What is the relationship between the directional derivative and the gradient vector?

Conditions

• The function $f(x,y)$ is differentiable.
• The direction vector is $w=[a,b]$.
• The gradient ∇f = [∂f/∂x, ∂f/∂y] exists.

Reasoning, step by step

1. Identify the components of the direction vector w.
2. Identify the components of the gradient vector ∇f.
3. Compute the dot product of w and ∇f.
4. Recognize that this dot product equals the directional derivative.

Example

The directional derivative ∇_w f can be written compactly as w·∇f.

Common misconceptions

• Confusing the gradient vector with the directional derivative.
• Thinking the dot product requires normalizing the direction vector first (the video presents the unnormalized form).

Watch the explanation

Explore next

• Why can the directional derivative be written as w·∇f?
• How does the dot-product formula generalize to higher dimensions?
• What is the geometric meaning of the gradient vector?