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 = x₀ + rcos, y = y₀ + rsin) and consider the function of r (with everything else in it fixed): f(x₀ + rcos, y₀ + rsin).

is then called the directional derivative of f at (x₀, y₀) in the direction in the xy plane having slope tan.

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

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

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.