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Suppose $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ is such that the Implicit Function Theorem applies to $F(x,y,z) = 0$ to determine $z = f(x,y)$, $x=g(y,z)$ and $y=h(x,z)$ in a neighborhood of a point $a \in \mathbb{R}^3$. Use the Implicit Function Theorem to prove $\frac{\partial f}{\partial x} \frac{\partial g}{\partial y} \frac{\partial h}{\partial z} = -1$ at a

I haven't used the implicit function on functions that aren't explicitly defined, so I'm a little thrown off as to how I can show those partial fractions equal -1. I know I can show things like

given $F(x, y, f(x,y)) = 0$ I can take the partial with respect to x and get

$\frac{\partial F}{\partial x} \frac{\partial x}{\partial x} + \frac{\partial F}{\partial y} \frac{\partial y}{\partial x} +\frac{\partial F}{\partial z} \frac{\partial z}{\partial x} = 0$

and I can show similar with the other two definitions, but I can't figure out how to use these to show those partials with respect to each of the functions equals -1.

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    $\begingroup$ Use this fact. $$f'_x=\frac{F'_x}{F'_z}$$ $\endgroup$ – pointer Oct 21 '14 at 19:35
  • $\begingroup$ @user121270: You forgot a minus sign. $\endgroup$ – Hans Lundmark Oct 21 '14 at 19:44
  • $\begingroup$ Oh, I am sorry. $\endgroup$ – pointer Oct 21 '14 at 20:03