Why can the directional derivative be written as the dot product of the direction vector and the gradient?

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. Write out the general formula for the directional derivative: a*(∂f/∂x) + b*(∂$f/y$).
2. Identify the components a and b as the vector w.
3. Identify the partial derivatives ∂f/∂x and ∂f/∂y as the gradient vector ∇f.
4. Recognize the sum of products as the dot product w·∇f.

Example

The expanded form a*(∂f/∂x) + b*(∂f/∂y) is equivalent to [a,b]·[∂f/∂x, ∂f/∂y].

Common misconceptions

• Thinking the dot product notation is just shorthand without mathematical equivalence.
• Confusing the gradient vector with the Hessian matrix.

Watch the explanation

Explore next

• How does the dot-product formula generalize to higher dimensions?
• What is the geometric meaning of the gradient vector?
• Why are there so many different notations for the directional derivative?