It's a known piece of math folklore that *e* was "discovered" by Jacob Bernoulli
in the 17th century, when he was pondering compound interest,
and defined thus [1]:

*e* is extremely important in mathematics for several reasons; one of them is
its useful behavior under derivation and integration; specifically, that:

In this post I want to present a couple of simple proofs of this fundamental fact.

## Proof using the limit definition

As a prerequisite for this proof, let's reorder the original definition of *e*
slightly. If we perform a change of variable replacing *n* by
, we get:

This equation will become useful a bit later.

Let's start our proof by spelling out the definition of a derivative:

A bit of algebra and observing that does not depend on *h* gives us:

At this point we're stuck; clearly as *h* approaches 0, both the numerator
and denominator approach 0 as well. The way out - as is often the case in such
scenarios - is a sneaky change of variable. Recall equation (1) - how could we
use it here?

The change of variable we'll use is , which implies that
. Note that as *h* approaches zero, so does *m*. Rewriting our
last expression, we get:

Equation (1) tells us that as *m* approaches zero,
approaches *e*. Substituting that into the denominator we get:

## Proof using power series expansion

It's always fun to prove the same thing in multiple ways; while I'm sure there
are *many* other techniques to find the derivative of , one I
particularly like for its simplicity uses its power series expansion.

Similarly to the way *e* itself was defined empirically, one can show that:

(For a proof of this equation, see the Appendix)

Let's use the Binomial theorem to open up the parentheses inside the limit:

We'll unroll the sum a bit, so it's easier to manipulate algebraically. We can
use the standard formula for "choose *n* out of *k*" and get:

Inside the limit, we can simplify all the *n-c* terms
with a constant *c* to just *n*, since compared to infinity *c* is negligible.
This means that all these terms can be simplified as ,
and so on. All these powers of *n* cancel out in
the numerator and denominator, and we get:

And since the contents of the limit don't actually depend on *n* any more, this
leaves us with a well-known formula for approximating [2]:

We can finally use this power series expansion to calculate the derivative of quite trivially. Since it's a sum of terms, the derivative is the sum of the derivatives of the terms:

Look at that, we've got back,

## Appendix

Let's see why:

We'll start with the limit and will arrive at . Using a change of variable :

Given our change of variable, since *n* approaches infinity, so does *m*.
Therefore, we get:

Nothing in the limit depends on *x*, so that exponent can be seen as applying to
the whole limit. And the limit is the definition of *e*; therefore, we get
,

[1] | What I love about this definition is that it's entirely empirical. Try
to substitute successively larger numbers for n in the equation,
and you'll see that the result approaches the value e more and more
closely. The limit of this process for an infinite n was called e.
Bernoulli did all of this by hand, which is rather tedious. His
best estimate was that e is "larger than 2 and a half but smaller than
3". |

[2] | Another way to get this formula is from the Maclaurin series expansion of , but we couldn't use that here since Maclaurin series require derivatives, while we're trying to figure out what the derivative of is. |