Suppose $X$ is a random variable with PDF and CDF, $f$ and $F$, respectively. Now, we define $Z(w)=F^{-1}left( frac{p+g-w}{p+g} right)$, and $Lambda(Z(w))=int_{0}^{Z(w)}(Z(w)-x)f(x)dx$ and $Theta(Z(w))=int_{Z(w)}^{infty}(x-Z(w))f(x)dx$.

Based on these definitions, I define a function $Pi(w)=(w-c)D+(w-c)left(Lambda(Z(w))-Theta(Z(w))right)$. I am wondering how to find the derivative of $Pi(w)$ with respect to $w$. How is it possible to define the inverse of CDF? Is it possible to determine the result based on hazard rate?