I have a question about the conditions for which the following identity is valid: $$\frac{d}{dx}(f*g)=\frac{df}{dx}*g$$ The wikipedia states that it is valid if, say, $f\in C^\infty_c$ and g is a tempered distribution.
Question: is this also valid if $f\in C^\infty_c$ and $g$ is not tempered, for example $e^x$?
(the wiki doesn't give any proof or reference and i haven't been able to find anything through search)
thanks