[00:14] I've introduced a few derivative formulas, but a
[00:17] really important one that I left out was exponentials.
[00:20] So here I want to talk about the derivatives of functions like 2 to the x, 7 to the x,
[00:26] and also to show why e to the x is arguably the most important of the exponentials.
[00:32] First of all, to get an intuition, let's just focus on the function 2 to the x.
[00:36] Let's think of that input as a time, t, maybe in days, and the output,
[00:41] 2 to the t, as a population size, perhaps of a particularly
[00:45] fertile band of pie creatures which doubles every single day.
[00:50] And actually, instead of population size, which grows in discrete little jumps with each
[00:55] new baby pie creature, maybe let's think of 2 to the t as the total mass of the
[01:00] population.
[01:02] I think that better reflects the continuity of this function, don't you?
[01:06] So for example, at time t equals 0, the total mass is 2 to the 0 equals 1,
[01:11] for the mass of one creature.
[01:14] At t equals 1 day, the population has grown to 2 to the 1 equals 2 creature masses.
[01:21] At day t equals 2, it's t squared, or 4, and in general it just keeps doubling every day.
[01:28] For the derivative, we want dm dt, the rate at which this population mass is growing,
[01:33] thought of as a tiny change in the mass, divided by a tiny change in time.
[01:39] Let's start by thinking of the rate of change over a full day,
[01:44] say between day 3 and day 4.
[01:46] In this case, it grows from 8 to 16, so that's 8
[01:50] new creature masses added over the course of one day.
[01:55] And notice, that rate of growth equals the population size at the start of the day.
[02:01] Between day 4 and day 5, it grows from 16 to 32,
[02:04] so that's a rate of 16 new creature masses per day,
[02:08] which again equals the population size at the start of the day.
[02:13] And in general, this rate of growth over a full day
[02:17] equals the population size at the start of that day.
[02:21] So it might be tempting to say that this means the derivative
[02:25] of 2 to the t equals itself, that the rate of change of this
[02:29] function at a given time t is equal to the value of that function.
[02:34] And this is definitely in the right direction, but it's not quite correct.
[02:39] What we're doing here is making comparisons over a full day,
[02:43] considering the difference between 2 to the t plus 1 and 2 to the t.
[02:48] But for the derivative, we need to ask what happens for smaller and smaller changes.
[02:53] What's the growth over the course of a tenth of a day,
[02:56] a hundredth of a day, one one billionth of a day?
[02:59] This is why I had us think of the function as representing population mass,
[03:04] since it makes sense to ask about a tiny change in mass over a tiny fraction of a day,
[03:09] but it doesn't make as much sense to ask about the tiny change in a discrete population
[03:14] size per second.
[03:15] More abstractly, for a tiny change in time, dt,
[03:19] we want to understand the difference between 2 to the t plus dt and 2 to the t,
[03:25] all divided by dt.
[03:27] The change in the function per unit time, but now we're looking very narrowly
[03:32] around a given point in time, rather than over the course of a full day.
[03:39] And here's the thing, I would love if there was some very clear geometric picture
[03:44] that made everything that's about to follow just pop out,
[03:47] some diagram where you could point to one value and say, see, that part,
[03:51] that is the derivative of 2 to the t.
[03:54] And if you know of one, please let me know.
[03:57] And while the goal here, as with the rest of the series,
[03:59] is to maintain a playful spirit of discovery, the type of play that
[04:03] follows will have more to do with finding numerical patterns rather than visual ones.
[04:08] So start by just taking a very close look at this term, 2 to the t plus dt.
[04:14] A core property of exponentials is that you can
[04:17] break this up as 2 to the t times 2 to the dt.
[04:21] That really is the most important property of exponents.
[04:24] If you add two values in that exponent, you can
[04:27] break up the output as a product of some kind.
[04:30] This is what lets you relate additive ideas, things like tiny steps in time,
[04:34] to multiplicative ideas, things like rates and ratios.
[04:38] I mean, just look at what happens here.
[04:40] After that move, we can factor out the term 2 to the t,
[04:44] which is now just multiplied by 2 to the dt minus 1, all divided by dt.
[04:50] And remember, the derivative of 2 to the t is whatever
[04:54] this whole expression approaches as dt approaches 0.
[04:58] And at first glance, that might seem like an unimportant manipulation.
[05:02] But a tremendously important fact is that this term on the right,
[05:06] where all of the dt stuff lives, is completely separate from the t term itself.
[05:11] It doesn't depend on the actual time where we started.
[05:14] You can go off to a calculator and plug in very small values for dt here,
[05:20] for example, maybe typing in 2 to the 0.001 minus 1 divided by 0.001.
[05:27] What you'll find is that for smaller and smaller choices of dt,
[05:32] this value approaches a very specific number, around 0.6931.
[05:38] Don't worry if that number seems mysterious, the
[05:41] central point is that this is some kind of constant.
[05:44] Unlike derivatives of other functions, all of the stuff
[05:48] that depends on dt is separate from the value of t itself.
[05:52] So the derivative of 2 to the t is just itself, but multiplied by some constant.
[05:59] And that should make sense, because earlier it felt like the derivative for 2 to the t
[06:03] should be itself, at least when we were looking at changes over the course of a full day.
[06:09] And evidently, the rate of change for this function over much smaller
[06:13] timescales is not quite equal to itself, but it's proportional to itself,
[06:18] with this very peculiar proportionality constant of 0.6931.
[06:29] And there's not too much special about the number 2 here.
[06:32] If instead we had dealt with the function 3 to the t,
[06:36] the exponential property would also have led us to the conclusion that the derivative
[06:41] of 3 to the t is proportional to itself, but this time it would have had a
[06:45] proportionality constant of 1.0986.
[06:49] And for other bases to your exponent, you can have fun trying to see what the
[06:53] various proportionality constants are, maybe seeing if you can find a pattern in them.
[06:58] For example, if you plug in 8 to the power of a very tiny number,
[07:03] minus 1, and divide by that same tiny number, you'd find that
[07:08] the relevant proportionality constant is around 2.079.
[07:12] And maybe, just maybe, you would notice that this number happens
[07:17] to be exactly 3 times the constant associated with the base for 2.
[07:22] So these numbers certainly aren't random, there is some kind of pattern, but what is it?
[07:28] What does 2 have to do with the number 0.6931,
[07:31] and what does 8 have to do with the number 2.079?
[07:36] Well, a second question that is ultimately going to explain these mystery constants is
[07:42] whether there's some base where that proportionality constant is 1,
[07:46] where the derivative of a to the power t is not just proportional to itself,
[07:51] but actually equal to itself.
[07:53] And there is!
[07:55] It's the special constant e around 2.71828.
[08:00] In fact, it's not just that the number e happens to show up here,
[08:04] this is in a sense what defines the number e.
[08:08] If you ask why does e of all numbers have this property,
[08:11] it's a little like asking why does pi of all numbers happen
[08:14] to be the ratio of the circumference of a circle to its diameter.
[08:18] This is at its heart what defines this value.
[08:22] All exponential functions are proportional to their own derivative,
[08:26] but e alone is the special number so that proportionality constant is 1,
[08:30] meaning e to the t actually equals its own derivative.
[08:35] One way to think of that is that if you look at the graph of e to the t,
[08:39] it has the peculiar property that the slope of a tangent line to any point
[08:43] on this graph equals the height of that point above the horizontal axis.
[08:48] The existence of a function like this answers the question of the mystery constants,
[08:52] and it's because it gives a different way to think about functions
[08:56] that are proportional to their own derivative.
[08:59] The key is to use the chain rule.
[09:01] For example, what is the derivative of e to the 3t?
[09:06] Well, you take the derivative of the outermost function,
[09:09] which due to this special nature of e is just itself,
[09:13] and multiply by the derivative of that inner function 3t, which is the constant 3.
[09:19] Or rather than just applying a rule blindly, you could take this moment
[09:23] to practice the intuition for the chain rule that I talked through last video,
[09:28] thinking about how a slight nudge to t changes the value of 3t,
[09:31] and how that intermediate change nudges the final value of e to the 3t.
[09:38] Either way, the point is e to the power of some constant
[09:42] times t is equal to that same constant times itself.
[09:47] And from here, the question of those mystery constants
[09:51] really just comes down to a certain algebraic manipulation.
[09:56] The number 2 can also be written as e to the natural log of 2.
[10:01] There's nothing fancy here, this is just the definition of the natural log,
[10:06] it asks the question e to the what equals 2.
[10:10] So the function 2 to the t is the same as the function
[10:14] e to the power of the natural log of 2 times t.
[10:20] And from what we just saw, combining the fact that e to the t is its own
[10:24] derivative with the chain rule, the derivative of this function is proportional
[10:28] to itself, with a proportionality constant equal to the natural log of 2.
[10:34] And indeed, if you go plug in the natural log of 2 to a calculator,
[10:38] you'll find that it's 0.6931, the mystery constant we ran into earlier.
[10:43] And the same goes for all the other bases.
[10:46] The mystery proportionality constant that pops up when taking derivatives is just
[10:49] the natural log of the base. The answer to the question e to the what equals that base.
[10:53] In fact, throughout applications of calculus, you rarely
[11:00] see exponentials written as some base to a power t.
[11:08] Instead, you almost always write the exponential
[11:10] as e to the power of some constant times t.
[11:14] It's all equivalent, I mean any function like 2 to the t or
[11:18] 3 to the t can also be written as e to some constant times t.
[11:24] At the risk of staying overfocused on the symbols here,
[11:27] I want to emphasize that there are many ways to write down any particular
[11:32] exponential function.
[11:34] And when you see something written as e to some constant times t,
[11:38] that's a choice we make to write it that way, and the number e is not fundamental to
[11:43] that function itself.
[11:45] What is special about writing exponentials in terms of e like this is
[11:49] that it gives that constant in the exponent a nice readable meaning.
[11:54] Here, let me show you what I mean.
[11:56] All sorts of natural phenomena involve some rate of
[11:59] change that's proportional to the thing that's changing.
[12:03] For example, the rate of growth of a population actually does
[12:06] tend to be proportional to the size of the population itself,
[12:10] assuming there isn't some limited resource slowing things down.
[12:14] And if you put a cup of hot water in a cool room,
[12:17] the rate at which the water cools is proportional to the difference in temperature
[12:22] between the room and the water, or said a little differently,
[12:26] the rate at which that difference changes is proportional to itself.
[12:31] If you invest your money, the rate at which it grows is
[12:35] proportional to the amount of money there at any time.
[12:39] In all of these cases, where some variable's rate of change is proportional to itself,
[12:45] the function describing that variable over time is going to
[12:48] look like some kind of exponential.
[12:51] And even though there are lots of ways to write any exponential function,
[12:56] it's very natural to choose to express these functions as e to the power
[13:00] of some constant times t, since that constant carries a very natural meaning.
[13:04] It's the same as the proportionality constant between
[13:08] the size of the changing variable and the rate of change.
[13:14] And as always, I want to thank those who have made this series possible.
[13:34] Thank you.