As we know, we could write mathematical expectation based on cumulative distribution function $(F)$ as follow: $E(x)=\int[1-F(x)]d(x) $
I have the mathematical expectation of a function $p(t)$ based on cumulative distribution function $(F)$ as follow: $E(p(t))=\int[1-F(p(t))]d(p(t)) $
($p(t)$ is a positive and differentiable function and monotonically increasing in variable $t$)
Now I want to differentiate above equation based on $t$.
Would you tell me what the first and second derivative will be based on t?
$\frac{dE(p(t))}{dt}=?$ and $\frac{d^2E(p(t))}{dt^2}=?$