6.4 Derivatives in Two Dimensions: Directional Derivative and Partial
Derivatives

Suppose we have a function of two variables,
$f(x,y)$
.

Such things are sometimes called
scalar fields
. (Scalar to indicate they are not vectors, and fields to indicate that there are two or more variables.)

We can choose a particular line in
$xy$
plane, (for example
$x={x}_{0}+r\mathrm{cos}\theta ,y={y}_{0}+r\mathrm{sin}\theta $
) and consider the function of
$r$
(with everything else in it fixed):
$f({x}_{0}+r\mathrm{cos}\theta ,{y}_{0}+r\mathrm{sin}\theta )$
.

$\frac{df}{dr}$
is then called
the directional derivative of
$f$ at
$({x}_{0},{y}_{0})$ in the direction in the
$xy$ plane having slope
$\mathrm{tan}\theta $.

In other words we can, by picking out any particular
line in the
$xy$ plane, reduce
$f$ to a function of a single value defined on that line and define the derivative of that one variable function with respect to distance
on that line.

This derivative is called
the directional derivative of
$f$ in the direction of the line.
(You may examine the directional derivatives of functions of two variables in the
applet
.)

The directional derivative in the direction of the x-axis is called the
partial
derivative of
$f$ with respect to $x$
, and is written as
$\frac{\partial f}{\partial x}$
.

Similarly the directional derivative of
$f$
in the direction of the y-axis is called the
partial derivative of
$f$ with respect to$y$
, and is written as
$\frac{\partial f}{\partial y}$
.

These partial derivatives are computable exactly as ordinary one dimensional derivatives are. When computing the partial derivative with respect to
$x$
, you treat
$y$
as a constant, and differentiate with respect to
$x$
exactly as you do in one dimension.