Real power series/Derivative by formal derivative/Fact

Let

${}g(x):=\sum _{n=0}^{\infty }a_{n}x^{n}\,$

denote a power series which converges on the open interval ${}]-r,r[$, and represents there a function ${}f\colon ]-r,r[\rightarrow \mathbb {R}$.

Then the formally differentiated power series

${}{\tilde {g}}(x):=\sum _{n=1}^{\infty }na_{n}x^{n-1}\,$

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

${}f'(x)={\tilde {g}}(x)\,$

holds.

Write a proof