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Table 4.2.1 lists the four secondorder partial derivatives for a function two variables. There are three common styles of notation for these derivatives, the subscript notation, the operator notation, and the Doperator notation.
Notation

SecondOrder Partial Derivatives for $f\left(x\,y\right)$

Subscript

${f}_{\mathrm{xx}}\,{f}_{\mathrm{xy}}\,{f}_{\mathrm{yx}}\,{f}_{\mathrm{yy}}$

Operator

$\frac{{\partial}^{2}f}{{\partial}^{}{x}^{2}}\,\frac{{\partial}^{2}f}{\partial y\partial x}comma;\frac{{\partial}^{2}f}{\partial x\partial y}comma;\frac{{\partial}^{2}f}{{\partial}^{}{y}^{2}}$

Doperator

${\mathrm{D}}_{1\,1}f\,{\mathrm{D}}_{1\,2}f\,{\mathrm{D}}_{2\,1}f\,{\mathrm{D}}_{2\,2}f$

Table 4.2.1 Secondorder partial derivatives for $f\left(x\,y\right)$



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For subscript notation, the lexical order of the subscripts is the order in which the derivatives are taken. Thus, in the derivative ${f}_{\mathrm{xy}}$, the derivative with respect to $x$ is taken first, while the derivative with respect to $y$ is taken second.
However, for operator notation, the operators are applied from the left, so that in the derivative $\frac{{\partial}^{2}f}{\partial y\partial x}$, the derivative with respect to $x$ is taken first, with the derivative with respect to $y$ taken second. Hence
${f}_{\mathrm{xy}}\=\frac{{\partial}^{2}f}{\partial y\partial x}$ and ${f}_{\mathrm{yx}}\=\frac{{\partial}^{2}f}{\partial x\partial y}$
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The subscripts in Doperator notation refer to the variable in that position in the argument list for the function. This, for $f\left(x\,y\right)$, the subscript 1 refers to the first variable, namely, $x$; whereas the subscript 2 refers to the second, namely, $y$. Thus
${\mathrm{D}}_{1\,2}f\={f}_{\mathrm{xy}}\=\frac{{\partial}^{2}f}{\partial y\partial x}$ and ${\mathrm{D}}_{2\,1}f\={f}_{\mathrm{yx}}\=\frac{{\partial}^{2}f}{\partial x\partial y}$
At any point at which $f\left(x\,y\right)$ is sufficiently well behaved, the mixed partial derivatives ${f}_{\mathrm{xy}}$ and ${f}_{\mathrm{yx}}$ are equal. Section 4.11 explores the precise conditions under which the mixed partial derivatives are equal. Example 4.2.5 explores a function $f\left(x\,y\right)$ for which the mixed partials are not equal at the origin.
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Table 4.2.2 lists the eight thirdorder partial derivatives for $f\left(x\,y\right)$.
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${f}_{\mathrm{xxx}}\=\frac{{\partial}^{3}f}{{\partial}^{}{x}^{3}}\={\mathrm{D}}_{1\,1\,1}f$

${f}_{\mathrm{yyy}}\=\frac{{\partial}^{3}f}{{\partial}^{}{y}^{3}}\={\mathrm{D}}_{2\,2\,2}f$

${f}_{\mathrm{xxy}}\=\frac{{\partial}^{3}f}{\partial y\partial {x}^{2}}equals;{\mathrm{D}}_{1comma;1comma;2}f$

${f}_{\mathrm{xyy}}\=\frac{{\partial}^{3}f}{\partial {y}^{2}\partial x}equals;{\mathrm{D}}_{1comma;2comma;2}f$

${f}_{\mathrm{xyx}}\=\frac{{\partial}^{3}f}{\partial x\partial y\partial x}equals;{\mathrm{D}}_{1comma;2comma;1}f$

${f}_{\mathrm{yxy}}\=\frac{{\partial}^{3}f}{\partial y\partial x\partial y}equals;{\mathrm{D}}_{2comma;1comma;2}f$

${f}_{\mathrm{yxx}}\=\frac{{\partial}^{3}f}{\partial {x}^{2}\partial y}equals;{\mathrm{D}}_{2comma;1comma;1}f$

${f}_{\mathrm{yyx}}\=\frac{{\partial}^{3}f}{\partial x\partial {y}^{2}}equals;{\mathrm{D}}_{2comma;2comma;1}f$

Table 4.2.2 Thirdorder partial derivatives for $f\left(x\,y\right)$



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Under the assumption of the equality of mixed partials, the following equalities hold.
${f}_{\mathrm{xxy}}\={f}_{\mathrm{xyx}}\={f}_{\mathrm{yxx}}$

${f}_{\mathrm{xyy}}\={f}_{\mathrm{yxy}}\={f}_{\mathrm{yyx}}$

$\frac{{\partial}^{3}f}{\partial y\partial {x}^{2}}equals;\frac{{\partial}^{3}f}{\partial x\partial y\partial x}equals;\frac{{\partial}^{3}f}{\partial {x}^{2}\partial y}$

$\frac{{\partial}^{3}f}{\partial {y}^{2}\partial x}equals;\frac{{\partial}^{3}f}{\partial y\partial x\partial y}equals;\frac{{\partial}^{3}f}{\partial x\partial {y}^{2}}$

${\mathrm{D}}_{1\,1\,2}f\={\mathrm{D}}_{1\,2\,1}f\={\mathrm{D}}_{2\,1\,1}f$

${\mathrm{D}}_{1\,2\,2}f\={\mathrm{D}}_{2\,1\,2}f\={\mathrm{D}}_{2\,2\,1}f$



The Calculus palette contains templates for both firstorder and secondorder partial derivatives of a function of two variables. These templates can be edited to fit other cases, and the symbol $\partial$, found in the Operators palette, can be used to build operatornotation templates.
Maple's Doperator acts on a function and returns derivatives as functions. Hence, parentheses are required, as in ${\mathrm{D}}_{1\,2}\left(f\right)$ and ${\mathrm{D}}_{1\,2}\left(f\right)\left(x\,y\right)$. In the first instance, the mixed partial ${f}_{\mathrm{xy}}$ is returned as a function; in the second, it is evaluated at the point $\left(x\,y\right)$, so is simply an expression in $x$ and $y$.
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