Table 4.2.1 lists the four second-order 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 D-operator notation.

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 $f_{\mathrm{yx}}\=\frac{{\partial }^{2}f}{\partial x \partial y}$ 

The subscripts in D-operator 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.

Table 4.2.2 lists the eight third-order partial derivatives for $f\left(x\,y\right)$.

Under the assumption of the equality of mixed partials, the following equalities hold.

The Calculus palette contains templates for both first-order and second-order 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 operator-notation templates.

Maple's D-operator 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 $y$.

For each $f$ and $\left(a\,b\right)$ in Examples 4.2.(1-5), obtain all second partial derivatives, both at $\left(x\,y\right)$ and at $\left(a\,b\right)$.