.820 --> 00:35:23.570 So here I have a 1/u, and here I have a du/dx. 00:35:23.570 --> 00:35:28.850 So I'm going to encode this on the next board here, 00:35:28.850 --> 00:35:31.290 which is sort of the main formula you always need 00:35:31.290 --> 00:35:39.530 to remember, which is that (ln u)' = u' / u. 00:35:39.530 --> 00:35:42.610 That's the one to remember here. 00:35:42.610 --> 00:35:47.320 STUDENT: [INAUDIBLE]. 00:35:47.320 --> 00:35:51.870 PROFESSOR: The question is how did I get this step here? 00:35:51.870 --> 00:35:58.500 So this is the chain rule. 00:35:58.500 --> 00:36:02.150 The rate of change of ln u with respect to x 00:36:02.150 --> 00:36:04.610 is the rate of change of ln u with respect u, 00:36:04.610 --> 00:36:07.730 times the rate of change of u with respect to x. 00:36:07.730 --> 00:36:18.560 That's the chain rule. 00:36:18.560 --> 00:36:22.840 So now I've worked out this identity here, 00:36:22.840 --> 00:36:30.730 and now let's show how it handles this case, d/dx a^x. 00:36:30.730 --> 00:36:31.740 Let's do this one. 00:36:31.740 --> 00:36:39.570 So in order to get that one, I would take u = a^x . 00:36:39.570 --> 00:36:51.460 And now let's just take a look at what ln u is. ln u = x ln a. 00:36:51.460 --> 00:36:55.010 Now I claim that this is pretty easy to differentiate. 00:36:55.010 --> 00:37:00.200 Again, it may seem hard, but it's actually quite easy. 00:37:00.200 --> 00:37:04.590 So maybe somebody can hazard a guess. 00:37:04.590 --> 00:37:11.530 What's the derivative of x ln a? 00:37:11.530 --> 00:37:14.870 It's just ln a. 00:37:14.870 --> 00:37:18.400 So this is the same thing that I was talking about before, which 00:37:18.400 --> 00:37:21.420 is if you've got 3x, and you're taking 00:37:21.420 --> 00:37:24.804 its derivative with respect to x here, that's just 3. 00:37:24.804 --> 00:37:26.220 That's the kind of thing you have. 00:37:26.220 --> 00:37:30.110 Again, don't be put off by this massive piece of junk here. 00:37:30.110 --> 00:37:33.260 It's a constant. 00:37:33.260 --> 00:37:38.220 So again, keep that in mind. 00:37:38.220 --> 00:37:42.460 It comes up regularly in this kind of question. 00:37:42.460 --> 00:37:46.980 So there's our formula, that the logarithmic derivative is this. 00:37:46.980 --> 00:37:50.360 But let's just rewrite that. 00:37:50.360 --> 00:37:58.600 That's the same thing as u' / u, which is (ln u)' = ln a, right? 00:37:58.600 --> 00:38:00.610 So this is our differentiation formula. 00:38:00.610 --> 00:38:01.800 So here we have u'. 00:38:01.800 --> 00:38:07.460 u' is equal to u times ln a, if I just multiply through by u. 00:38:07.460 --> 00:38:08.630 And that's what we wanted. 00:38:08.630 --> 00:38:16.660 That's d/dx a^x is equal to ln a (I'll reverse the order 00:38:16.660 --> 00:38:24.640 of the two, which is customary) times a^x. 00:38:24.640 --> 00:38:27.390 So this is the way that logarithmic differentiation 00:38:27.390 --> 00:38:27.890 works. 00:38:27.890 --> 00:38:32.680 It's the same arithmetic as the previous method, 00:38:32.680 --> 00:38:34.930 but we don't have to convert to base e. 00:38:34.930 --> 00:38:37.829 We're just keeping track of the exponents 00:38:37.829 --> 00:38:39.620 and doing differentiation on the exponents, 00:38:39.620 --> 00:38:44.330 and multiplying through at the end. 00:38:44.330 --> 00:38:49.660 Okay, so I'm going to do two trickier examples, which 00:38:49.660 --> 00:39:02.400 illustrate logarithmic differentiation. 00:39:02.400 --> 00:39:06.490 Again, these could be done equally well by using base e, 00:39:06.490 --> 00:39:07.730 but I won't do it. 00:39:07.730 --> 00:39:12.120 Method one and method two always both work. 00:39:12.120 --> 00:39:15.780 So here's a second example: again this 00:39:15.780 --> 00:39:23.220 is a problem when you have moving exponents. 00:39:23.220 --> 00:39:25.900 But this time, we're going to complicate matters 00:39:25.900 --> 00:39:30.520 by having both a moving exponent and a moving base. 00:39:30.520 --> 00:39:34.710 So we have a function u, which is, well maybe I'll call it v, 00:39:34.710 --> 00:39:38.640 since we already had a function u, which is x^x. 00:39:38.640 --> 00:39:41.670 A really complicated looking function here. 00:39:41.670 --> 00:39:44.490 So again you can handle this by converting 00:39:44.490 --> 00:39:47.220 to base e, method one. 00:39:47.220 --> 00:39:49.750 But we'll do the logarithmic differentiation version, 00:39:49.750 --> 00:39:51.110 alright? 00:39:51.110 --> 00:39:59.310 So I take the logs of both sides. 00:39:59.310 --> 00:40:04.370 And now I differentiate it. 00:40:04.370 --> 00:40:06.200 And now when I differentiate this here, 00:40:06.200 --> 00:40:07.654 I have to use the product rule. 00:40:07.654 --> 00:40:09.570 This time, instead of having ln a, a constant, 00:40:09.570 --> 00:40:10.980 I have a variable here. 00:40:10.980 --> 00:40:12.660 So I have two factors. 00:40:12.660 --> 00:40:15.416 I have ln x when I differentiate with respect to x. 00:40:15.416 --> 00:40:19.410 When I differentiate with respect to this factor here, 00:40:19.410 --> 00:40:21.800 I get that x times the derivative of that, 00:40:21.800 --> 00:40:26.910 which is 1/x. 00:40:26.910 --> 00:40:29.160 So, here's my formula. 00:40:29.160 --> 00:40:30.430 Almost finished. 00:40:30.430 --> 00:40:34.701 So I have here v' / v. I'm going to multiply these two things 00:40:34.701 --> 00:40:35.200 together. 00:40:35.200 --> 00:40:37.741 I'll put it on the other side, because I don't want to get it 00:40:37.741 --> 00:40:45.340 mixed up with ln(x+1), the quantity. 00:40:45.340 --> 00:40:47.100 And now I'm almost done. 00:40:47.100 --> 00:41:02.110 I have v' = v (1 + ln x), and that's just d/dx x^x = x^x (1 + 00:41:02.110 --> 00:41:04.310 ln x). 00:41:04.310 --> 00:41:13.450 That's it. 00:41:13.450 --> 00:41:31.989 So these two methods always work for moving exponents. 00:41:31.989 --> 00:41:33.530 So the next thing that I'd like to do 00:41:33.530 --> 00:41:36.220 is another fairly tricky example. 00:41:36.220 --> 00:41:45.880 And this one is not strictly speaking within calculus. 00:41:45.880 --> 00:41:48.800 Although we're going to use the tools that we just described 00:41:48.800 --> 00:41:52.820 to carry it out, in fact it will use some calculus 00:41:52.820 --> 00:41:55.990 in the very end. 00:41:55.990 --> 00:41:59.530 And what I'm going to do is I'm going to evaluate the limit 00:41:59.530 --> 00:42:02.020 as n goes to infinity of (1 + 1/n)^n. 00:42:11.430 --> 00:42:16.170 So now, the reason why I want to discuss this is, is it 00:42:16.170 --> 00:42:18.460 turns out to have a very interesting answer. 00:42:18.460 --> 00:42:23.520 And it's a problem that you can approach exactly 00:42:23.520 --> 00:42:24.490 by this method. 00:42:24.490 --> 00:42:28.580 And the reason is that it has a moving exponent. 00:42:28.580 --> 00:42:30.840 The exponent n here is changing. 00:42:30.840 --> 00:42:33.570 And so if you want to keep track of that, a good way to do that 00:42:33.570 --> 00:42:36.740 is to use logarithms. 00:42:36.740 --> 00:42:38.550 So in order to figure out this limit, 00:42:38.550 --> 00:42:40.040 we're going to take the log of it 00:42:40.040 --> 00:42:41.775 and figure out what the limit of the log 00:42:41.775 --> 00:42:43.280 is, instead of the log of the limit. 00:42:43.280 --> 00:42:44.990 Those will be the same thing. 00:42:44.990 --> 00:42:48.850 So we're going to take the natural log of this quantity 00:42:48.850 --> 00:42:56.200 here, and that's n ln(1 + 1/n). 00:43:02.640 --> 00:43:06.370 And now I'm going to rewrite this 00:43:06.370 --> 00:43:10.960 in a form which will make it more recognizable, 00:43:10.960 --> 00:43:20.040 so what I'd like to do is I'm going to write n, 00:43:20.040 --> 00:43:24.730 or maybe I should say it this way: delta x is equal to 1/n. 00:43:24.730 --> 00:43:29.880 So if n is going to infinity, then this delta x 00:43:29.880 --> 00:43:33.700 is going to be going to 0. 00:43:33.700 --> 00:43:37.500 So this is more familiar territory for us in this class, 00:43:37.500 --> 00:43:38.560 anyway. 00:43:38.560 --> 00:43:40.370 So let's rewrite it. 00:43:40.370 --> 00:43:42.980 So here, we have 1 over delta x. 00:43:42.980 --> 00:43:46.680 And then that is multiplied by ln(1 + delta x). 00:43:50.150 --> 00:43:54.860 So n is the reciprocal of delta x. 00:43:54.860 --> 00:43:58.330 Now I want to change this in a very, very minor way. 00:43:58.330 --> 00:44:01.150 I'm going to subtract 0 from it. 00:44:01.150 --> 00:44:02.470 So that's the same thing. 00:44:02.470 --> 00:44:06.420 So what I'm going to do is I'm going to subtract ln 1 from it. 00:44:06.420 --> 00:44:08.260 That's just equal to 0. 00:44:08.260 --> 00:44:10.210 So this is not a problem, and I'll 00:44:10.210 --> 00:44:14.800 put some parentheses around it. 00:44:14.800 --> 00:44:18.080 Now you're supposed to recognize, all of a sudden, 00:44:18.080 --> 00:44:21.280 what pattern this fits into. 00:44:21.280 --> 00:44:25.570 This is the thing which we need to calculate in order 00:44:25.570 --> 00:44:30.810 to calculate the derivative of the log function. 00:44:30.810 --> 00:44:33.540 So this is, in the limit as delta 00:44:33.540 --> 00:44:39.300 x goes to 0, equal to the derivative of ln x. 00:44:39.300 --> 00:44:39.890 Where? 00:44:39.890 --> 00:44:43.940 Well the base point is x=1. 00:44:43.940 --> 00:44:45.710 That's where we're evaluating it. 00:44:45.710 --> 00:44:46.730 That's the x_0. 00:44:46.730 --> 00:44:49.380 That's the base value. 00:44:49.380 --> 00:44:51.000 So this is the difference quotient. 00:44:51.000 --> 00:44:52.430 That's exactly what it is. 00:44:52.430 --> 00:44:57.630 And so this by definition tends to the limit here. 00:44:57.630 --> 00:45:01.470 But we know what the derivative of the log function is. 00:45:01.470 --> 00:45:03.520 The derivative of the log function is 1/x. 00:45:09.110 --> 00:45:17.470 So this limit is 1. 00:45:17.470 --> 00:45:18.500 So we got it. 00:45:18.500 --> 00:45:19.730 We got the limit. 00:45:19.730 --> 00:45:22.050 And now we just have to work backwards 00:45:22.050 --> 00:45:34.010 to figure out what this limit that we've got over here is. 00:45:34.010 --> 00:45:37.130 So let's do that. 00:45:37.130 --> 00:45:38.260 So let's see here. 00:45:38.260 --> 00:45:40.570 The log approached 1. 00:45:40.570 --> 00:45:45.960 So the limit as n goes to infinity of (1 + 1/n)^n. 00:45:49.530 --> 00:45:51.880 So sorry, the log of this. 00:45:51.880 --> 00:45:54.320 Yeah, let's write it this way. 00:45:54.320 --> 00:45:57.530 It's the same thing, as well, the thing that we know 00:45:57.530 --> 00:46:00.610 is the log of this. 00:46:00.610 --> 00:46:04.700 1 plus 1 over n to the n. 00:46:04.700 --> 00:46:06.390 And goes to infinity. 00:46:06.390 --> 00:46:08.190 That's the one that we just figured out. 00:46:08.190 --> 00:46:11.380 But now this thing is the exponential of that. 00:46:11.380 --> 00:46:16.550 So it's really e to this power here. 00:46:16.550 --> 00:46:19.060 So this guy is the same as the limit 00:46:19.060 --> 00:46:21.799 of the log of the limit of the thing, which is the same as log 00:46:21.799 --> 00:46:22.340 of the limit. 00:46:22.340 --> 00:46:25.268 The limit of the log and the log of the limit are the same. 00:46:25.268 --> 00:46:32.260 log lim equals lim log. 00:46:32.260 --> 00:46:33.754 Okay, so I take the logarithm, then 00:46:33.754 --> 00:46:35.170 I'm going to take the exponential. 00:46:35.170 --> 00:46:37.800 That just undoes what I did before. 00:46:37.800 --> 00:46:41.910 And so this limit is just 1, so this is e^1. 00:46:41.910 --> 00:46:52.000 And so the limit that we want here is equal to e. 00:46:52.000 --> 00:46:56.790 So I claim that with this step, we've actually closed 00:46:56.790 --> 00:46:58.270 the loop, finally. 00:46:58.270 --> 00:47:03.620 Because we have an honest numerical way to calculate e. 00:47:03.620 --> 00:47:04.140 The first. 00:47:04.140 --> 00:47:05.400 There are many such. 00:47:05.400 --> 00:47:07.640 But this one is a perfectly honest numerical way 00:47:07.640 --> 00:47:08.680 to calculate e. 00:47:08.680 --> 00:47:09.950 We had this thing. 00:47:09.950 --> 00:47:12.170 We didn't know exactly what it was. 00:47:12.170 --> 00:47:14.872 It was this M(e), there was M(a), the logarithm, and so on. 00:47:14.872 --> 00:47:15.830 We have all that stuff. 00:47:15.830 --> 00:47:18.920 But we really need to nail down what this number e is. 00:47:18.920 --> 00:47:20.920 And this is telling us, if you take 00:47:20.920 --> 00:47:25.620 for example 1 plus 1 over 100 to the 100th power, that's 00:47:25.620 --> 00:47:28.910 going to be a very good, perfectly decent anyway, 00:47:28.910 --> 00:47:30.730 approximation to e. 00:47:30.730 --> 00:47:36.330 So this is a numerical approximation, 00:47:36.330 --> 00:47:39.320 which is all we can ever do with just 00:47:39.320 --> 00:47:42.700 this kind of irrational number. 00:47:42.700 --> 00:47:46.270 And so that closes the loop, and we now 00:47:46.270 --> 00:47:49.452 have a coherent family of functions, 00:47:49.452 --> 00:47:51.660 which are actually well defined and for which we have 00:47:51.660 --> 00:47:54.550 practical methods to calculate. 00:47:54.550 --> 00:47:56.312 Okay, see you next time.