# calculus and analysis – Leibniz Rule with Inverse of CDF

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?