I'm trying to solve
A = $\frac{d}{dy} \int_{0}^{y} \int_{0}^{k(x_1,y)} (g(x_1)+h(x_2))f(x_2)dx_2 f(x_1)dx_1$. Here is my approach:
let $j(x_1, y) = \int_{0}^{k(x_1,y)} (g(x_1)+h(x_2))f(x_2)dx_2$, then $A=\frac{d}{dy}\int_{0}^{y}j(x_1, y)f(x_1)dx_1$. Applying Leibniz rule, we have $A= \int_{0}^{y}\frac{d}{dy} j(x_1,y)f(x_1)dx_1 + j(y,y)f_{x_1}(y)$
My question is, in $j(y,y)$, should I replace the $x_1$ in the upper limit (i.e., in the function $k(x_1,y)$, or only in the term $g(x_1)+h(x_2)$?
Thanks for your help!
p.s. $f(x_1)$ and $f(x_2)$ are probability density functions.