The Gradient and the Directional Derivative

For a differentiable function f, the directional derivative of f at the point (x, y) in the direction of the unit vector can be expressed as

where

and so that

and

We can find by using the formula for the increment .

where depends on and as .
Hence,

The term does not depend on . As and , and therefore .
Hence,

Thus the directional derivative of f at (x,y) in the direction of the unit vector is the dot product of the gradient of f at that point and the unit vector.
In short, is the component of in the direction of .

If f is differentiable, and we find the gradient at a given point, we can calculate the directional derivative of f at that point in any direction .

If , then and similarly so that in the case the directional derivative give rate of f in the i and j directions and coincide with the partial derivatives.

The concept of the directional derivative and its relation to the gradient can be extended to the functions three (or more) variables. If w=f(x,y,z) is a differentiable function, then the directional derivative at (x,y,z) in the direction of the unit vector is


where are the direction cosines a line through x,y,z in the direction .

In vector notation

and so that
and and


Thus where depends on and and tends to zero as tend to (0,0,0).

Since f is differentiable, the same mode of proof as for a function of two variables, shows that

Again the directional derivative is the dot product of the gradient vector and the unit vector .


Copyright 2001 Rensselaer Polytechnic Institute. All Rights Reserved.