Directional Derivative

analysis

Definition

Directional Derivative

Let be an open set, let be a totally differentiable function, let , and let .

The directional derivative of at in direction is

provided the limit exists.

It measures the first-order rate of change of at when one moves from in direction .

Gradient form

If is totally differentiable at , then the directional derivative is given by:

So the directional derivative is the dot product of the gradient with the direction vector.

Unit directions

If is a unit vector, then is the rate of change per unit distance.

If is not a unit vector, its length also scales the value:

Relation to partial derivatives

The partial derivatives are special cases of directional derivatives.

For the standard basis vectors ,

So partial derivatives only look along the coordinate axes, while directional derivatives allow arbitrary directions.

Relation to JVP

For a scalar-valued function , the Jacobian-vector product is exactly the directional derivative:

This is because the Jacobian of a scalar-valued function is the transpose of the gradient.

Geometric meaning

The gradient points in the direction of steepest increase.

• If , then increases in direction .
• If , then decreases in direction .
• If , then has no first-order change in that direction.

For a unit vector ,

where is the angle between and .

So the directional derivative is largest when points in the same direction as the gradient.

Examples

Quadratic function

Let

Then

At the point , we get

In direction

the directional derivative is

If we instead use the associated unit vector

then

The two values differ by the length .

Partial derivative as a directional derivative

Let

Choose the direction

Then

This is exactly