Derivative of convolution proof

Question

Using the definition of convolution: $$ (f*g)(x) = \frac {1}{2\pi} \int_{-\pi}^{\pi} f(x-t)g(t)dt$$ I need to prove the following:

Let $f$ be a continuously differentiable function. Then $f*g$ is also differentiable and it's derivative at point $t$ is $((f')*g)(t)$.

Any help would be appreciated.

Answer 1

Hint

Use the definition of derivative and swap the convolution integral with the $\lim$ in the definition of derivative.