Chapter 14.5, Problem 58E

Suppose that the equation $F\left(x,y,z\right)=0$ implicitly defines each of the three variables x, y, and z as functions of the other two: $z=f\left(x,y\right),y=g\left(x,z\right),x=h\left(y,z\right).$ If F is differentiable and $F_x,F_y$ and $F_z$ are all nonzero, show that

$\frac{\partial z}{\partial x}\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}=-1$