Name: What's so special about Euler's number e? | EoC #5
Uploaded: 2021-05-13T20:23:30.000Z
Duration: 13 min 50 s
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I've introduced a few derivative formulas

Adverbs of degree

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,

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,

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

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,

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

considering the difference between 2 to the t plus 1,

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

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.

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,

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

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

to multiplicative ideas, things like rates and ratios.

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

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

this value approaches a very specific number,

Don't worry if that number seems mysterious,

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

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,

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

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.

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

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