WEBVTT 00:00:14.759 --> 00:00:17.327 I've introduced a few derivative formulas, but a 00:00:17.327 --> 00:00:20.160 really important one that I left out was exponentials. 00:00:20.839 --> 00:00:26.059 So here I want to talk about the derivatives of functions like 2 to the x, 7 to the x, 00:00:26.059 --> 00:00:31.039 and also to show why e to the x is arguably the most important of the exponentials. 00:00:32.240 --> 00:00:36.120 First of all, to get an intuition, let's just focus on the function 2 to the x. 00:00:36.920 --> 00:00:41.505 Let's think of that input as a time, t, maybe in days, and the output, 00:00:41.505 --> 00:00:45.380 2 to the t, as a population size, perhaps of a particularly 00:00:45.380 --> 00:00:49.320 fertile band of pie creatures which doubles every single day. 00:00:50.560 --> 00:00:55.978 And actually, instead of population size, which grows in discrete little jumps with each 00:00:55.978 --> 00:01:00.850 new baby pie creature, maybe let's think of 2 to the t as the total mass of the 00:01:00.850 --> 00:01:01.520 population. 00:01:02.219 --> 00:01:05.319 I think that better reflects the continuity of this function, don't you? 00:01:06.379 --> 00:01:11.643 So for example, at time t equals 0, the total mass is 2 to the 0 equals 1, 00:01:11.644 --> 00:01:13.680 for the mass of one creature. 00:01:14.409 --> 00:01:20.200 At t equals 1 day, the population has grown to 2 to the 1 equals 2 creature masses. 00:01:21.159 --> 00:01:27.119 At day t equals 2, it's t squared, or 4, and in general it just keeps doubling every day. 00:01:28.260 --> 00:01:33.989 For the derivative, we want dm dt, the rate at which this population mass is growing, 00:01:33.989 --> 00:01:38.920 thought of as a tiny change in the mass, divided by a tiny change in time. 00:01:39.840 --> 00:01:44.146 Let's start by thinking of the rate of change over a full day, 00:01:44.146 --> 00:01:46.060 say between day 3 and day 4. 00:01:46.500 --> 00:01:50.207 In this case, it grows from 8 to 16, so that's 8 00:01:50.207 --> 00:01:54.219 new creature masses added over the course of one day. 00:01:55.060 --> 00:01:59.840 And notice, that rate of growth equals the population size at the start of the day. 00:02:01.480 --> 00:02:04.850 Between day 4 and day 5, it grows from 16 to 32, 00:02:04.849 --> 00:02:08.425 so that's a rate of 16 new creature masses per day, 00:02:08.425 --> 00:02:12.759 which again equals the population size at the start of the day. 00:02:13.520 --> 00:02:17.090 And in general, this rate of growth over a full day 00:02:17.090 --> 00:02:20.659 equals the population size at the start of that day. 00:02:21.680 --> 00:02:25.760 So it might be tempting to say that this means the derivative 00:02:25.759 --> 00:02:29.774 of 2 to the t equals itself, that the rate of change of this 00:02:29.775 --> 00:02:34.120 function at a given time t is equal to the value of that function. 00:02:34.120 --> 00:02:38.879 And this is definitely in the right direction, but it's not quite correct. 00:02:39.460 --> 00:02:43.365 What we're doing here is making comparisons over a full day, 00:02:43.365 --> 00:02:47.719 considering the difference between 2 to the t plus 1 and 2 to the t. 00:02:48.560 --> 00:02:53.340 But for the derivative, we need to ask what happens for smaller and smaller changes. 00:02:53.960 --> 00:02:56.741 What's the growth over the course of a tenth of a day, 00:02:56.741 --> 00:02:59.220 a hundredth of a day, one one billionth of a day? 00:02:59.960 --> 00:03:04.229 This is why I had us think of the function as representing population mass, 00:03:04.229 --> 00:03:09.117 since it makes sense to ask about a tiny change in mass over a tiny fraction of a day, 00:03:09.116 --> 00:03:14.060 but it doesn't make as much sense to ask about the tiny change in a discrete population 00:03:14.061 --> 00:03:14.960 size per second. 00:03:15.900 --> 00:03:19.594 More abstractly, for a tiny change in time, dt, 00:03:19.594 --> 00:03:25.753 we want to understand the difference between 2 to the t plus dt and 2 to the t, 00:03:25.753 --> 00:03:27.139 all divided by dt. 00:03:27.659 --> 00:03:32.204 The change in the function per unit time, but now we're looking very narrowly 00:03:32.204 --> 00:03:36.400 around a given point in time, rather than over the course of a full day. 00:03:39.580 --> 00:03:44.139 And here's the thing, I would love if there was some very clear geometric picture 00:03:44.139 --> 00:03:47.364 that made everything that's about to follow just pop out, 00:03:47.364 --> 00:03:51.421 some diagram where you could point to one value and say, see, that part, 00:03:51.421 --> 00:03:53.479 that is the derivative of 2 to the t. 00:03:54.379 --> 00:03:56.639 And if you know of one, please let me know. 00:03:57.020 --> 00:03:59.908 And while the goal here, as with the rest of the series, 00:03:59.907 --> 00:04:03.352 is to maintain a playful spirit of discovery, the type of play that 00:04:03.353 --> 00:04:07.659 follows will have more to do with finding numerical patterns rather than visual ones. 00:04:08.680 --> 00:04:13.560 So start by just taking a very close look at this term, 2 to the t plus dt. 00:04:14.360 --> 00:04:17.607 A core property of exponentials is that you can 00:04:17.607 --> 00:04:20.720 break this up as 2 to the t times 2 to the dt. 00:04:21.259 --> 00:04:24.120 That really is the most important property of exponents. 00:04:24.660 --> 00:04:27.458 If you add two values in that exponent, you can 00:04:27.458 --> 00:04:30.140 break up the output as a product of some kind. 00:04:30.819 --> 00:04:34.851 This is what lets you relate additive ideas, things like tiny steps in time, 00:04:34.851 --> 00:04:37.679 to multiplicative ideas, things like rates and ratios. 00:04:38.420 --> 00:04:39.960 I mean, just look at what happens here. 00:04:40.839 --> 00:04:44.807 After that move, we can factor out the term 2 to the t, 00:04:44.807 --> 00:04:49.839 which is now just multiplied by 2 to the dt minus 1, all divided by dt. 00:04:50.720 --> 00:04:54.184 And remember, the derivative of 2 to the t is whatever 00:04:54.184 --> 00:04:57.460 this whole expression approaches as dt approaches 0. 00:04:58.540 --> 00:05:02.080 And at first glance, that might seem like an unimportant manipulation. 00:05:02.699 --> 00:05:06.377 But a tremendously important fact is that this term on the right, 00:05:06.377 --> 00:05:10.780 where all of the dt stuff lives, is completely separate from the t term itself. 00:05:11.259 --> 00:05:13.920 It doesn't depend on the actual time where we started. 00:05:14.620 --> 00:05:20.684 You can go off to a calculator and plug in very small values for dt here, 00:05:20.684 --> 00:05:26.340 for example, maybe typing in 2 to the 0.001 minus 1 divided by 0.001. 00:05:27.759 --> 00:05:32.817 What you'll find is that for smaller and smaller choices of dt, 00:05:32.817 --> 00:05:37.560 this value approaches a very specific number, around 0.6931. 00:05:38.639 --> 00:05:41.036 Don't worry if that number seems mysterious, the 00:05:41.036 --> 00:05:43.579 central point is that this is some kind of constant. 00:05:44.500 --> 00:05:48.252 Unlike derivatives of other functions, all of the stuff 00:05:48.252 --> 00:05:52.139 that depends on dt is separate from the value of t itself. 00:05:52.839 --> 00:05:58.119 So the derivative of 2 to the t is just itself, but multiplied by some constant. 00:05:59.300 --> 00:06:03.817 And that should make sense, because earlier it felt like the derivative for 2 to the t 00:06:03.817 --> 00:06:08.439 should be itself, at least when we were looking at changes over the course of a full day. 00:06:09.029 --> 00:06:13.777 And evidently, the rate of change for this function over much smaller 00:06:13.778 --> 00:06:18.797 timescales is not quite equal to itself, but it's proportional to itself, 00:06:18.797 --> 00:06:22.800 with this very peculiar proportionality constant of 0.6931. 00:06:29.040 --> 00:06:32.200 And there's not too much special about the number 2 here. 00:06:32.839 --> 00:06:36.139 If instead we had dealt with the function 3 to the t, 00:06:36.139 --> 00:06:41.395 the exponential property would also have led us to the conclusion that the derivative 00:06:41.396 --> 00:06:45.980 of 3 to the t is proportional to itself, but this time it would have had a 00:06:45.980 --> 00:06:48.120 proportionality constant of 1.0986. 00:06:49.199 --> 00:06:53.156 And for other bases to your exponent, you can have fun trying to see what the 00:06:53.156 --> 00:06:57.519 various proportionality constants are, maybe seeing if you can find a pattern in them. 00:06:58.399 --> 00:07:03.382 For example, if you plug in 8 to the power of a very tiny number, 00:07:03.382 --> 00:07:08.062 minus 1, and divide by that same tiny number, you'd find that 00:07:08.062 --> 00:07:12.139 the relevant proportionality constant is around 2.079. 00:07:12.660 --> 00:07:17.145 And maybe, just maybe, you would notice that this number happens 00:07:17.144 --> 00:07:21.699 to be exactly 3 times the constant associated with the base for 2. 00:07:22.459 --> 00:07:27.959 So these numbers certainly aren't random, there is some kind of pattern, but what is it? 00:07:28.180 --> 00:07:31.713 What does 2 have to do with the number 0.6931, 00:07:31.713 --> 00:07:35.399 and what does 8 have to do with the number 2.079? 00:07:36.779 --> 00:07:42.245 Well, a second question that is ultimately going to explain these mystery constants is 00:07:42.245 --> 00:07:46.519 whether there's some base where that proportionality constant is 1, 00:07:46.519 --> 00:07:51.357 where the derivative of a to the power t is not just proportional to itself, 00:07:51.357 --> 00:07:53.180 but actually equal to itself. 00:07:53.720 --> 00:07:54.680 And there is! 00:07:55.079 --> 00:07:59.300 It's the special constant e around 2.71828. 00:08:00.319 --> 00:08:04.421 In fact, it's not just that the number e happens to show up here, 00:08:04.422 --> 00:08:07.220 this is in a sense what defines the number e. 00:08:08.600 --> 00:08:11.581 If you ask why does e of all numbers have this property, 00:08:11.581 --> 00:08:14.720 it's a little like asking why does pi of all numbers happen 00:08:14.720 --> 00:08:18.120 to be the ratio of the circumference of a circle to its diameter. 00:08:18.670 --> 00:08:21.280 This is at its heart what defines this value. 00:08:22.060 --> 00:08:26.271 All exponential functions are proportional to their own derivative, 00:08:26.271 --> 00:08:30.793 but e alone is the special number so that proportionality constant is 1, 00:08:30.793 --> 00:08:34.139 meaning e to the t actually equals its own derivative. 00:08:35.440 --> 00:08:39.488 One way to think of that is that if you look at the graph of e to the t, 00:08:39.488 --> 00:08:43.647 it has the peculiar property that the slope of a tangent line to any point 00:08:43.647 --> 00:08:47.640 on this graph equals the height of that point above the horizontal axis. 00:08:48.759 --> 00:08:52.855 The existence of a function like this answers the question of the mystery constants, 00:08:52.855 --> 00:08:56.082 and it's because it gives a different way to think about functions 00:08:56.082 --> 00:08:58.299 that are proportional to their own derivative. 00:08:59.200 --> 00:09:01.000 The key is to use the chain rule. 00:09:01.919 --> 00:09:05.299 For example, what is the derivative of e to the 3t? 00:09:06.340 --> 00:09:09.907 Well, you take the derivative of the outermost function, 00:09:09.907 --> 00:09:13.287 which due to this special nature of e is just itself, 00:09:13.287 --> 00:09:18.420 and multiply by the derivative of that inner function 3t, which is the constant 3. 00:09:19.460 --> 00:09:23.552 Or rather than just applying a rule blindly, you could take this moment 00:09:23.552 --> 00:09:28.043 to practice the intuition for the chain rule that I talked through last video, 00:09:28.043 --> 00:09:31.682 thinking about how a slight nudge to t changes the value of 3t, 00:09:31.682 --> 00:09:35.720 and how that intermediate change nudges the final value of e to the 3t. 00:09:38.419 --> 00:09:42.801 Either way, the point is e to the power of some constant 00:09:42.802 --> 00:09:46.800 times t is equal to that same constant times itself. 00:09:47.960 --> 00:09:51.182 And from here, the question of those mystery constants 00:09:51.182 --> 00:09:54.639 really just comes down to a certain algebraic manipulation. 00:09:56.299 --> 00:10:01.059 The number 2 can also be written as e to the natural log of 2. 00:10:01.059 --> 00:10:06.391 There's nothing fancy here, this is just the definition of the natural log, 00:10:06.392 --> 00:10:09.480 it asks the question e to the what equals 2. 00:10:10.820 --> 00:10:14.896 So the function 2 to the t is the same as the function 00:10:14.895 --> 00:10:18.379 e to the power of the natural log of 2 times t. 00:10:20.320 --> 00:10:24.415 And from what we just saw, combining the fact that e to the t is its own 00:10:24.414 --> 00:10:28.903 derivative with the chain rule, the derivative of this function is proportional 00:10:28.903 --> 00:10:33.000 to itself, with a proportionality constant equal to the natural log of 2. 00:10:34.080 --> 00:10:38.403 And indeed, if you go plug in the natural log of 2 to a calculator, 00:10:38.403 --> 00:10:42.919 you'll find that it's 0.6931, the mystery constant we ran into earlier. 00:10:43.980 --> 00:10:46.220 And the same goes for all the other bases. 00:10:46.759 --> 00:10:49.990 The mystery proportionality constant that pops up when taking derivatives is just 00:10:49.990 --> 00:10:53.419 the natural log of the base. The answer to the question e to the what equals that base. 00:10:53.419 --> 00:11:00.786 In fact, throughout applications of calculus, you rarely 00:11:00.787 --> 00:11:07.379 see exponentials written as some base to a power t. 00:11:08.059 --> 00:11:10.860 Instead, you almost always write the exponential 00:11:10.860 --> 00:11:13.319 as e to the power of some constant times t. 00:11:14.200 --> 00:11:18.285 It's all equivalent, I mean any function like 2 to the t or 00:11:18.284 --> 00:11:22.439 3 to the t can also be written as e to some constant times t. 00:11:24.519 --> 00:11:27.938 At the risk of staying overfocused on the symbols here, 00:11:27.938 --> 00:11:32.456 I want to emphasize that there are many ways to write down any particular 00:11:32.456 --> 00:11:33.740 exponential function. 00:11:34.500 --> 00:11:38.505 And when you see something written as e to some constant times t, 00:11:38.505 --> 00:11:43.664 that's a choice we make to write it that way, and the number e is not fundamental to 00:11:43.664 --> 00:11:44.939 that function itself. 00:11:45.559 --> 00:11:49.728 What is special about writing exponentials in terms of e like this is 00:11:49.729 --> 00:11:53.780 that it gives that constant in the exponent a nice readable meaning. 00:11:54.440 --> 00:11:55.540 Here, let me show you what I mean. 00:11:56.279 --> 00:11:59.158 All sorts of natural phenomena involve some rate of 00:11:59.158 --> 00:12:02.259 change that's proportional to the thing that's changing. 00:12:03.259 --> 00:12:06.648 For example, the rate of growth of a population actually does 00:12:06.648 --> 00:12:10.036 tend to be proportional to the size of the population itself, 00:12:10.035 --> 00:12:13.479 assuming there isn't some limited resource slowing things down. 00:12:14.100 --> 00:12:17.278 And if you put a cup of hot water in a cool room, 00:12:17.278 --> 00:12:22.555 the rate at which the water cools is proportional to the difference in temperature 00:12:22.554 --> 00:12:26.495 between the room and the water, or said a little differently, 00:12:26.495 --> 00:12:30.819 the rate at which that difference changes is proportional to itself. 00:12:31.960 --> 00:12:35.584 If you invest your money, the rate at which it grows is 00:12:35.583 --> 00:12:39.079 proportional to the amount of money there at any time. 00:12:39.940 --> 00:12:45.054 In all of these cases, where some variable's rate of change is proportional to itself, 00:12:45.053 --> 00:12:48.581 the function describing that variable over time is going to 00:12:48.581 --> 00:12:50.639 look like some kind of exponential. 00:12:51.759 --> 00:12:56.100 And even though there are lots of ways to write any exponential function, 00:12:56.100 --> 00:13:00.383 it's very natural to choose to express these functions as e to the power 00:13:00.383 --> 00:13:04.900 of some constant times t, since that constant carries a very natural meaning. 00:13:04.899 --> 00:13:08.216 It's the same as the proportionality constant between 00:13:08.216 --> 00:13:11.720 the size of the changing variable and the rate of change. 00:13:14.759 --> 00:13:17.860 And as always, I want to thank those who have made this series possible. 00:13:34.899 --> 00:13:49.500 Thank you.