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
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: Exercise $11$