I'm working on a problem, inspired by options theory/Black-Scholes, that requires one to find a function, $ g\left(\frac{f(x)}{x}\right)$, which has the property
$$f'(x) = \left(1+\exp\left\{c\cdot g\left(\frac{f(x)}{x}\right)\right\}\right)^{-1},$$
has a closed-form solution. Any suggestions on $ g\left(\frac{f(x)}{x}\right)$, where $g$ is a monotonically increasing function, that would lead to a neat closed-form solution for $f(x)$?