Context: $F$ denotes cumulative distribution function, $f$ denotes probability density function. For continuous random variables $X,Y,Z,$ and $W$ we have the following relationship:$$F_{Z,W}(z,w)=F_{X,Y}(z,w)+F_{X,Y}(w,z)-F_{X,Y}(w,w)$$ and wish to find $f_{Z,W}$ in terms of $f_{X,Y}$ (here, we know the joint density of $f_{X,Y}$ only)
I wish to compute $f_{Z,W}$ by taking partial derivatives: $$f_{Z,W}(z,w) = \frac{\partial^2} {\partial{z}\partial{w}} F_{Z,W}(z,w)=\frac{\partial^2} {\partial{z}\partial{w}}(F_{X,Y}(z,w)+F_{X,Y}(w,z)-F_{X,Y}(w,w))$$
How do I evaluate the right hand side? I am used to doing this with one variable - here is how I would do it with a specific example where we know that, for example, $F_Z(z)=F_{X,Y}(2z,3z)$: $$f_Z(z) = \frac{\partial}{\partial{z}}{F_Z(z)} = \frac{\partial}{\partial{z}}{F_{X,Y}(2z,3z)}={f_{X,Y}(2z,3z)}*(2)*(3) = 6{f_{X,Y}(2z,3z)}$$
As $$\frac{\partial^2} {\partial{z}\partial{w}}(F_{X,Y}(z,w)+F_{X,Y}(w,z)-F_{X,Y}(w,w))=f_{X,Y}(z,w)+f_{X,Y}(w,z) + 0$$ since $F_{X,Y}(w,w)$ does not depend on $z$,
$$f_{Z,W}(z,w)=f_{X,Y}(z,w)+f_{X,Y}(w,z)$$