In Wikipedia it is written that $$\frac{\partial}{\partial x_i}(f\ast g)=\frac{\partial f}{\partial x_i}\ast g=f\ast \frac{\partial g}{\partial x_i} $$ "hold under the precise condition that $f$ and $g$ are absolutely integrable and at least one of them has an absolutely integrable ($L^1$) weak derivative, as a consequence of Young's inequality".
My question: is that partial derivative for the convolution, $$\frac{\partial}{\partial x_i}(f\ast g),$$ in the classical sense? If so, what is the exact proof of this statement?