Partial Derivative/Examples/v + ln u = x y, u + ln v = x - y

Example of Partial Derivative
Consider the simultaneous equations:


 * $\begin {cases} v + \ln u = x y \\ u + \ln v = x - y \end {cases}$

Then:

Proof
and so combining $(1)$ and $(2)$ into matrix form:


 * $\begin {pmatrix} 1 & u \\ v & 1 \end {pmatrix} \begin {pmatrix} \dfrac {\partial u} {\partial x} \\ \dfrac {\partial v} {\partial x} \end {pmatrix} = \begin {pmatrix} y u \\ v \end {pmatrix}$

Hence by Cramer's Rule:

and: