Directional Derivative — Mathematics Knowledge System

Overview

The directional derivative D_u f generalizes the partial derivative: instead of measuring change along just the x-axis or y-axis, it measures the rate of change of f in an arbitrary direction u. Partial derivatives ∂f/∂x and ∂f/∂y are just the special cases where u = (1,0) or u = (0,1).

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Formal Definition

Definition

The directional derivative of f at point P in the direction of unit vector u:

D_{\mathbf u}f(P) = \lim_{h\to 0} \frac{f(P+h\mathbf u) - f(P)}{h}

Limit definition

D_{\mathbf u}f = \nabla f \cdot \mathbf u

Valid when f is differentiable; u must be a unit vector (|u|=1)

Gradient formula

Notation

Notation	Meaning
$D_{\mathbf u}f$	Directional derivative of f in the direction of unit vector u
$\frac{\partial f}{\partial \mathbf{u}}$	Alternative notation for the directional derivative

Worked Examples

Normalize v to a unit vector: |v|=5, so u=(3/5, 4/5).
$\mathbf{u} = \left(\frac{3}{5},\frac{4}{5}\right)$
Compute the gradient at (1,2).
$\nabla f = (2x+3y, 3x) \implies \nabla f(1,2) = (8, 3)$
Dot the gradient with the unit vector.
$D_{\mathbf u}f = (8,3)\cdot\left(\frac{3}{5},\frac{4}{5}\right) = \frac{24}{5}+\frac{12}{5} = \frac{36}{5}$

Answer: D_u f(1,2) = 36/5 = 7.2

Practice Problems

Difficulty 6/10

Find the directional derivative of f(x,y)=xy² at (2,1) in the direction of (1,1) (unnormalized).

Difficulty 5/10

A hiker stands on terrain z = f(x,y). In which direction is the ground steepest uphill, and how does the directional derivative express that?

Common Mistakes

Common Mistake

Plugging a non-unit direction vector directly into ∇f·v without normalizing.

The formula D_u f = ∇f·u requires u to be a UNIT vector. If given an arbitrary vector v, first divide by |v| to get u=v/|v|, otherwise the result is scaled incorrectly by |v|.

Quiz

The directional derivative D_u f is largest when the unit vector u points:

opposite to the gradientalong the gradient ∇fperpendicular to the gradientin the x-direction

Along a level curve (contour), the directional derivative of f is:

maximalzeronegative infinityequal to |∇f|

Summary

The directional derivative D_u f measures the rate of change of f in an arbitrary unit-vector direction u.
D_u f = ∇f · u, requiring u to be a unit vector.
Partial derivatives are special cases: ∂f/∂x = D_u f with u=(1,0).
The direction of maximum D_u f is u = ∇f/|∇f|, matching the steepest-ascent property of the gradient.

References

WebsiteWikipedia — Directional derivative