Exercise 58

Suppose that the equation \(F(x, y, z) = 0\) implicitly defines each of the three variables \(x, y\), and \(z\) as functions of the other two: \(z = f(x, y)\), \(y = g(x, z)\), \(x = h(y, z)\). If \(F\) is differentiable and \(F_x, F_y\), and \(F_z\) are all nonzero, show that \[ \frac{\partial z}{\partial x} \frac{\partial x}{\partial y} \frac{\partial y}{\partial z} = -1 \]