Partial Derivative/Examples/x^(x y)
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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:
\(\ds \dfrac {\partial f} {\partial x}\) | \(=\) | \(\ds x^{x y} \paren {y \ln x + y}\) | ||||||||||||
\(\ds \dfrac {\partial f} {\partial y}\) | \(=\) | \(\ds x^{x y + 1} \ln x\) |
Proof
With Respect to $x$
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.
$\blacksquare$
With Respect to $y$
By definition, the partial derivative with respect to $y$ 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:
\(\ds \map {D_y} {x^{x y} }\) | \(=\) | \(\ds x \map {D_{x y} } {x^{x y} }\) | Derivative of Function of Constant Multiple | |||||||||||
\(\ds \) | \(=\) | \(\ds x \paren {x^{x y} } \ln x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^{x y + 1} \ln x\) |
$\blacksquare$
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: $1.1$ Partial Derivatives: Example $\text A$