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59.2: Higher-Order Partial Derivatives

Last updated

Aug 8, 2024
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- 59.1: First Partial Derivatives
- 60: Lagrangian Mechanics

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It is similarly possible to take higher-order partial derivatives. For a function of two variables $f(x, y)$ , there are three possible second derivatives:

$\frac{\partial^{2} f}{\partial x^{2}}=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) ; \quad \frac{\partial^{2} f}{\partial x \partial y}=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) ; \quad \text { and } \quad \frac{\partial^{2} f}{\partial y^{2}}=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) .$

In the second case, the order of differentiation doesn't matter: $\partial^{2} f /(\partial x \partial y) \equiv \partial^{2} f /(\partial y \partial x)$ . This property is known as Clairaut's theorem.

For example, suppose $f(x, y)$ is as given by Eq. 59.1.3. Then the second partial derivatives of $f$ are found by taking partial derivatives of Eqs. 59.1.4 and 59.1.5:

$\begin{align} \frac{\partial^{2} f}{\partial x^{2}} & =30 x y^{5} \\[8pt] \frac{\partial^{2} f}{\partial x \partial y} & =75 x^{2} y^{4}-42 y^{5} \\[8pt] \frac{\partial^{2} f}{\partial y^{2}} & =100 x^{3} y^{3}+8-210 x y^{4} \end{align}$

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