I was wondering whether anybody could explain how you derive this implicit differentiation rule:

$\frac{\mathrm{\partial}z}{\mathrm{\partial}x}=\frac{-\mathrm{\partial}f/\mathrm{\partial}x}{\mathrm{\partial}f/\mathrm{\partial}z}$

if you have a function $z$ implicity defined by $f(x,y,z)=0$?

$\frac{\mathrm{\partial}z}{\mathrm{\partial}x}=\frac{-\mathrm{\partial}f/\mathrm{\partial}x}{\mathrm{\partial}f/\mathrm{\partial}z}$

if you have a function $z$ implicity defined by $f(x,y,z)=0$?