Suppose \(f\in C^1_c(\mathbb{R}^d)\) and \(g\in L^1_{\text{loc}}\). Then the convolution \(f\ast g\) is also differentiable , i.e \(f\ast g\in C^1(\mathbb{R}^d)\) with

\[ \nabla (f\ast g)= \nabla f \ast g. \]

Proof Link to heading

This can be proved by applying the dominated convergence theorem.