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

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:
 * $\dfrac {\partial f} {\partial y} = x^{x y + 1} \ln x$

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

From Derivative of Power of Constant:
 * $\map {D_y} {x^y} = x^y \ln x$

for constant $a$.

Then: