In my function spaces course I was asked to prove that given if $f\in C^k_c(\mathbb{R^d})$ and $g\in L^1_{loc}(\mathbb{R^d})$ then $f*g\in C^k(\mathbb{R^d})$ and $\partial^\alpha(f*g)(x)=((\partial^\alpha f)*g)(x)$ for $x\in\mathbb{R}^d$ where $\alpha\in\mathbb{N}^d, |\alpha|\le k$
When applying induction we know that $C^k_c(\mathbb{R^d})\subset C^k_c(\mathbb{R^{d-1}})$ so we have that $\partial^\alpha(f*g)(x)=\partial_i((\partial^{\alpha'} f)*g)(x)$ for some $i=1,2,\dots,d$ and $\alpha'\in\mathbb{N}^{d-1}$. Here $\partial^{\alpha'} f\in C^1_c(\mathbb{R^d})$ so now we need only solve for the case of $k=1$
I've had no problem solving this for $k=0$ but I am struggling to solve $k=1$. I tried writing out both sides of of the above using the definition of partial derivative and of convolution, I see that somehow I need to justify being able to switch the limit of the derivative with the integral from the convolution but I do not know how.