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  • I've introduced a few derivative formulas

  • but a really important one that Ieft out was exponentials.

  • So here, I want to talk about the derivatives of functions like

  • Two to the x, seven to the x, and also to show why

  • e to the x is arguably the most important of the exponentials.

  • First of all, to get an intuition, let's just focus on the function two to the x.

  • And let's think of that input as a time, "t," maybe in days,

  • and the output, 2 to the t, as a population size

  • perhaps of a particularly fertile band of pi creatures which doubles every single day.

  • And actually, instead of population size,

  • which grows in discrete little jumps with each new baby pi creature,

  • maybe let's think of 2 to the t as the total mass of the population.

  • I think that better reflects the continuity of this function, don't you?

  • So, for example, at time t=0, the total mass is 2 to the 0 equals 1,

  • for the mass of one creature.

  • At t=1 day, the population has grown to 2 to the 1 = 2 creature masses.

  • At day t=2, it's t squared, or 4, and in general, it just keeps doubling every day.

  • For the derivative, we want dm/dt, the rate at which this population mass is growing,

  • thought of as a tiny change in the mass divided by a tiny change in time.

  • And let's start by thinking of the rate of change over a full day,

  • say, between day 3 and day 4.

  • Well, in this case it grows from 8 to 16, so that's 8 new creature masses

  • added over the course of one day.

  • And notice, that rate of growth equals the population size at the start of the day.

  • Between day 4 and day 5, it grows from 16 to 32.

  • So that's a rate of 16 new creature masses per day.

  • Which, again, equals the population size at the start of the day.

  • And in general, this rate of growth over a full day

  • equals the population size at the start of that day.

  • So it might be tempting to say that this means

  • the derivative of 2 to the t equals itself.

  • That the rate of change of this function at a given time t,

  • is equal to, well, the value of that function.

  • And this is definitely in the right direction,

  • but it's not quite correct.

  • What we're doing here is making comparisons over a full day,

  • considering the difference between 2 to the t plus 1,

  • and 2 to the t.

  • but for the derivative, we need to ask what happens for smaller and smaller changes.

  • What's the growth over the course of a tenth of a day? A hundredth of a day? One one-billionth of a day?

  • This is why I had us think of the function as representing population mass

  • since it makes sense to ask about a tiny change in mass over a tiny fraction of a day

  • but it doesn't make as much sense to ask about the tiny change in a discrete population size per second.

  • More abstractly, for a tiny change in time, dt, we want to understand

  • the difference between 2 to the t plus dt

  • and 2 to the t,

  • all divided by dt.

  • A change in the function per unit time, but now we're looking very narrowly around a given point in time,

  • rather than over the course of a full day.

  • And here's the thing:

  • I would love if there was some very clear geometric picture

  • that made everything that's about to follow just pop out,

  • some diagram where you could point to one value,

  • and say, "See! *that* part. That is the derivative of 2 to the t."

  • And if you know of one, please let me know.

  • And while the goal here as with the rest of the series

  • is to maintain a playful spirit of discover,

  • the type of play that follows will have more to do with finding numerical patterns,

  • rather than visual ones.

  • So start by just taking a very close look at this term

  • 2 to the t, plus dt

  • A core property of exponentials is that you can break this up as 2 to the t times 2 to the dt.

  • That really is the most important property of exponents.

  • If you add two values in that exponent, you can break up the output as a product of some kind.

  • This is what lets you relate additive ideas

  • things like tiny steps in time,

  • to multiplicative ideas, things like rates and ratios.

  • I mean, just look at what happens here.

  • After that move, we can factor out the term 2 to the t.

  • which is now just multiplied by 2 to the dt minus 1, all divided by dt.

  • And remember, the derivative of 2 to the t

  • is whatever this whole expression approaches as dt approaches 0.

  • And at first glance that might seem like an unimportant manipulation,

  • but a tremendously important fact is that this term on the right,

  • where all of the dt stuff lives, is completely separate from

  • the t term itself. It doesn't depend on the actual time where we started.

  • You can go off to a calculator and plug in very small values for dt here,

  • for example, maybe typing in 2 to the 0.001

  • minus 1, divided by 0.001

  • What you'll find is that for smaller and smaller choices of dt,

  • this value approaches a very specific number,

  • around 0.6931.

  • Don't worry if that number seems mysterious,

  • The central point is that this is some kind of constant.

  • Unlike derivatives of other functions,

  • all of the stuff that depends on dt is separate from the value of t itself.

  • So the derivative of 2 to the t is just itself,

  • but multiplied by some constant

  • And that should kind of make sense,

  • because earlier, it felt like the derivative for 2 to the t should be itself,

  • at least when we were looking at changes over the course of a full day.

  • And evidently, the rate of change for this function over much smaller time scales

  • is not quite equal to itself,

  • but it's proportional to itself,

  • with this very peculiar proportionality constant of 0.6931

  • And there's not too much special about the number 2 here,

  • if instead we had dealt with the function 3 to the t,

  • the exponential property would also have led us to the conclusion that

  • the derivative of 3 to the t is proportional to itself.

  • But this time it would have had a proportionality constant 1.0986.

  • And for other bases to your exponent you can have fun trying to see what the various

  • proportionality constants are, maying seeing if you can find a pattern in them.

  • For example, if you plug in 8 to the power of a very tiny number

  • minus 1, and divide by that same tiny number,

  • what you'd find is that the relevant proportionality constant is around 2.079,

  • and maybe, just maybe you would notice that this number happens

  • to be exactly three times the constant associated with the base for 2,

  • so these numbers certainly aren't random, there is some kind of pattern,

  • but what is it?

  • What does 2 have to do with the number 0.6931?

  • And what does 8 have to do with the number 2.079?

  • Well, a second question that is ultimately going to explain these mystery constants

  • is whether there's some base where that proportionality constant is one (1),

  • where the derivative of "a"to the power t is not just proportional to itself,

  • but actually equal to itself.

  • And there is!

  • It's the special constant "e,"

  • around 2.71828.

  • In fact, it's not just that the number e happens to show up here,

  • this is, in a sense, what defines the number e.

  • If you ask, "why does e, of all numbers, have this property?"

  • It's a little like asking "why does pi, of all numbers happen to be the ratio of the circumference of a circle to its diameter?"

  • This is, at its heart, what defines this value.

  • All exponential functions are proportional to their own derivative,

  • but e along is the special number so that that proportionality constant is one,

  • meaning e to the t actually equals its own derivative.

  • One way to think of that is that if you look at the graph of e to the t,

  • it has the peculiar property that the slope of a tangent line to any point on this graph

  • equals the height of that point above the horizontal axis.

  • The existence of a function like this answers the question of the mystery constants

  • and it's because it gives a different way to think about functions

  • that are proportional to their own derivative.

  • The key is to use the chain rule.

  • For example, what is the derivative of e to the 3t?

  • Well,

  • you take the derivative of the outermost function, which due to this special nature of e

  • is just itself and then multipliy it by the derivative of that inner function, 3t

  • which is the constant, 3.

  • Or, rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule

  • that I talked through last video, thinking about how a slight nudge to t changes the value of 3t

  • and how that intermediate change nudges the final value of e to the 3t.

  • Either way, the point is, e to the power of some constant times t

  • is equal to that same constant times itself.

  • And from here, the question of those mystery constants really just comes down to a certain algebraic manipulation.

  • The number 2 can also be written as e to the natural log of 2.

  • There's nothing fancy here, this is just the definition of the natural log,

  • it asks the question, "e to what equals 2?"

  • So, the function 2 to the t

  • is the same as the function e to the power of the natural log of 2 times t.

  • And from what we just saw, combining the facts that e to the t is its own derivative

  • with the chain rule, the derivative of this function is proportional to itself,

  • with a proportionality constant equal to the natural log of 2.

  • And indeed, if you go plug in the natural log of two to a calculator,

  • you'll find that it's 0.6931,

  • the mystery constant that we ran into earlier.

  • And the same goes for all of the other bases.

  • The mystery proportionality constant that pops up when taking derivatives

  • is just the natural log of the base,

  • the answer to the question, "e to the what equals that base?"

  • In fact, throughout applications of calculus, you rarely see exponentials written as some base to a power t,

  • instead you almost always write the exponential as e to the power of some constant times t.

  • It's all equivalent. I mean any function like 2 to the t

  • or 3 to the t can also be written as e to some constant time t.

  • At the risk of staying over-focused on the symbols here,

  • Ireally want to emphasize that there are many many ways to write down any particular exponential function,

  • and when you see something written as e to some constant time t,

  • that's a choice that we make to write it that way, and the number e is not fundamental to that function itself.

  • What is special about writing exponentials in terms of e like this,

  • is that it gives that constant in the exponent a nice, readable meaning.

  • Here, let me show you what I mean.

  • All sorts of natural phenomena involve some rate of change that's proportional to the thing that's changing.

  • For example, the rate of growth of a population actually does tend to be proportional

  • to the size of the population itself,

  • assuming there isn't some limited resource slowing things down.

  • And if you put a cup of hot water in a cool room,

  • the rate at which the water cools is proportional to the difference in temperature

  • between the room and the water.

  • Or, said a little differently

  • the rate at which that difference changes is proportional to itself.

  • If you invest your money, the rate at which it grows

  • is proportional to the amount of money there at any time.

  • In all of these cases, where some variable's rate of change

  • is proportional to itself

  • the function describing that variable over time is going to look like some kind of exponential.

  • And even though there are lots of ways to write any exponential function,

  • it's very natural to choose to express these functions

  • as e to the power of some constant times t

  • since that constant carries a very natural meaning.

  • It's the same as the proportionality constant between the size of the changing variable

  • and the rate of change.

  • And, as always, I want to thank those who have made this series possible.

I've introduced a few derivative formulas

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What's so special about Euler's number e? | EoC #5

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    tai に公開 2021 年 05 月 13 日
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