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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

• 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