Real power series/Derivative by formal derivative/Fact

Let

$g(x):=\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$

denote a power series which converges on the open interval $]-r,r[$, and represents there a function $f:]-r,r[\to ℝ$.

Then the formally differentiated power series

$\tilde{g}(x):=\sum _{n=1}^{\mathrm{\infty }}n{a}_n{x}^{n-1}$

is convergent on $]-r,r[$. The function $f$ is differentiable in every point of the interval, and

$f^{\prime }(x)=\tilde{g}(x)$

holds.

Write a proof