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- Jan 26, 2012

- 995

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**Problem**: 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: $z=f(x,y)$, $y=g(x,z)$ and $x=h(y,z)$. If $F$ is differentiable and $F_x$, $F_y$, and $F_z$ are nonzero, show that

\[\frac{\partial z}{\partial x}\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}=-1.\]

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