Derivative of Exponential Function/Proof 4

Theorem
Let $\exp$ be the exponential function.

Then:
 * $D_x \left({\exp x}\right) = \exp x$

Proof
This proof assumes the series definition of $\exp$.

That is, let:
 * $\displaystyle \exp x = \sum_{k \mathop = 0}^{\infty} \frac{x^k}{k!}$

From Series of Power over Factorial Converges, the interval of convergence of $\exp$ is the entirety of $\R$.

So we may apply Differentiation of Power Series to $\exp$ for all $x \in \R$.

Thus we have:

Hence the result.