# Suppose z = z(x,y) is given implicitly by the equation xyz = (x+y) z. Find z x and ...

## Question:

Suppose{eq}\displaystyle \ z = z(x,y) {/eq} is given implicitly by the equation{eq}\displaystyle \ \frac {xy}{z} = (x+y) \ln z. {/eq}

Find{eq}\displaystyle \ \frac {\partial z}{\partial x} {/eq} and{eq}\displaystyle \ \frac {\partial z}{\partial y}. {/eq}

## Implicit Derivation:

The functions, ideally could be given in explicit form, but this is not always possible.

For the case of the implicit derivation, the partial derivatives are calculated, in two variables such as:

{eq}F\left( {x,y,z} \right) = 0 \to \left\{ \begin{array}{l} \frac{{\partial z}}{{\partial x}} = - \frac{{\frac{{\partial F}}{{\partial x}}}}{{\frac{{\partial F}}{{\partial z}}}}\\ \frac{{\partial z}}{{\partial y}} = - \frac{{\frac{{\partial F}}{{\partial y}}}}{{\frac{{\partial F}}{{\partial z}}}} \end{array} \right. {/eq}