# I was wondering whether anybody could explain how you derive this implicit differentiation rule: (del z)/(del x)=−(del f/ del x)/(del f/ del z) if you have a function z implicity defined by f(x,y,z)=0?

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 $f\left(x,y,z\right)=0$?
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abortargy
Take $z$ a function of $x,y$. Then it is always true that
$f\left(x,y,z\left(x,y\right)\right)=0$
Its derivative is also 0. So
$\frac{\mathrm{\partial }f}{\mathrm{\partial }x}dx+\frac{\mathrm{\partial }f}{\mathrm{\partial }x}dy+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\cdot \left(\frac{\mathrm{\partial }z}{\mathrm{\partial }x}dx+\frac{\mathrm{\partial }f}{\mathrm{\partial }x}dy\right)=0.$
If we consider the case $dy=0$, $dy\ne 0$, then divide by $dx$ and cross out $dy$,
$\frac{\mathrm{\partial }f}{\mathrm{\partial }x}+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\frac{\mathrm{\partial }z}{\mathrm{\partial }x}=0.$
Or,
$\frac{\mathrm{\partial }z}{\mathrm{\partial }x}=-\frac{\mathrm{\partial }f/\mathrm{\partial }x}{\mathrm{\partial }f/\mathrm{\partial }z}$