A closer look at why ex is its own derivative

The constant e is often introduced through Jacob Bernoulli's 17th-century work on compound interest, where it arises as the limit:

One of the properties that makes e so central to calculus is that the exponential function ex is its own derivative. That fact is usually stated early and proven late, if at all. Here are two straightforward proofs, starting from different definitions of e.

Proof via the limit definition of the derivative

First, rearrange the definition of e with a change of variable. Replace n by 1/m:

This form will be useful shortly. Now write the derivative of ex using the standard limit definition:

Factor out ex (it does not depend on h):

As h approaches 0, both the numerator and denominator tend to 0, so we need a substitution. Let m = eh − 1, which implies h = ln(1 + m). As h → 0, m also → 0. Rewriting:

From the rearranged definition of e, as m → 0, (1 + m)1/m → e. Substituting into the denominator gives:

So the derivative of ex is indeed ex.

Proof via power series

An alternative route uses the power series expansion of ex. It can be derived from the limit definition using the binomial theorem:

Expanding the binomial inside the limit and writing out the binomial coefficients:

In the limit as n → ∞, terms like n − c for constant c behave like n. After simplifying, all powers of n cancel, leaving:

Since the limit no longer depends on n, this becomes the familiar series:

Differentiating term by term — each term xk/k! is a simple power rule application — yields:

The series on the right is exactly the original expansion of ex, so the derivative is the function itself.

Why the power series holds

For completeness, here is a derivation of the power series from the limit. Start with the limit definition and substitute m = n/x:

As n → ∞, m → ∞ as well, so the limit becomes the definition of e raised to the x power:

Nothing inside the limit depends on x, so the exponent can be pulled outside. The inner limit is the classic definition of e, yielding ex.

[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.