Derivative of Exponential Function/Proof 4

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

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

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.