Derivative of Uniformly Convergent Series of Continuously Differentiable Functions

Theorem
Let $\left\langle{f_n}\right\rangle$ be a sequence of real functions.

Let each of $\left\langle{f_n}\right\rangle$ be continuously differentiable on the interval $\left[{a \,.\,.\, b}\right]$.

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

be pointwise convergent for all $x \in \left[{a \,.\,.\, b}\right]$.

Let the series:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac {\mathrm d}{\mathrm dx} f_n \left({x}\right)$

be uniformly convergent for all $x \in \left[{a \,.\,.\, b}\right]$.

Then:
 * $\displaystyle \frac {\mathrm d}{\mathrm dx} f \left({x}\right) := \sum_{n \mathop = 1}^\infty \frac {\mathrm d}{\mathrm dx} f_n \left({x}\right)$