What's so special about Euler's number e? | Chapter 5, Essence of calculus

I've introduced a few derivative formulas, but a

really important one that I left out was exponentials.

So here I want to talk about the derivatives of functions like 2 to the x, 7 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 2 to the x.

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 pie creatures which doubles every single day.

And actually, instead of population size, which grows in discrete little jumps with each

new baby pie 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 equals 0, the total mass is 2 to the 0 equals 1,

for the mass of one creature.

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

At day t equals 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.

Let's start by thinking of the rate of change over a full day,

say between day 3 and day 4.

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 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?