If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$ respectively. I want to compute the joint density of $U=X+Y$ and $V=\frac{X}{X+Y}$ without using jacobian transformation.

Hint provided is to differentiate the following equation with respect to u and v.

$P(U\leq u, V\leq v)=\iint_{(x,y):-(x+y)\leq u,\frac{x}{x+y}\leq v} f_{X,Y}(x,y)dxdy$

Now how to differentiate the above equation with respect to u and v?