Derivative of Uniformly Convergent Series of Continuously Differentiable Functions

Theorem
Let $\sequence {f_n}$ be a sequence of real functions.

Let each of $\sequence {f_n}$ be continuously differentiable on the interval $\closedint a b$.

Let the series:
 * $\displaystyle \map f x := \sum_{n \mathop = 1}^\infty \map {f_n} x$

be pointwise convergent for all $x \in \closedint a b$.

Let the series:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac \d {\d x} \map {f_n} x$

be uniformly convergent for all $x \in \closedint a b$.

Then:
 * $\displaystyle \frac \d {\d x} \map f x := \sum_{n \mathop = 1}^\infty \frac \d {\d x} \map {f_n} x$