WEBVTT 00:00:00.040 --> 00:00:02.320 The following content is provided under a Creative 00:00:02.320 --> 00:00:03.590 Commons license. 00:00:03.590 --> 00:00:05.930 Your support will help MIT OpenCourseWare 00:00:05.930 --> 00:00:09.990 continue to offer high quality educational resources for free. 00:00:09.990 --> 00:00:12.500 To make a donation, or to view additional material 00:00:12.500 --> 00:00:16.120 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.120 --> 00:00:21.700 at ocw.mit.edu. 00:00:21.700 --> 00:00:25.000 PROFESSOR: Right now, we're finishing up 00:00:25.000 --> 00:00:27.740 with the first unit, and I'd like 00:00:27.740 --> 00:00:31.890 to continue in this lecture, lecture seven, 00:00:31.890 --> 00:00:45.510 with some final remarks about exponents. 00:00:45.510 --> 00:00:49.560 So what I'd like to do is just review something 00:00:49.560 --> 00:00:51.910 that I did quickly last time, and make 00:00:51.910 --> 00:00:53.920 a few philosophical remarks about it. 00:00:53.920 --> 00:00:57.660 I think that the steps involved were maybe a little tricky, 00:00:57.660 --> 00:01:00.680 and so I'd like to go through it one more time. 00:01:00.680 --> 00:01:03.900 Remember, that we were talking about this number a_k, 00:01:03.900 --> 00:01:05.510 which is (1 + 1/k)^k. 00:01:08.270 --> 00:01:11.400 And what we showed was that the limit as k 00:01:11.400 --> 00:01:16.920 goes to infinity of a_k was e. 00:01:16.920 --> 00:01:19.590 So the first thing that I'd like to do 00:01:19.590 --> 00:01:22.890 is just explain the proof a little bit more clearly 00:01:22.890 --> 00:01:28.440 than I did last time with fewer symbols, 00:01:28.440 --> 00:01:31.830 or at least with this abbreviation of the symbol 00:01:31.830 --> 00:01:35.050 here, to show you what we actually did. 00:01:35.050 --> 00:01:41.580 So I'll just remind you of what we did last time, 00:01:41.580 --> 00:01:45.726 and the first observation was to check, 00:01:45.726 --> 00:01:47.350 rather than the limit of this function, 00:01:47.350 --> 00:01:49.550 but to take the log first. 00:01:49.550 --> 00:01:51.640 And this is typically what's done 00:01:51.640 --> 00:01:54.810 when you have an exponential, when you have an exponent. 00:01:54.810 --> 00:01:57.750 And what we found was that the limit here 00:01:57.750 --> 00:02:03.500 was 1 as k goes to infinity. 00:02:03.500 --> 00:02:05.160 So last time, this is what we did. 00:02:05.160 --> 00:02:07.970 And I just wanted to be careful and show you 00:02:07.970 --> 00:02:09.830 exactly what the next step is. 00:02:09.830 --> 00:02:14.360 If you exponentiate this fact; you take e to this power, 00:02:14.360 --> 00:02:21.230 that's going to tend to e^1, which is just e. 00:02:21.230 --> 00:02:26.570 And then, we just observe that this is the same as a_k. 00:02:26.570 --> 00:02:32.180 So the basic ingredient here is that e^ln a = a. 00:02:32.180 --> 00:02:36.750 That's because the log function is the inverse 00:02:36.750 --> 00:02:38.030 of the exponential function. 00:02:38.030 --> 00:02:38.860 Yes, question? 00:02:38.860 --> 00:02:54.190 STUDENT: [INAUDIBLE] 00:02:54.190 --> 00:02:58.180 PROFESSOR: So the question was, wouldn't the log of this 00:02:58.180 --> 00:03:01.530 be 0 because a_k is tending to 1. 00:03:01.530 --> 00:03:03.830 But a_k isn't tending to 1. 00:03:03.830 --> 00:03:06.640 Who said it was? 00:03:06.640 --> 00:03:10.040 If you take the logarithm, which is what we did last time, 00:03:10.040 --> 00:03:12.880 logarithm of a_k is indeed k * ln(1 + 1/k). 00:03:17.030 --> 00:03:18.520 That does not tend to 0. 00:03:18.520 --> 00:03:22.610 This part of it tends to 0, and this part tends to infinity. 00:03:22.610 --> 00:03:26.460 And they balance each other, 0 times infinity. 00:03:26.460 --> 00:03:28.460 We don't really know yet from this expression, 00:03:28.460 --> 00:03:32.189 in fact we did some cleverness with limits and derivatives, 00:03:32.189 --> 00:03:33.230 to figure out this limit. 00:03:33.230 --> 00:03:34.354 It was a very subtle thing. 00:03:34.354 --> 00:03:37.560 It turned out to be 1. 00:03:37.560 --> 00:03:38.850 All right? 00:03:38.850 --> 00:03:40.645 Now, the thing that I'd like to say 00:03:40.645 --> 00:03:43.550 - I'm sorry I'm going to erase this aside here 00:03:43.550 --> 00:03:46.100 - but you need to go back to your notes 00:03:46.100 --> 00:03:48.824 and remember that this is what we did last time. 00:03:48.824 --> 00:03:50.490 Because I want to have room for the next 00:03:50.490 --> 00:03:54.420 comment that I want to make on this little blackboard here. 00:03:54.420 --> 00:03:57.090 What we just derived was this property here, 00:03:57.090 --> 00:04:01.810 but I made a claim yesterday, and I just 00:04:01.810 --> 00:04:04.120 want to emphasize it again so that we realized 00:04:04.120 --> 00:04:07.786 what it is that we're doing. 00:04:07.786 --> 00:04:08.910 I looked at this backwards. 00:04:08.910 --> 00:04:11.430 One way you can think of this is we're evaluating this limit 00:04:11.430 --> 00:04:13.000 and getting an answer. 00:04:13.000 --> 00:04:16.524 But all equalities can be read both directions. 00:04:16.524 --> 00:04:17.940 And we can write it the other way: 00:04:17.940 --> 00:04:25.881 e equals the limit, as k goes to infinity, of this expression 00:04:25.881 --> 00:04:26.380 here. 00:04:26.380 --> 00:04:28.640 So that's just the same thing. 00:04:28.640 --> 00:04:30.840 And if we read it backwards, what we're saying 00:04:30.840 --> 00:04:35.700 is that this limit is a formula for e. 00:04:35.700 --> 00:04:38.325 So this is very typical of mathematics. 00:04:38.325 --> 00:04:40.700 You want to always reverse your perspective all the time. 00:04:40.700 --> 00:04:43.710 Equations work both ways, and in this case, 00:04:43.710 --> 00:04:46.300 we have two different things here. 00:04:46.300 --> 00:04:49.430 This e was what we defined as the base, 00:04:49.430 --> 00:04:54.461 which when you graph e^x, you get slope 1 at 0. 00:04:54.461 --> 00:04:56.710 And then it turns out to be equal to this limit, which 00:04:56.710 --> 00:04:59.060 we can calculate numerically. 00:04:59.060 --> 00:05:02.100 If you do this on your calculators, you, of course, 00:05:02.100 --> 00:05:05.420 will have a way of programming in this number 00:05:05.420 --> 00:05:07.290 and evaluating it for each k. 00:05:07.290 --> 00:05:10.880 And you'll have another button available to evaluate this one. 00:05:10.880 --> 00:05:12.432 So another way of saying it is it 00:05:12.432 --> 00:05:14.640 that there's a relationship between these two things. 00:05:14.640 --> 00:05:19.210 And all of calculus is a matter of getting these relationships. 00:05:19.210 --> 00:05:21.860 So we can look at these things in several different ways. 00:05:21.860 --> 00:05:23.350 And indeed, that's what we're going 00:05:23.350 --> 00:05:25.832 to be doing at least at the end of today 00:05:25.832 --> 00:05:27.040 in talking about derivatives. 00:05:27.040 --> 00:05:29.480 A lot of times when we talk about derivatives, 00:05:29.480 --> 00:05:32.230 we're trying to look at them from several perspectives 00:05:32.230 --> 00:05:34.780 at once. 00:05:34.780 --> 00:05:37.280 Okay, so I have to keep on going with exponents, 00:05:37.280 --> 00:05:40.130 because I have one loose end. 00:05:40.130 --> 00:05:44.490 One loose end that I did not cover yet. 00:05:44.490 --> 00:05:48.330 There's one very important formula that's left, 00:05:48.330 --> 00:05:51.920 and it's the derivative of the powers. 00:05:51.920 --> 00:05:54.070 We actually didn't do this - well we 00:05:54.070 --> 00:05:57.120 did it for rational numbers r. 00:05:57.120 --> 00:06:00.070 So this is the formula here. 00:06:00.070 --> 00:06:06.200 But now we're going to check this for all real numbers, r. 00:06:06.200 --> 00:06:09.200 So including all the irrational ones as well. 00:06:09.200 --> 00:06:13.690 This is also good practice for using base e 00:06:13.690 --> 00:06:16.480 and using logarithmic differentiation. 00:06:16.480 --> 00:06:20.810 So let me do this by our two methods 00:06:20.810 --> 00:06:26.160 that we can use to handle exponential type problems. 00:06:26.160 --> 00:06:32.070 So method one was base e. 00:06:32.070 --> 00:06:34.040 So if I just rewrite this base e again, 00:06:34.040 --> 00:06:36.910 that's just this formula over here. 00:06:36.910 --> 00:06:50.350 x^r = (e^ln x)^r, which is e^r ln x. 00:06:50.350 --> 00:06:55.320 Okay, so now I can differentiate this. 00:06:55.320 --> 00:07:04.650 So I get that d/dx (x^r), now I'm going to use prime 00:07:04.650 --> 00:07:07.460 notation, because I don't want to keep on writing that d/dx 00:07:07.460 --> 00:07:10.580 here; (e^(r ln x))'. 00:07:13.890 --> 00:07:18.457 And now, what I can do is I can use the chain rule. 00:07:18.457 --> 00:07:20.290 The chain rule says that it's the derivative 00:07:20.290 --> 00:07:24.380 of this times the derivative of the function. 00:07:24.380 --> 00:07:29.500 So the derivative of the exponential is just itself. 00:07:29.500 --> 00:07:31.470 And the derivative of this guy here, 00:07:31.470 --> 00:07:35.400 well I'll write it out once, is (r ln x)'. 00:07:39.700 --> 00:07:42.250 So what's that equal to? 00:07:42.250 --> 00:07:45.890 Well, e^(r ln x) is is just x^r. 00:07:45.890 --> 00:07:53.870 And this derivative here is-- Well the derivative of r is 0. 00:07:53.870 --> 00:07:54.860 This is a constant. 00:07:54.860 --> 00:07:56.680 It just factors out. 00:07:56.680 --> 00:08:02.070 And ln x now has derivative-- What's the derivative of ln x? 00:08:02.070 --> 00:08:06.570 1/x, so this is going to be times r/x. 00:08:06.570 --> 00:08:10.180 And now, we rewrite it in the customary form, which is r, 00:08:10.180 --> 00:08:13.040 we put the r in front, x^(r-1). 00:08:13.040 --> 00:08:13.820 Okay? 00:08:13.820 --> 00:08:19.060 So I just derived the formula for you. 00:08:19.060 --> 00:08:23.310 And it didn't matter whether r was rational or irrational, 00:08:23.310 --> 00:08:25.320 it's the same proof. 00:08:25.320 --> 00:08:29.440 Okay so now I have to show you how method two works as well. 00:08:29.440 --> 00:08:34.550 So let's do method two, which we call 00:08:34.550 --> 00:08:39.280 logarithmic differentiation. 00:08:39.280 --> 00:08:43.764 And so here I'll use a symbol, say u, for x^r, 00:08:43.764 --> 00:08:44.930 and I'll take its logarithm. 00:08:44.930 --> 00:08:50.480 That's r ln x. 00:08:50.480 --> 00:08:51.830 And now I differentiate it. 00:08:51.830 --> 00:08:54.160 I'll leave that in the middle, because I 00:08:54.160 --> 00:08:55.979 want to remember the key property 00:08:55.979 --> 00:08:57.270 of logarithmic differentiation. 00:08:57.270 --> 00:08:58.747 But first I'll differentiate it. 00:08:58.747 --> 00:09:01.080 Later on, what I'm going to use is that this is the same 00:09:01.080 --> 00:09:02.620 as u'/u. 00:09:02.620 --> 00:09:06.470 This is one way of evaluating a logarithmic derivative. 00:09:06.470 --> 00:09:08.660 And then the other is to differentiate 00:09:08.660 --> 00:09:10.980 the explicit function that we have over here. 00:09:10.980 --> 00:09:16.790 And that is just, as we said, r/x. 00:09:16.790 --> 00:09:25.470 So now, I multiply through, and I get u' = ur/x which is just 00:09:25.470 --> 00:09:29.750 x^r r/x, which is just what we did before. 00:09:29.750 --> 00:09:30.520 It's r x^(r-1). 00:09:33.410 --> 00:09:36.790 Again, you can now see by comparing 00:09:36.790 --> 00:09:40.110 these two pieces of arithmetic that they're basically 00:09:40.110 --> 00:09:41.470 the same. 00:09:41.470 --> 00:09:43.510 Pretty much every time you convert to base e 00:09:43.510 --> 00:09:45.170 or you do logarithmic differentiation, 00:09:45.170 --> 00:09:46.836 it'll amount to the same thing, provided 00:09:46.836 --> 00:09:48.270 you don't get mixed up. 00:09:48.270 --> 00:09:51.720 You generally have to introduce a new symbol here. 00:09:51.720 --> 00:09:55.840 On the other hand, you're dealing with exponents there. 00:09:55.840 --> 00:10:00.990 It's worth it to know both points of view. 00:10:00.990 --> 00:10:07.470 All right, so now I want to make one last remark before we 00:10:07.470 --> 00:10:09.910 finish with exponents. 00:10:09.910 --> 00:10:16.120 And, I'll try to sell this to you in a lot of ways 00:10:16.120 --> 00:10:19.490 as the course goes on, but one thing that I 00:10:19.490 --> 00:10:23.370 want to try to emphasize is that the natural logarithm really 00:10:23.370 --> 00:10:27.210 is natural. 00:10:27.210 --> 00:10:39.920 So, I claim that the natural log is natural. 00:10:39.920 --> 00:10:45.900 And the example that we're going to use for this illustration 00:10:45.900 --> 00:10:53.340 is economics. 00:10:53.340 --> 00:10:54.090 Okay? 00:10:54.090 --> 00:10:58.520 So let me explain to why the natural log is the one that's 00:10:58.520 --> 00:11:00.820 natural for economics. 00:11:00.820 --> 00:11:06.040 If you are imagining the price of a stock that you own 00:11:06.040 --> 00:11:11.160 goes down by a dollar, that's a totally meaningless statement. 00:11:11.160 --> 00:11:13.439 It depends on a lot of things. 00:11:13.439 --> 00:11:15.730 In particular, it depends on whether the original price 00:11:15.730 --> 00:11:18.300 was a dollar or 100 dollars. 00:11:18.300 --> 00:11:22.130 So there's not much meaning to these absolute numbers. 00:11:22.130 --> 00:11:25.080 It's always the ratios that matter. 00:11:25.080 --> 00:11:29.280 So, for example, I just looked up an hour ago, 00:11:29.280 --> 00:11:42.050 the London Exchange closed, and it was down 27.9, 00:11:42.050 --> 00:11:44.480 which as I said, is pretty meaningless 00:11:44.480 --> 00:11:50.050 unless you know what the actual total of this index is. 00:11:50.050 --> 00:11:54.200 It turns out it was 6,432. 00:11:54.200 --> 00:11:57.070 So the change in the price, divided 00:11:57.070 --> 00:12:03.980 by the price, which in this case is 27.9 / 6,432, 00:12:03.980 --> 00:12:07.550 is what matters. 00:12:07.550 --> 00:12:11.791 And, in this case, it happens to be .43%. 00:12:11.791 --> 00:12:12.290 All right? 00:12:12.290 --> 00:12:14.270 That's what happened today. 00:12:14.270 --> 00:12:18.410 And similarly, if you take the infinitesimal of this, 00:12:18.410 --> 00:12:21.260 people think of days as being relatively small increments 00:12:21.260 --> 00:12:23.900 when you're investing in a stock, 00:12:23.900 --> 00:12:27.240 you would be interested in the infinitesimal sense, 00:12:27.240 --> 00:12:28.690 you would be interested in p'/p. 00:12:28.690 --> 00:12:33.080 The derivative of p divided by p. 00:12:33.080 --> 00:12:35.530 That's just (ln p)'. 00:12:38.160 --> 00:12:42.275 So this is the - let me put a little box around it 00:12:42.275 --> 00:12:45.460 - the formula of logarithmic differentiation. 00:12:45.460 --> 00:12:49.700 But let me just emphasize that it has an actual significance, 00:12:49.700 --> 00:12:52.430 and it's the one that's used by economists and people who 00:12:52.430 --> 00:12:54.450 are modeling prices of things all the time. 00:12:54.450 --> 00:12:58.620 They never use absolute prices when there are large swings. 00:12:58.620 --> 00:13:01.010 They always use the log of the price. 00:13:01.010 --> 00:13:07.010 And there's no point in using log base 10, or log base 2. 00:13:07.010 --> 00:13:08.180 Those give you junk. 00:13:08.180 --> 00:13:11.190 They give you an extra factor of log 2. 00:13:11.190 --> 00:13:14.870 It's the natural log that's the obvious one to use. 00:13:14.870 --> 00:13:18.000 It's completely straightforward that this 00:13:18.000 --> 00:13:21.010 is a simpler expression than using log base 10 00:13:21.010 --> 00:13:24.030 and having a factor of natural log of 10 there, 00:13:24.030 --> 00:13:26.800 which would just mess everything up. 00:13:26.800 --> 00:13:29.360 All right, so this is just one illustration. 00:13:29.360 --> 00:13:31.680 Anything that has to do with ratios 00:13:31.680 --> 00:13:36.160 is going to encounter logarithms. 00:13:36.160 --> 00:13:41.270 All right, so that's pretty much it. 00:13:41.270 --> 00:13:45.822 That's all I want to say for now anyway. 00:13:45.822 --> 00:13:47.280 There's lots more to say, but we'll 00:13:47.280 --> 00:13:50.459 be saying it when we do applications of derivatives 00:13:50.459 --> 00:13:51.250 in the second unit. 00:13:51.250 --> 00:13:54.450 So now, what I'd like to do is to start a review. 00:13:54.450 --> 00:13:57.790 I'm just going to run through what we did in this unit. 00:13:57.790 --> 00:13:59.700 I'll tell you approximately what I 00:13:59.700 --> 00:14:06.150 expect from you on the test that's coming up tomorrow. 00:14:06.150 --> 00:14:14.640 And, well, so let's get started with that. 00:14:14.640 --> 00:14:27.050 So this is a review of Unit One. 00:14:27.050 --> 00:14:32.610 And I'm just going to put on the board all of the things 00:14:32.610 --> 00:14:35.750 that you need to think about, anyway, keep in your head. 00:14:35.750 --> 00:14:41.750 And there are what are called general formulas 00:14:41.750 --> 00:14:45.070 for derivatives. 00:14:45.070 --> 00:14:51.970 And then there are the specific ones. 00:14:51.970 --> 00:14:55.920 And let me just remind you what the general formulas are. 00:14:55.920 --> 00:14:58.750 There's what you do to differentiate 00:14:58.750 --> 00:15:04.190 a sum, a multiple of a function, the product 00:15:04.190 --> 00:15:08.190 rule, the quotient rule. 00:15:08.190 --> 00:15:11.550 Those are several general formulas. 00:15:11.550 --> 00:15:13.080 And then there's one more, which is 00:15:13.080 --> 00:15:15.780 the chain rule, which I'm going to say just 00:15:15.780 --> 00:15:17.450 a little bit more about. 00:15:17.450 --> 00:15:21.200 So the derivative of a function of a function 00:15:21.200 --> 00:15:26.380 is the derivative of the function times the derivative 00:15:26.380 --> 00:15:27.430 of the other function. 00:15:27.430 --> 00:15:33.780 So here, I've abbreviated u is u(x). 00:15:33.780 --> 00:15:36.630 Right, so this is one of two ways of writing it. 00:15:36.630 --> 00:15:39.850 The other way is also one that you can keep in mind 00:15:39.850 --> 00:15:42.470 and you might find easier to remember. 00:15:42.470 --> 00:15:46.690 It's probably a good idea to remember both formulas. 00:15:46.690 --> 00:15:49.660 And then the last type of general formula 00:15:49.660 --> 00:15:56.950 that we did was implicit differentiation. 00:15:56.950 --> 00:15:59.200 Okay? 00:15:59.200 --> 00:16:03.190 So when you do implicit differentiation, 00:16:03.190 --> 00:16:06.530 you have an equation and you don't 00:16:06.530 --> 00:16:09.270 try to solve for the unknown function. 00:16:09.270 --> 00:16:13.110 You just put it in its simplest form and you differentiate. 00:16:13.110 --> 00:16:20.440 So, we actually did this, in particular, for inverses. 00:16:20.440 --> 00:16:23.520 That was a very, very key method for calculating 00:16:23.520 --> 00:16:25.180 the inverses of functions. 00:16:25.180 --> 00:16:28.600 And it's also true that logarithmic differentiation 00:16:28.600 --> 00:16:31.420 is of this type. 00:16:31.420 --> 00:16:33.299 This is a transformation. 00:16:33.299 --> 00:16:34.840 We're differentiating something else. 00:16:34.840 --> 00:16:37.920 We're transforming the equation by taking its logarithm 00:16:37.920 --> 00:16:40.980 and then differentiating. 00:16:40.980 --> 00:16:45.200 Okay, so there are a number of different ways this is applied. 00:16:45.200 --> 00:16:48.450 It can also be applied, anyway, these are two of them. 00:16:48.450 --> 00:16:50.320 So maybe in parenthesis. 00:16:50.320 --> 00:16:53.120 These are just examples. 00:16:53.120 --> 00:16:54.350 All right. 00:16:54.350 --> 00:16:59.461 I'll try to give examples of at least a few of these rules 00:16:59.461 --> 00:16:59.960 later. 00:16:59.960 --> 00:17:05.670 So now, the specific functions that you know how 00:17:05.670 --> 00:17:08.360 to differentiate: well you know how to differentiate now x^r 00:17:08.360 --> 00:17:11.410 thanks to what I just did. 00:17:11.410 --> 00:17:15.030 We have the sine and the cosine functions, 00:17:15.030 --> 00:17:17.910 which you're responsible for knowing 00:17:17.910 --> 00:17:19.500 what their derivatives are. 00:17:19.500 --> 00:17:26.490 And then other trig functions like tan and secant. 00:17:26.490 --> 00:17:29.810 We generally don't bother with cosecants and cotangents, 00:17:29.810 --> 00:17:32.710 because everything can be expressed in terms of these 00:17:32.710 --> 00:17:33.759 anyway. 00:17:33.759 --> 00:17:35.550 Actually, you can really express everything 00:17:35.550 --> 00:17:36.906 in terms of sines and cosines. 00:17:36.906 --> 00:17:38.280 But what you'll find is that it's 00:17:38.280 --> 00:17:41.660 much more convenient to remember the derivatives of these 00:17:41.660 --> 00:17:42.730 as well. 00:17:42.730 --> 00:17:45.870 So memorize all of these. 00:17:45.870 --> 00:17:49.810 All right, and then we had e^x and ln x. 00:17:49.810 --> 00:17:53.920 And we had the inverses of the trig functions. 00:17:53.920 --> 00:18:00.010 These were the two that we did: the arctangent and the arcsine. 00:18:00.010 --> 00:18:02.220 So those are the ones you're responsible for. 00:18:02.220 --> 00:18:06.970 You should have enough time, anyway, to work out anything 00:18:06.970 --> 00:18:09.390 else, if you know these. 00:18:09.390 --> 00:18:11.210 All right, so basically the idea is 00:18:11.210 --> 00:18:13.070 you have a bunch of special formulas. 00:18:13.070 --> 00:18:14.820 You have a bunch of general formulas. 00:18:14.820 --> 00:18:16.620 You put them together, and you can 00:18:16.620 --> 00:18:20.970 generate pretty much anything. 00:18:20.970 --> 00:18:24.810 Okay, so let's do a few examples before going on 00:18:24.810 --> 00:18:41.290 with the review. 00:18:41.290 --> 00:18:48.230 Okay, so I do want to do a few examples in sort of increasing 00:18:48.230 --> 00:18:50.170 level of difficulty in how you would 00:18:50.170 --> 00:18:51.420 combine these things together. 00:18:51.420 --> 00:18:55.980 So first of all, you should remember 00:18:55.980 --> 00:19:00.885 that if you differentiate the secant function, that's just 00:19:00.885 --> 00:19:03.780 - oh I just realized that I wanted to say something 00:19:03.780 --> 00:19:06.630 else before - so forget that. 00:19:06.630 --> 00:19:08.060 We'll do that in a second. 00:19:08.060 --> 00:19:10.980 I wanted to make some general remarks. 00:19:10.980 --> 00:19:17.600 So there's one rule that you discussed in my absence, which 00:19:17.600 --> 00:19:19.070 is the chain rule. 00:19:19.070 --> 00:19:21.780 And I do want to make just a couple of remarks 00:19:21.780 --> 00:19:26.160 about the chain rule now to remind you of what it is, 00:19:26.160 --> 00:19:30.160 and also to present some consequences. 00:19:30.160 --> 00:19:39.190 So, a little bit of extra on the chain rule. 00:19:39.190 --> 00:19:43.720 The first thing that I want say is that we didn't really 00:19:43.720 --> 00:19:46.660 fully explain why it's true. 00:19:46.660 --> 00:19:54.140 And I do want to just explain it by example, okay? 00:19:54.140 --> 00:19:59.720 So imagine that you have a function which is, say, 00:19:59.720 --> 00:20:02.000 10x + b. 00:20:02.000 --> 00:20:02.500 All right? 00:20:02.500 --> 00:20:04.980 So y = 10x + b. 00:20:04.980 --> 00:20:09.970 Then obviously, y is changing 10 times as fast as b, right? 00:20:09.970 --> 00:20:18.060 The issue is this number here, dy/dx, is 10. 00:20:18.060 --> 00:20:20.460 And now if x is a function of something, 00:20:20.460 --> 00:20:34.290 say t, shifted by some other constant here, then dx/dt = 5. 00:20:34.290 --> 00:20:38.800 Now all the chain rule is saying is that if y is going 10 times 00:20:38.800 --> 00:20:44.610 as fast as t, I'm sorry as x, and x is going 5 times 00:20:44.610 --> 00:20:50.620 as fast as t, then y is going 50 times as fast as t. 00:20:50.620 --> 00:20:53.530 And algebraically, all this means is if I plug 00:20:53.530 --> 00:20:57.100 in and substitute, which is what the composition of the two 00:20:57.100 --> 00:21:04.760 functions amounts to, 10(5t + a) + b and I multiply it out, 00:21:04.760 --> 00:21:09.200 I get 50t + 10a + b. 00:21:09.200 --> 00:21:11.697 Now these terms don't matter. 00:21:11.697 --> 00:21:13.030 The constant terms don't matter. 00:21:13.030 --> 00:21:14.800 The rate is 50. 00:21:14.800 --> 00:21:17.130 And so the consequence, if we put them together, 00:21:17.130 --> 00:21:30.170 is that dy/dt = 10*5, which is 50. 00:21:30.170 --> 00:21:31.990 All right, so this is in a nutshell 00:21:31.990 --> 00:21:33.630 why the chain rule works. 00:21:33.630 --> 00:21:39.450 And why these rates multiply. 00:21:39.450 --> 00:21:42.460 The second thing that I wanted to say about the chain rule 00:21:42.460 --> 00:21:45.050 is that it has a few consequences that 00:21:45.050 --> 00:21:47.380 make some of the other rules a little easier 00:21:47.380 --> 00:21:50.220 to remember or possibly to avoid. 00:21:50.220 --> 00:21:54.510 The messiest rule in my humble opinion 00:21:54.510 --> 00:21:59.430 is the quotient rule, which is kind of a nuisance to remember. 00:21:59.430 --> 00:22:01.150 So let me just remind you, if you 00:22:01.150 --> 00:22:03.840 take just the reciprocal of a function, 00:22:03.840 --> 00:22:05.990 and you differentiate it, there's 00:22:05.990 --> 00:22:08.170 another way of looking at this. 00:22:08.170 --> 00:22:09.720 And it's actually the way that I use, 00:22:09.720 --> 00:22:12.800 so I want to encourage you to think about it this way too. 00:22:12.800 --> 00:22:15.670 This is the same as (v^(-1))'. 00:22:15.670 --> 00:22:16.700 . 00:22:16.700 --> 00:22:18.840 And now instead of using the quotient rule, which 00:22:18.840 --> 00:22:23.840 we could've used, we can use the chain rule here 00:22:23.840 --> 00:22:29.720 with the power -1, which works by the power law. 00:22:29.720 --> 00:22:30.960 So what is this equal to? 00:22:30.960 --> 00:22:33.640 This is equal to -v^(-2) v'. 00:22:38.930 --> 00:22:42.730 So here, I've applied the chain rule rather than 00:22:42.730 --> 00:22:47.370 the quotient rule. 00:22:47.370 --> 00:22:54.060 And similarly, suppose I wanted to derive the full quotient 00:22:54.060 --> 00:22:54.560 rule. 00:22:54.560 --> 00:22:57.240 Well, now this may or may not be easier. 00:22:57.240 --> 00:22:59.770 But this is one way of remembering what's going on. 00:22:59.770 --> 00:23:05.300 If you convert it to uv^(-1) and you differentiate that, 00:23:05.300 --> 00:23:09.180 now I can use the product rule on this. 00:23:09.180 --> 00:23:11.970 Of course, I have to use the chain rule and this rule 00:23:11.970 --> 00:23:13.030 as well. 00:23:13.030 --> 00:23:15.620 So what do I get? 00:23:15.620 --> 00:23:21.519 I get u', the inverse, plus u, and then I have 00:23:21.519 --> 00:23:22.810 to differentiate the v inverse. 00:23:22.810 --> 00:23:24.490 That's the formula right up here. 00:23:24.490 --> 00:23:25.290 That's -v^(-2) v'. 00:23:30.300 --> 00:23:33.230 So that's one way of doing it. 00:23:33.230 --> 00:23:35.760 This actually explains the funny minus sign 00:23:35.760 --> 00:23:38.560 when you differentiate v in the formula. 00:23:38.560 --> 00:23:41.280 The other formula, the other way that we did it, 00:23:41.280 --> 00:23:44.370 was by putting this over a common denominator. 00:23:44.370 --> 00:23:49.330 The common denominator was v^2. 00:23:49.330 --> 00:23:51.580 This comes from this v v^(-2). 00:23:51.580 --> 00:23:54.730 And then the second term is -uv'. 00:23:57.250 --> 00:24:00.020 And the first term, we have to multiply by an extra factor 00:24:00.020 --> 00:24:02.190 of v, because we have a v^2 in the denominator. 00:24:02.190 --> 00:24:07.720 So it's u'v. All right, so this is the quotient rule as we 00:24:07.720 --> 00:24:11.247 wrote it down in lecture, and this is just another way 00:24:11.247 --> 00:24:13.580 of remembering it or deriving it without remembering it, 00:24:13.580 --> 00:24:16.700 if you just remember the chain rule and the product rule. 00:24:16.700 --> 00:24:19.710 Okay, so you'll find that in many contexts, 00:24:19.710 --> 00:24:25.910 it's easier to do one or the other. 00:24:25.910 --> 00:24:29.210 Okay, so now I'm ready to differentiate the secant 00:24:29.210 --> 00:24:30.990 and a few such functions. 00:24:30.990 --> 00:24:36.200 So we'll do some examples here here. 00:24:36.200 --> 00:24:39.030 So here's the secant function, and I 00:24:39.030 --> 00:24:44.820 want to use that formula up there for the reciprocal. 00:24:44.820 --> 00:24:48.090 This is the way I think of it. 00:24:48.090 --> 00:24:53.150 This is the cosine function to the power -1. 00:24:53.150 --> 00:24:58.750 And so, the formula here is just what? 00:24:58.750 --> 00:25:04.030 It's just -(cos x)^(-2) times -sin x. 00:25:20.280 --> 00:25:22.550 So now this is usually written in a different fashion, 00:25:22.550 --> 00:25:25.170 so that's why I'm doing this for a reason actually. 00:25:25.170 --> 00:25:27.810 Which is although there are several formulas for things, 00:25:27.810 --> 00:25:29.810 with trig functions, there are usually 00:25:29.810 --> 00:25:31.854 five ways of writing something. 00:25:31.854 --> 00:25:33.520 So I'm writing this one down so that you 00:25:33.520 --> 00:25:36.780 know what the standard way of presenting it is. 00:25:36.780 --> 00:25:39.760 So what happens here is that we have two minus signs 00:25:39.760 --> 00:25:40.300 cancelling. 00:25:40.300 --> 00:25:44.360 And we get sin x / cos^2 x. 00:25:44.360 --> 00:25:46.430 That's a perfectly acceptable answer, 00:25:46.430 --> 00:25:49.470 but there's a customary way in which is written. 00:25:49.470 --> 00:25:55.890 It's written (1 / cos x) (sin x / cos x). 00:25:55.890 --> 00:25:57.530 And then we get rid of the denominators 00:25:57.530 --> 00:26:00.710 by rewriting it in terms of secant and tangent, 00:26:00.710 --> 00:26:04.100 so sec x tan x. 00:26:04.100 --> 00:26:07.680 So this is the form that's generally 00:26:07.680 --> 00:26:11.790 used when you see these formulas written in textbooks. 00:26:11.790 --> 00:26:14.440 And so you know, you need to watch out, 00:26:14.440 --> 00:26:16.810 because if you ever want to use this kind of calculus, 00:26:16.810 --> 00:26:22.840 you'll have not be put off by all the secants and tangents. 00:26:22.840 --> 00:26:26.830 All right, so getting slightly more complicated, 00:26:26.830 --> 00:26:28.750 how about if we differentiate ln(sec x)? 00:26:37.400 --> 00:26:39.560 If you differentiate the natural log, 00:26:39.560 --> 00:26:49.450 that's just going to be (sec x)' / sec x. 00:26:49.450 --> 00:26:51.250 And plugging in the formula that we 00:26:51.250 --> 00:27:00.330 had before, that's sec x tan x / sec x, which is tan x. 00:27:00.330 --> 00:27:03.850 So this one also has a very nice form. 00:27:03.850 --> 00:27:07.940 And you might say that this is kind of an ugly function, 00:27:07.940 --> 00:27:14.120 but the strange thing is that the natural log was invented 00:27:14.120 --> 00:27:19.030 before the exponential by a guy named Napier, exactly 00:27:19.030 --> 00:27:21.720 in order to evaluate functions like this. 00:27:21.720 --> 00:27:25.930 These are the functions that people cared about a lot, 00:27:25.930 --> 00:27:28.890 because they were used in navigation. 00:27:28.890 --> 00:27:32.640 You wanted to multiply sines and cosines together 00:27:32.640 --> 00:27:34.030 to do navigation. 00:27:34.030 --> 00:27:38.794 And the multiplication he encoded using a logarithm. 00:27:38.794 --> 00:27:40.710 So these were invented long before people even 00:27:40.710 --> 00:27:42.794 knew about exponents. 00:27:42.794 --> 00:27:44.460 And it was a surprise, actually, that it 00:27:44.460 --> 00:27:46.100 was connected to exponents. 00:27:46.100 --> 00:27:48.650 So the natural log was invented before the log base 10 00:27:48.650 --> 00:27:52.650 and everything else, exactly for this kind of purpose. 00:27:52.650 --> 00:27:54.550 Anyway, so this is a nice function, 00:27:54.550 --> 00:27:58.010 which was very important, so that your ships wouldn't 00:27:58.010 --> 00:28:03.770 crash into the reef. 00:28:03.770 --> 00:28:05.570 Okay, let's continue here. 00:28:05.570 --> 00:28:08.870 So there's another kind of function 00:28:08.870 --> 00:28:10.490 that I want to discuss with you. 00:28:10.490 --> 00:28:12.460 And these are the kinds in which there's 00:28:12.460 --> 00:28:19.380 a choice as to which of these rules to apply. 00:28:19.380 --> 00:28:25.130 And I'll just give a couple of examples of that. 00:28:25.130 --> 00:28:27.620 There usually is a better and a worse way, 00:28:27.620 --> 00:28:38.430 so let me illustrate that. 00:28:38.430 --> 00:28:41.120 Okay, yet another example. 00:28:41.120 --> 00:28:43.830 I hope you've seen some of these before. 00:28:43.830 --> 00:28:46.660 Say (x^10 + 8x)^6. 00:28:51.010 --> 00:28:52.990 So it's a little bit more complicated than what 00:28:52.990 --> 00:29:00.330 we had before, because there were several more symbols here. 00:29:00.330 --> 00:29:03.210 So what should we do at this point? 00:29:03.210 --> 00:29:06.210 There's one choice which I claim is a bad idea, 00:29:06.210 --> 00:29:10.810 and that is to expand this out to the 6th power. 00:29:10.810 --> 00:29:13.530 That's a bad idea, because it's very long. 00:29:13.530 --> 00:29:15.990 And then your answer will also be very long. 00:29:15.990 --> 00:29:19.521 It will fill the entire exam paper, for instance. 00:29:19.521 --> 00:29:20.020 Yeah? 00:29:20.020 --> 00:29:21.380 STUDENT: Can you use the chain rule? 00:29:21.380 --> 00:29:21.970 PROFESSOR: Chain rule. 00:29:21.970 --> 00:29:22.580 That's it. 00:29:22.580 --> 00:29:23.500 We use the chain rule. 00:29:23.500 --> 00:29:26.620 So fortunately, this is relatively easy 00:29:26.620 --> 00:29:27.620 using the chain rule. 00:29:27.620 --> 00:29:30.790 We just think of this box as being the function. 00:29:30.790 --> 00:29:34.560 And we take 6 times this guy to the 5th, 00:29:34.560 --> 00:29:37.570 times the derivative of this guy, which is 10x^9 + 8. 00:29:41.500 --> 00:29:43.910 And this is, filling this in, it's x^10 + 8x. 00:29:43.910 --> 00:29:46.140 And that's it. 00:29:46.140 --> 00:29:50.270 That's all you need to do differentiate things like this. 00:29:50.270 --> 00:29:55.140 The chain rule is very effective. 00:29:55.140 --> 00:29:59.864 STUDENT: [INAUDIBLE] 00:29:59.864 --> 00:30:01.280 PROFESSOR: That's a good question. 00:30:01.280 --> 00:30:04.330 So I'm not really willing to answer too many questions 00:30:04.330 --> 00:30:07.500 about what's going to be on the exam. 00:30:07.500 --> 00:30:09.070 But the question that was just asked 00:30:09.070 --> 00:30:13.200 is exactly the kind of question I'm very happy to answer. 00:30:13.200 --> 00:30:18.350 Okay, the question was, in what form is-- what 00:30:18.350 --> 00:30:20.090 form is an acceptable answer? 00:30:20.090 --> 00:30:23.560 Now in real life, that is a really serious question. 00:30:23.560 --> 00:30:25.200 When you ask a computer a question 00:30:25.200 --> 00:30:28.930 and it gives you 500 million sheets of printout, 00:30:28.930 --> 00:30:31.380 it's useless. 00:30:31.380 --> 00:30:33.960 And you really care what form answers are in, 00:30:33.960 --> 00:30:35.730 and indeed, somebody might really 00:30:35.730 --> 00:30:39.000 care what this thing to the 6th power is, 00:30:39.000 --> 00:30:42.090 and then you would be forced to discuss things in terms 00:30:42.090 --> 00:30:46.110 of that other functional form. 00:30:46.110 --> 00:30:50.410 For the purposes of this exam, this is okay form. 00:30:50.410 --> 00:30:54.490 And, in fact, any correct form is an okay form. 00:30:54.490 --> 00:30:57.770 I recommend strongly that you not try to simplify things 00:30:57.770 --> 00:30:59.700 unless we tell you to. 00:30:59.700 --> 00:31:04.860 Sometimes it will be to your advantage to simplify things. 00:31:04.860 --> 00:31:08.010 Sometimes we'll say simplify. 00:31:08.010 --> 00:31:10.390 It takes a good deal of experience 00:31:10.390 --> 00:31:13.121 to know when it's really worth it to simplify expressions. 00:31:13.121 --> 00:31:13.620 Yes? 00:31:13.620 --> 00:31:19.530 STUDENT: [INAUDIBLE] 00:31:19.530 --> 00:31:23.590 PROFESSOR: Right, so turning to this example. 00:31:23.590 --> 00:31:25.520 The question is what is this derivative? 00:31:25.520 --> 00:31:27.240 And here's an answer. 00:31:27.240 --> 00:31:29.500 That's the end of the problem. 00:31:29.500 --> 00:31:31.810 This is a more customary form. 00:31:31.810 --> 00:31:37.160 But this is answer is okay. 00:31:37.160 --> 00:31:38.610 Same issue. 00:31:38.610 --> 00:31:40.970 That's exactly the point. 00:31:40.970 --> 00:31:41.660 Yes? 00:31:41.660 --> 00:31:51.460 STUDENT: [INAUDIBLE] 00:31:51.460 --> 00:31:59.032 PROFESSOR: The question is, do you have to show the work? 00:31:59.032 --> 00:32:00.240 Do you have to show the work? 00:32:00.240 --> 00:32:04.870 Well if I ask you what is d/dx of sec x, 00:32:04.870 --> 00:32:06.650 then if you wrote down this answer 00:32:06.650 --> 00:32:09.510 or you wrote down this answer showing no work, 00:32:09.510 --> 00:32:11.200 that would be acceptable. 00:32:11.200 --> 00:32:15.950 If the question was derive the formula for this 00:32:15.950 --> 00:32:18.650 from the formula for the derivative of the cosine 00:32:18.650 --> 00:32:21.160 or something like that, then it would not be acceptable. 00:32:21.160 --> 00:32:24.340 You'd have to carry out this arithmetic. 00:32:24.340 --> 00:32:28.470 So, in other words, typically this 00:32:28.470 --> 00:32:32.290 will come up, for instance, in various contexts. 00:32:32.290 --> 00:32:34.830 You just basically have to follow directions. 00:32:34.830 --> 00:32:35.330 Yes? 00:32:35.330 --> 00:32:41.424 STUDENT: [INAUDIBLE] 00:32:41.424 --> 00:32:43.090 PROFESSOR: The next question is, are you 00:32:43.090 --> 00:32:44.465 expected to be able to prove what 00:32:44.465 --> 00:32:46.180 the derivative of the sine function is? 00:32:46.180 --> 00:32:49.580 The short answer to that is yes. 00:32:49.580 --> 00:32:51.630 But I will be getting to that when I discuss 00:32:51.630 --> 00:32:54.240 the rest of the material here. 00:32:54.240 --> 00:32:58.430 We're almost there. 00:32:58.430 --> 00:33:02.640 Okay, so let me just finish these examples 00:33:02.640 --> 00:33:04.880 with one last one. 00:33:04.880 --> 00:33:06.880 And then we'll talk about this question 00:33:06.880 --> 00:33:10.630 of things like the derivative of the sine function, 00:33:10.630 --> 00:33:12.060 and deriving it. 00:33:12.060 --> 00:33:15.620 So the last example that I'd like to write down is the one 00:33:15.620 --> 00:33:18.940 that I promised you in the first lecture, 00:33:18.940 --> 00:33:26.172 namely to differentiate e^(x tan^(-1) x). 00:33:26.172 --> 00:33:28.380 Basically you're supposed to be able to differentiate 00:33:28.380 --> 00:33:29.350 any function. 00:33:29.350 --> 00:33:32.390 So this is the one that we mentioned at the beginning. 00:33:32.390 --> 00:33:34.130 So here it is. 00:33:34.130 --> 00:33:37.280 Let's do it. 00:33:37.280 --> 00:33:38.170 So what is it? 00:33:38.170 --> 00:33:45.742 Well, it's just equal to - I have to differentiate. 00:33:45.742 --> 00:33:47.200 I have to use the chain rule - it's 00:33:47.200 --> 00:33:52.930 equal to the exponential times the derivative 00:33:52.930 --> 00:33:58.200 of this expression here. 00:33:58.200 --> 00:33:59.260 That's the chain rule. 00:33:59.260 --> 00:34:01.700 That's the first step. 00:34:01.700 --> 00:34:06.440 And now I have to apply the product rule here. 00:34:06.440 --> 00:34:10.820 So I have e^(x tan^(-1) x). 00:34:10.820 --> 00:34:15.809 And I differentiate the first factor, so I get tan^(-1) x. 00:34:15.809 --> 00:34:17.600 Add to it what happens when I differentiate 00:34:17.600 --> 00:34:19.670 the second factor, leaving alone the x. 00:34:19.670 --> 00:34:21.450 So that's x / (1+x^2). 00:34:24.310 --> 00:34:26.300 And that's it. 00:34:26.300 --> 00:34:28.780 That's the end of the problem. 00:34:28.780 --> 00:34:30.590 It wasn't that hard. 00:34:30.590 --> 00:34:35.330 Of course, it requires you to remember all of the rules, 00:34:35.330 --> 00:34:37.300 and a lot of formulas underlying them. 00:34:37.300 --> 00:34:39.560 So that's consistent with what I just told you. 00:34:39.560 --> 00:34:42.060 I told you that you wanted to know this. 00:34:42.060 --> 00:34:44.740 I told you that you needed to know this product rule, 00:34:44.740 --> 00:34:50.419 and that you needed to know the chain rule. 00:34:50.419 --> 00:34:51.960 And I guess there was one more thing, 00:34:51.960 --> 00:34:55.260 the derivative of e^x came into play there. 00:34:55.260 --> 00:34:59.040 So of these formulas, we used four of them 00:34:59.040 --> 00:35:03.810 in this one calculation. 00:35:03.810 --> 00:35:15.880 Okay, so now what other things did we talk about in Unit One? 00:35:15.880 --> 00:35:23.590 So the main other thing that we talked about 00:35:23.590 --> 00:35:33.120 was the definition of a derivative. 00:35:33.120 --> 00:35:40.020 And also there was sort of a goal 00:35:40.020 --> 00:35:51.050 which was to get to the meaning of the derivative. 00:35:51.050 --> 00:35:56.520 So these are things - so we had a couple of ways of looking 00:35:56.520 --> 00:35:59.170 at it, or at least a couple that I'm 00:35:59.170 --> 00:36:01.780 going to emphasize right now. 00:36:01.780 --> 00:36:06.270 But first, let me remind you what the formula is. 00:36:06.270 --> 00:36:13.900 The derivative is the limit as delta x goes to 0 of (f(x + 00:36:13.900 --> 00:36:19.040 delta x) - f(x)) / delta x. 00:36:19.040 --> 00:36:22.430 So that's it, and this is certainly 00:36:22.430 --> 00:36:25.640 going to be a central focus here. 00:36:25.640 --> 00:36:29.600 And you want to be able to recognize this formula 00:36:29.600 --> 00:36:42.760 in a number of ways. 00:36:42.760 --> 00:36:44.520 So, how do we use this? 00:36:44.520 --> 00:36:48.950 Well one thing we did was we calculated a bunch 00:36:48.950 --> 00:36:51.450 of these rates of change. 00:36:51.450 --> 00:36:53.580 In fact, more or less, they're the ones which 00:36:53.580 --> 00:36:55.760 are written right over here. 00:36:55.760 --> 00:36:57.210 This list of functions here. 00:36:57.210 --> 00:37:01.470 Now, which ones did we start out with just straight 00:37:01.470 --> 00:37:03.800 from the definition here? 00:37:03.800 --> 00:37:04.840 Which of these things? 00:37:04.840 --> 00:37:06.215 There were a whole bunch of them. 00:37:06.215 --> 00:37:09.180 So we started out with a function 1/x. 00:37:09.180 --> 00:37:11.530 We did x^n. 00:37:11.530 --> 00:37:14.530 We did sine x. 00:37:14.530 --> 00:37:16.880 We did cosine x. 00:37:16.880 --> 00:37:19.205 Now there was a little bit of subtlety with sine 00:37:19.205 --> 00:37:21.110 x and cosine x. 00:37:21.110 --> 00:37:25.210 We got them using something else. 00:37:25.210 --> 00:37:26.880 We didn't quite get them all the way. 00:37:26.880 --> 00:37:31.790 We got them using the case x = 0. 00:37:31.790 --> 00:37:34.530 We got them from the derivative at x = 0, 00:37:34.530 --> 00:37:37.680 we got the formulas for the derivatives of sine and cosine. 00:37:37.680 --> 00:37:40.890 But that was an argument which involved plugging in sin 00:37:40.890 --> 00:37:44.460 (x + delta x), and running through. 00:37:44.460 --> 00:37:45.840 So that's one example. 00:37:45.840 --> 00:37:50.630 We also did a^x. 00:37:50.630 --> 00:37:53.270 And that may be it. 00:37:53.270 --> 00:37:58.350 Oh yeah, I think that's about it. 00:37:58.350 --> 00:38:00.450 That may be about it. 00:38:00.450 --> 00:38:00.950 No. 00:38:00.950 --> 00:38:01.620 It isn't. 00:38:01.620 --> 00:38:03.910 Okay, so let me make a connection here which you 00:38:03.910 --> 00:38:07.770 probably haven't yet made, which is that we did it for (u v)'. 00:38:10.520 --> 00:38:15.690 And we also did it for (u / v)'. 00:38:15.690 --> 00:38:17.460 So sorry, I shouldn't write primes, 00:38:17.460 --> 00:38:20.500 because that's not consistent with the claim there. 00:38:20.500 --> 00:38:24.900 I differentiated the product; I differentiated the quotient 00:38:24.900 --> 00:38:28.110 using the same delta x notation. 00:38:28.110 --> 00:38:32.760 I guess I forgot that because I wasn't there when it happened. 00:38:32.760 --> 00:38:36.550 So look, these are the ones that you do by this. 00:38:36.550 --> 00:38:39.310 And, of course, you might have to reduce them to other things. 00:38:39.310 --> 00:38:42.190 These involve using something else. 00:38:42.190 --> 00:38:46.610 This one involves using the slope of this function at 0, 00:38:46.610 --> 00:38:48.600 just the way the sine and the cosine did. 00:38:48.600 --> 00:38:52.410 This one involves the slopes of the individual functions, u 00:38:52.410 --> 00:38:54.827 and v. And this one also involves the individual-- 00:38:54.827 --> 00:38:56.410 So, in other words, it doesn't get you 00:38:56.410 --> 00:38:58.390 all the way through to the end, but it's 00:38:58.390 --> 00:39:03.010 expressed in terms of something simpler in each of these cases. 00:39:03.010 --> 00:39:05.840 And I could go on. 00:39:05.840 --> 00:39:09.280 We didn't do these in class, but you're certainly-- 00:39:09.280 --> 00:39:12.170 e^x is a perfectly okay one on one of the exams. 00:39:12.170 --> 00:39:14.561 We ask you for 1/x^2. 00:39:14.561 --> 00:39:16.310 In other words, I'm not claiming that it's 00:39:16.310 --> 00:39:18.380 going to be one on this list, but it certainly 00:39:18.380 --> 00:39:19.671 can be any one of these. 00:39:19.671 --> 00:39:21.170 But we're not going to ask you to go 00:39:21.170 --> 00:39:26.380 all the way through to the beginning in these formulas. 00:39:26.380 --> 00:39:28.940 There are also some fundamental limits that I certainly 00:39:28.940 --> 00:39:31.210 want you to know about. 00:39:31.210 --> 00:39:34.680 And these you can derive in reverse. 00:39:34.680 --> 00:39:58.880 So I will describe that now. 00:39:58.880 --> 00:40:06.800 So let me also emphasize the following thing: I want 00:40:06.800 --> 00:40:18.590 to read this backwards now. 00:40:18.590 --> 00:40:21.370 This is the theme from the very beginning of this lecture. 00:40:21.370 --> 00:40:25.210 Namely, if you're given the function f, 00:40:25.210 --> 00:40:27.811 you can figure out its derivative by this formula 00:40:27.811 --> 00:40:28.310 here. 00:40:28.310 --> 00:40:29.860 That is the formula for this in terms of what's 00:40:29.860 --> 00:40:30.920 on the right hand side. 00:40:30.920 --> 00:40:34.400 On the other hand, you can also use 00:40:34.400 --> 00:40:45.280 the formula in that direction, and if you 00:40:45.280 --> 00:40:48.310 know the slope of something, you can figure out 00:40:48.310 --> 00:40:49.170 what the limit is. 00:40:49.170 --> 00:40:54.570 For example, I'll use the letter x here, 00:40:54.570 --> 00:40:56.040 even though it's cheating. 00:40:56.040 --> 00:40:59.530 Maybe I'll call it delta x so it's clearer to you. 00:40:59.530 --> 00:41:06.900 Maybe I'll call it u. 00:41:06.900 --> 00:41:10.370 Suppose you look at this limit here. 00:41:10.370 --> 00:41:14.800 Well, I claim that you should recognize that is 00:41:14.800 --> 00:41:19.870 the derivative with respect to u of the function e^u at u = 0, 00:41:19.870 --> 00:41:22.660 which of course we know to be 1. 00:41:22.660 --> 00:41:25.420 So this is reading this formula in reverse. 00:41:25.420 --> 00:41:27.940 It's recognizing that one of these limits - 00:41:27.940 --> 00:41:35.160 let me rewrite this again here - one of these so-called 00:41:35.160 --> 00:41:39.390 difference quotient limits is a derivative. 00:41:39.390 --> 00:41:42.190 And since we know a formula for that derivative, 00:41:42.190 --> 00:41:49.940 we can evaluate it. 00:41:49.940 --> 00:41:54.150 And lastly, there's one other type of thing 00:41:54.150 --> 00:41:57.550 which I think you should know. 00:41:57.550 --> 00:41:59.767 These are the ones you do with difference quotients. 00:41:59.767 --> 00:42:01.350 There are also other formulas that you 00:42:01.350 --> 00:42:03.000 want to be able to derive. 00:42:03.000 --> 00:42:19.740 You want to be able to derive formulas 00:42:19.740 --> 00:42:27.670 by implicit differentiation. 00:42:27.670 --> 00:42:30.220 In other words, the basic idea is 00:42:30.220 --> 00:42:32.150 to take whatever equation you've got 00:42:32.150 --> 00:42:36.560 and simplify it as much as possible, 00:42:36.560 --> 00:42:41.260 without insisting that you solve for y. 00:42:41.260 --> 00:42:44.080 That's not necessarily the most appropriate way 00:42:44.080 --> 00:42:45.630 to get the rate of change. 00:42:45.630 --> 00:42:51.910 The much simpler formula is sin y = x. 00:42:51.910 --> 00:42:59.780 And that one is easier to differentiate implicitly. 00:42:59.780 --> 00:43:02.900 So I should say, do this kind of thing. 00:43:02.900 --> 00:43:05.550 So that's, if you like, a typical derivation 00:43:05.550 --> 00:43:08.390 that you might see. 00:43:08.390 --> 00:43:13.070 And then there's one last type of problem that you'll face, 00:43:13.070 --> 00:43:21.590 and it's the other thing that I claim we discussed. 00:43:21.590 --> 00:43:26.580 And it goes all the way back to the first lecture. 00:43:26.580 --> 00:43:33.700 So the last thing that we'll be talking about is tangent lines. 00:43:33.700 --> 00:43:34.200 All right? 00:43:34.200 --> 00:43:38.760 The geometric point of view of a derivative. 00:43:38.760 --> 00:43:41.900 And we'll be doing more of this in next the unit. 00:43:41.900 --> 00:43:44.850 So first of all, you'll be expected 00:43:44.850 --> 00:43:52.380 to be able to compute the tangent line. 00:43:52.380 --> 00:43:56.400 That's often fairly straightforward. 00:43:56.400 --> 00:44:03.100 And the second thing is to graph y' , 00:44:03.100 --> 00:44:07.370 the derivative of a function. 00:44:07.370 --> 00:44:09.350 And the third thing, which I'm going 00:44:09.350 --> 00:44:11.550 to throw in here, because I regard it 00:44:11.550 --> 00:44:14.900 in a sort of geometric vein, although it's got 00:44:14.900 --> 00:44:16.690 an analytical aspect to it. 00:44:16.690 --> 00:44:18.870 So this is a picture. 00:44:18.870 --> 00:44:20.710 This is a computation. 00:44:20.710 --> 00:44:23.080 And if you combine the two together, 00:44:23.080 --> 00:44:24.270 you get something else. 00:44:24.270 --> 00:44:37.870 And so this is to recognize differentiable functions. 00:44:37.870 --> 00:44:40.190 Alright, so how do you do this? 00:44:40.190 --> 00:44:43.600 Well, we really only have one way of doing it. 00:44:43.600 --> 00:44:54.580 We're going to check the left and right tangents. 00:44:54.580 --> 00:44:59.450 They must be equal. 00:44:59.450 --> 00:45:04.280 So again, this is a property that you 00:45:04.280 --> 00:45:06.830 should be familiar with from some of your exercises. 00:45:06.830 --> 00:45:09.660 And the idea is simply, that if the tangent line exists, 00:45:09.660 --> 00:45:14.770 it's the same from the right and from the left. 00:45:14.770 --> 00:45:20.200 Okay, now I'm going to just do one example here 00:45:20.200 --> 00:45:25.450 from this sort of qualitative sketching skill 00:45:25.450 --> 00:45:27.289 to give you an example here. 00:45:27.289 --> 00:45:28.830 And what I'm going to do is I'm going 00:45:28.830 --> 00:45:34.750 to draw a graph of a function like this. 00:45:34.750 --> 00:45:38.490 And what I want to do underneath is draw 00:45:38.490 --> 00:45:41.600 the graph of the derivative. 00:45:41.600 --> 00:45:45.900 So this is the function y = f(x), 00:45:45.900 --> 00:45:48.350 and here I'm going to draw the graph of the function y = 00:45:48.350 --> 00:45:56.490 f'(x) right underneath it. 00:45:56.490 --> 00:46:00.330 So now, let's think about what it's supposed to look like. 00:46:00.330 --> 00:46:05.840 And the one step that you need to make in order to do this, 00:46:05.840 --> 00:46:08.660 is to draw a few tangent lines. 00:46:08.660 --> 00:46:13.210 I'm just going to draw one down here. 00:46:13.210 --> 00:46:18.730 And I'm going to draw one up here. 00:46:18.730 --> 00:46:22.740 Now, the tangent lines here - notice 00:46:22.740 --> 00:46:27.266 that the slope of these tangent lines are all positive. 00:46:27.266 --> 00:46:28.640 So everything I draw down here is 00:46:28.640 --> 00:46:33.880 going to be above the x-axis. 00:46:33.880 --> 00:46:36.190 Furthermore, as I go further to the left, 00:46:36.190 --> 00:46:37.830 they get steeper and steeper. 00:46:37.830 --> 00:46:39.500 So they're getting higher and higher. 00:46:39.500 --> 00:46:44.020 So the function is coming down like this. 00:46:44.020 --> 00:46:45.350 It starts up there. 00:46:45.350 --> 00:46:50.570 Maybe I'll draw it in green to illustrate the graph here. 00:46:50.570 --> 00:46:56.910 So that's this function here. 00:46:56.910 --> 00:46:59.750 As we get farther out, it's getting flatter and flatter. 00:46:59.750 --> 00:47:06.270 So it's leveling off, but above the axis like that. 00:47:06.270 --> 00:47:08.280 So one of the things to emphasize 00:47:08.280 --> 00:47:10.830 is, you should not expect the derivative 00:47:10.830 --> 00:47:12.090 to look like the function. 00:47:12.090 --> 00:47:13.320 It's totally different. 00:47:13.320 --> 00:47:17.280 It's keeping track at each point of its tangent line. 00:47:17.280 --> 00:47:19.780 On the other hand, you should get some kind of physical feel 00:47:19.780 --> 00:47:23.619 for it, and we'll be practicing this more in the next unit. 00:47:23.619 --> 00:47:25.660 So let me give you an example of a function which 00:47:25.660 --> 00:47:27.800 does exactly this. 00:47:27.800 --> 00:47:33.240 And it's the function y = ln x. 00:47:33.240 --> 00:47:38.560 If you differentiate it, you get y' = 1/x. 00:47:38.560 --> 00:47:44.630 And this plot above is, roughly speaking, the logarithm. 00:47:44.630 --> 00:47:50.230 And this plot underneath is the function 1/x. 00:47:50.230 --> 00:47:53.230 We still have time for one question. 00:47:53.230 --> 00:47:58.580 And so, fire away. 00:47:58.580 --> 00:48:03.207 Yes? 00:48:03.207 --> 00:48:04.040 STUDENT: [INAUDIBLE] 00:48:04.040 --> 00:48:05.580 PROFESSOR: The question is, can you 00:48:05.580 --> 00:48:09.770 show how you derive the inverse tangent of x. 00:48:09.770 --> 00:48:13.350 So that's in a lecture. 00:48:13.350 --> 00:48:17.060 I'm happy to do it right now, but it's going 00:48:17.060 --> 00:48:20.420 to take me a whole two minutes. 00:48:20.420 --> 00:48:27.560 So, here's how you do it. y = tan^(-1) x. 00:48:27.560 --> 00:48:30.230 And now this is hopeless to differentiate, 00:48:30.230 --> 00:48:34.720 so I rewrite it as tan y = x. 00:48:34.720 --> 00:48:38.440 And now I have to differentiate that. 00:48:38.440 --> 00:48:42.360 So when I differentiate it, I get 00:48:42.360 --> 00:48:44.260 the derivative of tan y with respect 00:48:44.260 --> 00:48:46.560 to x-- with respect to y. 00:48:46.560 --> 00:48:48.210 That's 1 / (1 + y^2) times y'. 00:48:51.120 --> 00:48:52.850 So this is a hard step. 00:48:52.850 --> 00:48:53.930 That's the chain rule. 00:48:53.930 --> 00:48:55.860 And on the left side I get 1. 00:48:55.860 --> 00:48:57.650 So I'm doing this super fast because we 00:48:57.650 --> 00:49:00.720 have thirty seconds left. 00:49:00.720 --> 00:49:02.840 But this is the hard step right here. 00:49:02.840 --> 00:49:04.900 And it needs for you to know that d/dy tan 00:49:04.900 --> 00:49:14.432 y is equal to one over-- Oh, bad bad bad, secant squared. 00:49:14.432 --> 00:49:22.810 I was ahead of myself so fast. 00:49:22.810 --> 00:49:24.920 So here's the identity. 00:49:24.920 --> 00:49:28.500 So you need have known this in advance. 00:49:28.500 --> 00:49:30.740 And that's the input into this equation. 00:49:30.740 --> 00:49:44.000 So now, what we have is that y' = 1 / sec^2 y y, 00:49:44.000 --> 00:49:51.380 which is the same thing as cos^2 y. 00:49:51.380 --> 00:49:54.170 Now, the last bit of the problem is 00:49:54.170 --> 00:49:57.930 to rewrite this in terms of x. 00:49:57.930 --> 00:50:02.664 And that you have to do with a right triangle. 00:50:02.664 --> 00:50:05.618 If this is x and this is 1, then the angle 00:50:05.618 --> 00:50:09.416 is y, because the tangent of y is x. 00:50:09.416 --> 00:50:14.902 So this expresses the fact that the tangent of y is x. 00:50:14.902 --> 00:50:18.700 And then the hypotenuse is the square root of 1 + x^2. 00:50:21.654 --> 00:50:27.140 And so the cosine is 1 divided by that. 00:50:27.140 --> 00:50:30.938 So this thing is 1 divided by the square root of 1 + x^2, 00:50:30.938 --> 00:50:36.424 the quantity squared. 00:50:36.424 --> 00:50:40.644 So, and then the last little bit here, since I'm racing along, 00:50:40.644 --> 00:50:45.286 is that it's 1 / (1 + x^2), which I incorrectly wrote over 00:50:45.286 --> 00:50:46.130 here. 00:50:46.130 --> 00:50:48.662 Okay, so good luck on the test. 00:50:48.662 --> 00:50:50.756 See you tomorrow.