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 with respect to $x$ 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.

$\blacksquare$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: $1.1$ Partial Derivatives: Example $\text A$