For the function $f\left(x\,y\right)$, the first partial derivative with respect to $x$ is the rate of change of $f$ along the cross section $y\=c$. On this cross section, $y$ is held constant, and the derivative of $f$ is taken with respect to $x$. See Figure 4.1.1.

For the function $f\left(x\,y\right)$, the first partial derivative with respect to $y$ is the rate of change of $f$ along the cross section $x\=c$. On this cross section, $x$ is held constant, and the derivative of $f$ is taken with respect to $y$. See Figure 4.1.2.

Recall the two most prevalent notations for ordinary derivatives, listed in Table 4.1.2.

The advantage of the Newtonian notation is that a derivative, evaluated at $x\=a$, can easily be expressed as $f\prime \left(a\right)$, whereas in the notation of Leibniz, it becomes the much more involved $\genfrac{}{}{0ex}{}{\frac{\mathrm{df}}{\mathrm{dx}}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{a\=a}$. Much the same issues prevail for the three common notations used for partial derivatives.

The $\frac{d}{\mathrm{dx}}$ operator of Leibniz becomes $\frac{\partial }{\partial x}$ for partial differentiation. The prime of Newtonian notation becomes the subscript seen in Table 4.1.1. And then there's the $\mathrm{D}$-operator notation, an alternate form of the Leibniz notation. Of the three, once again the Newtonian notation is the clearest and most compact for expressing the evaluation of a partial derivative at some fixed point. Table 4.1.3 clarifies these issues.

The subscripts 1 and 2 on the D-operator refer to the "first" and "second" variables, respectively, in the definition of the function $f\left(x\,y\right)$. The first variable is the first argument, here $x$; the second, is the second argument, here $y$.

Sometimes the Leibniz usage is abused when evaluating at a point by writing $\frac{\partial }{\partial x}f\left(a\,b\right)$. In a certain sense, if it is clear that $f$ is really a function of $x$ and $y$, and that $\left(a\,b\right)$ is a point of evaluation, then it is possible to allow this notation to mean "first differentiate, then evaluate." However, this ambiguity will not appear in the remainder of this work. The alert reader will already realize there is a bias towards the subscript notation, which will be even more apparent throughout the remainder of this work.

All derivatives in Maple are partial derivatives! The underlying diff command differentiates an expression with respect to a stated variable, holding all other variables constant. The D-operator uses a numeric subscript to indicate the variable of differentiation, and again, holds all other variables constant.

However, there are several options for obtaining a partial derivative in a syntax-free way. All these possibilities are summarized in Table 4.1.4, where, for the sake of clarity, $G$ is the name of an expression in the variables $x$ and $y$ and $g$ is the name of a function of $x$ and $y$.

To implement subscripts as differentiation operators, the name of the function (or expression) and the name for the derivative must be different. Maple does not see the names $f$ and $f_x$ as sufficiently distinct for these purposes. Hence, if the names are to be related, as in $f$ and $f_x$, then the subscripted name has to be an Atomic Identifier. It would "work" to use $f$ for the function (or expression) and $F$ for the derivative, without making $F$ an Atomic Identifier, but since there are two possible first partials, a third name would be required! The use of the Atomic Identifier for the subscripted name of the function representing the partial derivative seems to be the best choice.