Partial Derivative/Examples/x^(x y)/wrt x

Example of Partial Derivative
Let $\map f {x, y} = x^{x y}$ be a real function of $2$ variables such that $x, y \in \R_{>0}$.

Then:
 * $\dfrac {\partial f} {\partial x} = x^{x y} \paren {y \ln x + y}$

Proof
By definition, the partial derivative $x$ is obtained by holding $y$ constant.

Hence Derivative of $x^{a x}$ can be directly used:


 * $\dfrac \d {\d x} x^{y x} = y x^{y x} \paren {\ln x + 1}$

The result can then be rearranged to match the form given.