Is there an example of an expression using $f(x)$ and $g(x)$ (perhaps composed with other elementary functions) whose derivative can be factorized into multiplicatively separable expressions, one containing $f(x)$ and $f'(x)$, and the other containing $g(x)$ and $g'(x)$?
In other words, I'm looking for an example of a function $\Psi$ such that $$\left[\Psi(f(x), g(x))\right]' = \phi(f(x), f'(x)) \cdot \varphi(g(x), g'(x)).$$
For instance, if one considers \begin{align} \Psi &= f(x)\exp(g(x))\\ \Rightarrow \Psi' &= [f'(x) + f(x)g'(x)]\exp(g(x)) \end{align} this seems close, but doesn't work because the $g'(x)$ cannot be factored out of the square-bracketed expression.
edit: I'm trying to find a non-trivial example of $\Psi$, some formulation of $f(x)$ and $g(x)$ that results in a separable derivative with respect to a single variable, $x$.