Third Partial Derivatives of x^y

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Theorem

Let:

$u = x^y$

Then:

$\dfrac {\partial^3 u} {\partial x^2 \partial y} = \dfrac {\partial^3 u} {\partial x \partial y \partial x}$


Proof

\(\ds \dfrac {\partial u} {\partial y}\) \(=\) \(\ds x^y \ln x\) Derivative of General Logarithm Function keeping $x$ constant
\(\ds \leadsto \ \ \) \(\ds \dfrac {\partial^2 u} {\partial x \partial y}\) \(=\) \(\ds \map {\dfrac \partial {\partial x} } {\dfrac {\partial u} {\partial y} }\) Definition of Second Partial Derivative
\(\ds \) \(=\) \(\ds y x^{y - 1} \ln x + \dfrac 1 x x^y\) Power Rule for Derivatives, Derivative of Natural Logarithm keeping $y$ constant
\(\ds \) \(=\) \(\ds x^{y - 1} \paren {y \ln x + 1}\) rearranging
\(\ds \leadsto \ \ \) \(\ds \dfrac {\partial^3 u} {\partial x^2 \partial y}\) \(=\) \(\ds \map {\dfrac \partial {\partial x} } {\dfrac {\partial^2 u} {\partial x \partial y} }\) Definition of Third Partial Derivative
\(\ds \) \(=\) \(\ds x^{y - 1} \map {\dfrac \partial {\partial x} } {y \ln x + 1} + \paren {y \ln x + 1} \map {\dfrac \partial {\partial x} } {x^{y - 1} }\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds x^{y - 1} \paren {\dfrac y x} + \paren {y \ln x + 1} \paren {y - 1} x^{y - 2}\) Derivative of Natural Logarithm, Power Rule for Derivatives keeping $y$ constant
\(\ds \) \(=\) \(\ds x^{y - 2} \paren {y + \paren {y \ln x + 1} \paren {y - 1} }\) simplifying
\(\ds \) \(=\) \(\ds x^{y - 2} \paren {y + y^2 \ln x + y -y \ln x - 1}\) multiplying out
\(\ds \) \(=\) \(\ds x^{y - 2} \paren {2 y + y \paren {y - 1} \ln x - 1}\) simplifying

$\Box$


\(\ds \dfrac {\partial u} {\partial x}\) \(=\) \(\ds y x^{y - 1}\) Power Rule for Derivatives keeping $y$ constant
\(\ds \leadsto \ \ \) \(\ds \dfrac {\partial^2 u} {\partial y \partial x}\) \(=\) \(\ds \map {\dfrac \partial {\partial y} } {\dfrac {\partial u} {\partial x} }\) Definition of Second Partial Derivative
\(\ds \) \(=\) \(\ds x^{y - 1} \map {\dfrac \partial {\partial y} } y + y \map {\dfrac \partial {\partial y} } {x^{y - 1} }\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds x^{y - 1} + y x^{y - 1} \ln x\) Derivative of Identity Function, Derivative of General Logarithm Function keeping $x$ constant
\(\ds \leadsto \ \ \) \(\ds \dfrac {\partial^3 u} {\partial x \partial y \partial x}\) \(=\) \(\ds \map {\dfrac \partial {\partial x} } {\dfrac {\partial^2 u} {\partial y \partial x} }\) Definition of Third Partial Derivative
\(\ds \) \(=\) \(\ds \map {\dfrac \partial {\partial x} } {x^{y - 1} + y x^{y - 1} \ln x}\)
\(\ds \) \(=\) \(\ds \paren {y - 1} x^{y - 2} + y \paren {x^{y - 1} \map {\dfrac \partial {\partial x} } {\ln x} + \ln x \map {\dfrac \partial {\partial x} } {x^{y - 1} } }\) Power Rule for Derivatives, Product Rule for Derivatives
\(\ds \) \(=\) \(\ds \paren {y - 1} x^{y - 2} + y x^{y - 1} \paren {\dfrac 1 x} + y \ln x \paren {y - 1} x^{y - 2}\) Derivative of Natural Logarithm, Power Rule for Derivatives
\(\ds \) \(=\) \(\ds x^{y - 2} \paren {\paren {y - 1} + y + y \ln x \paren {y - 1} }\) factoring
\(\ds \) \(=\) \(\ds x^{y - 2} \paren {2 y + y \paren {y - 1} \ln x - 1}\) simplifying

$\Box$


The two expressions are equal.

Hence the result.

$\blacksquare$


Sources