I am trying to prove that if $f \in C_c^k(\mathbb{R}^n)$ and $g$ is Lebesgue integrable on $\mathbb{R}^n$, then the derivatives of $f * g$ equal $$D^\alpha(f * g)(x) = \int_{\mathbb{R}^n}(D^\alpha f)(x-y)g(y)dy$$ and are continuous for all multi-indexes of order up to $k$.
I know the version for $\mathbb{R}$, however I am interested in this $n$-dimensional generalization.
Well, since $f\in C_c^k$, you have that $D^{\alpha}f\in L^{\infty}$, and thus, $|D^{\alpha} f(x-y) g(y)|\leq \|D^{\alpha} f\|_{\infty} |g(y)|\in L^1$. Thus, the desired simply follows from the dominated convergence theorem for differentiation under the integral, since for every fixed $y$,
$$ D^{\alpha}( f(x-y)g(y))=D^{\alpha}(f)(x-y)g(y) $$