**(a)** Actually it reflects the following property of convolution which we never mentioned:

$\newcommand{\supp}{\operatorname{supp}}$

\begin{equation}

\supp (f*g) \subset \supp(f)+\supp(g)

\label{eq-1}

\end{equation}

where $\supp{f}$ is support of $f$ the smallest closed set outside of which $f=0$ and $+$ mean arithmetic sum of sets: $A+B= \{x+y :\,x\in A,\, y\in B\}$

**(b)** Ian is correct that one should not differentiate non-differentiable "factor" in convolution but instead first change variable of integration $y\mapsto x-y$

\begin{equation}

(f*g): = \int f(x-y)g(y)\,dy = \int f(y)g(x-y)\,dy

\label{eq-2}

\end{equation}

and then differentiate:

\begin{equation}

(f*g)'=\left\{\begin{aligned}

&f'*g,

&f*g'

\end{aligned}\right.

\label{eq-3}

\end{equation}

Note, I did not use $f*g=g*f$ as this would fail for (say) matrix valued functions while (\ref{eq-3}) would remain valid.

**PS. **Still, in the framework of the theory of distributions we can differentiate everything and

\begin{equation}

(f*g)'=f'*g=f*g'

\end{equation}

where both differentiation and convolution are extended to this framework.