Differentiating a parametrized piecewise function

Question

Here's my piecewise function in which $\eta$ is a parameter: \begin{align} h(\eta)=\begin{cases} f(x) & x\leq g(\eta) \\ 0 & x>g(\eta) \end{cases} \end{align} where $x$ is a positive continuous strictly increasing function of $\eta$, and $f,g,h,$ are positive continuous strictly increasing functions of their arguments. I wish to find the extremum of function $h$, for which I need $dh/d\eta$. I don't know how to even proceed with this one. This equation actually comes from an optimization problem modeling a physical system. Any help is appreciated. Thanks in advance.

Answer 1

Since it is given that $h$ is positive, the inequality $x>g(\eta)$ can never hold. Thus $h(\eta)=f(x)$. Does this help?