Second Partial Derivative/Examples/u + ln u = x y

Examples of Second Partial Derivative
Let $u + \ln u = x y$ be an implicit function.

Then:


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

Proof
From Partial Derivative Examples: u + ln u = x y:

Then:

Then we have that:


 * $\dfrac {\partial^2 u} {\partial x \partial y} = \map {\dfrac \partial {\partial x} } {\dfrac {u x} {u + 1} }$

which is exactly the same as:


 * $\dfrac {\partial^2 u} {\partial y \partial x} = \map {\dfrac \partial {\partial y} } {\dfrac {u y} {u + 1} }$

but with $y$ and $x$ exchanged.

Hence:

and it is seen that the second partial derivatives are indeed the same.