Suppose I have two functions that are Schwartz class, say $f,g \in S(\mathbb{R})$, and suppose I have another function $\psi(x)$ such that \begin{equation} g(x) = \psi(x)f(x) \end{equation} I would like to find a way to understand why this means that $\psi$ must be differentiable but I struggle to find a start as I am very new to Schwartz class functions.
I thought maybe smoothness is enough to begin with. In any case, away from points where $f(x) \neq 0$ I have no problems, but somehow I need to find a way to say something about the differentiability fo $\psi$ in general, provided $f,g$ are Schwartz (and not both identically zero).
Any help would be great !