Originally Posted by

**SlipEternal** It has been a long times since I have done multivariable calculus, so I may have this wrong, but I believe:

$\dfrac{df}{dz} = 0 = \dfrac{\partial f}{\partial x} \dfrac{\partial x}{\partial z} + \dfrac{\partial f}{\partial y} \dfrac{\partial y}{\partial z} + \dfrac{\partial f}{\partial z}$

Solving for $\dfrac{\partial f}{\partial z}$ gives

$\dfrac{\partial f}{\partial z} = -\dfrac{ \left( \dfrac{\partial f}{\partial x} \right) }{ \left( \dfrac{\partial z}{\partial x} \right) } - \dfrac{ \left( \dfrac{\partial f}{\partial y} \right) }{ \left( \dfrac{\partial z}{\partial y} \right) }$