 Reciprocity rule for constrained partial derivatives

On a regular level set $F(x,y,z)=0$, holding $z$ fixed gives $F_{x}dx+F_{y}dy=0$. If $F$ is continuously differentiable and both $F_x$ and $F_y$ are nonzero, the implicit function theorem gives $(\partial y/\partial x{)}_z=-F_x/F_y$ and $(\partial x/\partial y{)}_z=-F_y/F_x$. Multiplication proves the identity. The same held-fixed variable is essential, and a zero denominator invalidates the corresponding quotient rather than creating an infinite factor that may be cancelled formally.