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

Example of Partial Derivative
Consider the simultaneous equations:


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

Then:
 * $\dfrac {\partial^2 u} {\partial x^2} = \dfrac {u \paren {\paren {y - v}^2 - v \paren {\paren {1 - y u} + u \dfrac {y - v} {1 - u v} \paren {1 + y - v - y u} } } } {\paren {1 - u v}^2}$

Proof
We have from Partial Derivatives of $v + \ln u = x y$, $u + \ln v = x - y$ that:

Hence: