What if I tried to obtain a F(x,y)= 0 which I could then derive and get an expression for dg/du? The problem would be getting rid of u and v, but the truth is I suck at solving systems of equations. Maybe adding and multiplying the equations...

Another idea would be that y could be defined implicitly as a function of x.

It would be like F(x,y)=F(x,f(x))=0, where x is defined by u and v and keeps holding the condition.