If $f(x) = h(x)g(x)$, is $h$ differentiable if $f$ and $g$ are? I know that if I have two differentiable functions $f, g$ then the functions $(f + g)$ and $fg$ are also differentiable. 
I would like to find a way how to argue about the function $h$ where 
\begin{equation}
f(x) = (hg)(x) := h(x)g(x) \quad \text{and } f,g \text{ are differentiable}
\end{equation}
For a start I can conclude $h$ is differentiable at all points where $g(x) \neq 0$ since there I can express $h$ as
\begin{equation}
h = \frac{f}{g}
\end{equation}
But for the remaining points I am not sure, my guess is that $h$ is differentiable, any hints how I can make this into a formal argument ? Or am I probably wrong ? In that case, would it help to impose further smoothness on $f$ and $g$, say both are $C^\infty$ ?
Many thanks!

No. Let $f(x)=g(x)=0$ for all $x$. There are no functions more regular than constant functions. Then $f=gh$ holds for any function $h$, even if it is nowhere-differentiable.
