59.2: Higher-Order Partial Derivatives
- Page ID
- 92384
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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:
\[
\begin{align}
\frac{\partial^{2} f}{\partial x^{2}} & =30 x y^{5} \\[8pt]
\frac{\partial^{2} f}{\partial x \partial y} & =75 x^{2} y^{4}-42 y^{5} \\[8pt]
\frac{\partial^{2} f}{\partial y^{2}} & =100 x^{3} y^{3}+8-210 x y^{4}
\end{align}
\]