termwise differentiation

Theorem.

If in the open interval $I$,all the of the series

$f_1(x)+f_2(x)+\mathrm{\cdots }$(1)

have continuous derivatives,the series converges having sum $S(x)$ andthe differentiated series  $f_1^{\prime }(x)+f_2^{\prime }(x)+\mathrm{\cdots }$ converges uniformly (http://planetmath.org/SumFunctionOfSeries) on the interval $I$,then the series (1) can be differentiated termwise, i.e. in every point of $I$ the sum function $S(x)$ is differentiable and

$$\frac{dS(x)}{dx}=f_1^{\prime }(x)+f_2^{\prime }(x)+\mathrm{\cdots }$$

The situation implies also that the series (1) converges uniformly on $I$.