Elementary proof that $2^x$ is derivable

There is a proof based entirely on the methods of differential calculus; see this

Differentiability of Exponential Functions
by Philip M. Anselone and John W. Lee

In that paper you will find the following.

Theorem 1. Let $f (x) = a^x$ with any $a > 1$. Then f is differentiable at $ x = 0$ and $f'(0) > 0$.

Theorem 2. Let $f (x) = a^x$ with any $a > 1$. Then f is differentiable for all $ x$ and $f'(x) = f'(0)a^x$.

The authors continue, justifying the claim that there is one and only one $e > 0$ satisfying

$\tag 1 \frac{d}{dx} e^x = e^x$

They then introduce the natural log function.