Directional Derivatives

The central idea

If $f:\mathbb R^n\to\mathbb R$ is differentiable at $\mathbf a$, its directional derivative in a unit direction $\mathbf u$ is $D_{\mathbf u}f(\mathbf a)=\lim_{h\to0}[f(\mathbf a+h\mathbf u)-f(\mathbf a)]/h =\nabla f(\mathbf a)\cdot\mathbf u$. A nonunit direction changes the rate's scale.

Definitions, hypotheses, and notation

The formula from the gradient requires differentiability. Existence of all directional derivatives alone does not guarantee differentiability or a single linear map controlling every direction. At differentiable points, dependence on $\mathbf u$ is linear before unit normalization and the Cauchy–Schwarz inequality proves the maximum-rate statement.

A phrase such as 'toward point B' means subtract base point A from B; reversing the subtraction changes the derivative's sign. If coordinates have physical units or unequal scaling, the meaning of a Euclidean unit direction should be interpreted in that coordinate model.

Conceptual meaning

The gradient projects onto the chosen direction. Its largest possible directional derivative is $\|\nabla f\|$, attained along the gradient; the most negative is its opposite. Directions tangent to a level surface give zero first-order change.

A dependable method and decision rule

1. Compute the displacement direction from the given geometric information.
2. Normalize it to length one unless the problem explicitly defines otherwise.
3. Compute and evaluate the gradient at the base point.
4. Take the dot product.
5. Interpret sign and units as change per unit distance in that direction.

Fully worked example

Graphical or geometric meaning

On a contour map, the gradient crosses level curves perpendicularly toward higher labels. A direction making angle $\theta$ with the gradient has rate $\|\nabla f\|\cos\theta$, so alignment controls the projection.

Common mistakes and why they fail

Verification and reasonableness checks

• Confirm $\|\mathbf u\|=1$.
• Verify $|D_\mathbf u f|\le\|\nabla f\|$.
• Use a small finite difference along the specified ray.

Normalize the direction before measuring change

For differentiable $f$, the directional derivative in a unit direction $\mathbf u$ is $D_{\mathbf u}f=\nabla f\cdot\mathbf u$. If a problem supplies a vector $\mathbf v$, first replace it by $\mathbf v/\|\mathbf v\|$ unless rate per unit parameter rather than per unit distance is intended. Cauchy–Schwarz shows the largest directional derivative is $\|\nabla f\|$ in the gradient direction, and the most negative is its opposite. A zero gradient makes every first-order directional derivative zero, though higher-order behavior may remain. Evaluate the gradient at the specified point before taking the dot product. The result is scalar and has output units per unit distance, which helps detect an unnormalized direction.

Practice

1. For $f=x+y$, find the derivative in direction $\langle1,1\rangle$ normalized.
2. Which direction maximizes increase?
3. What is every directional derivative when $\nabla f=0$?

Connections and next steps

• Gradient
• Partial Derivatives
• Tangent Planes