I'm attempting to differentiate the following double integral with respect to $u$:
$$I(u) = \int_a^u\int_b^v [(y-u) + (v - x)] f(x,y)\,dx \,dy$$
where $f(x,y)$ is the joint density function of RV $X$ and $Y$.
I'm trying to find $I'(u)$, that is, the derivative of $I$ with respect to $u$. I know this involves Leibniz integral rule, however I'm getting stuck on the double integral part.
If I let $g(x,y,u) = \int_b^v [(y-u) + (v - x)] f(x,y)\,dx$ then I think the problem to solve is:
$$\frac{d}{du}\bigg(\int_a^ug(u,x,y)\,dy\bigg) = g(u,x,u) + \int_a^u\frac{\partial}{\partial u}g(u,x,y)\,dy$$
However I'm getting stuck evaluating the two expressions on the RHS.
Any help?