---
title: "Directional Derivatives and the Gradient"
description: "For a differentiable function, dot its gradient with a unit direction to find a rate per unit distance. A nonzero gradient gives the direction of fastest increase and is perpendicular to level-curve t"
canonical: https://lightmysky.com/learn/mathematics/directional-derivatives-and-the-gradient-mt_LW_KckY5Ad
source: https://lightmysky.com/learn/mathematics/directional-derivatives-and-the-gradient-mt_LW_KckY5Ad.md
retrieved: 2026-11-12
---

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# Directional Derivatives and the Gradient

For a differentiable function, dot its gradient with a unit direction to find a rate per unit distance. A nonzero gradient gives the direction of fastest increase and is perpendicular to level-curve tangents.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/directional-derivatives-and-the-gradient-mt_LW_KckY5Ad

## Ready when they can

- Compute and interpret gradient components and length; use unit directions for rates per unit distance and compare them with rates along a parameterized path
- Find maximum and minimum directional derivatives and their directions, including the zero-gradient case
- Use a nonzero gradient to check level-curve tangents and normals, and distinguish zero instantaneous change from a constant-value path

## Lesson: The gradient points uphill

For differentiable f(x, y), write fx and fy for its partial derivatives. Hold the other variable fixed when finding each partial. The gradient ∇f = (fx, fy) lists them in x, y order. Its components give the rates along (1, 0) and (0, 1). Evaluate both partials at the point. The gradient length is sqrt(fx² + fy²); it need not be 1.

D_u f(P) is the instantaneous rate per unit distance in the input plane, in unit direction u = (u1, u2). For f differentiable at P, find its partials there and calculate D_u f(P) = fx(P)u1 + fy(P)u2 = ∇f(P) dot u. Multiply matching components and add. A unit vector has length 1. For a nonzero direction v = (v1, v2), first find |v| = sqrt(v1² + v2²), then use u = v/|v|.

*(drawing: The direction (0, 2) has length 2. Using (0, 1) gives the rate per unit distance.)*

For f differentiable at P with nonzero gradient g, D_u f(P) = |g| cos(theta). Here theta is the angle between g and unit u, and |g| = sqrt(g1² + g2²). Cosine ranges from -1 to 1. The maximum rate is |g| along g/|g|; the minimum is -|g| along -g/|g|. If g = (0, 0), every unit direction has derivative 0, so there is no unique fastest direction.

On a level curve f(x, y) = c, f stays constant. If f is differentiable, the chain rule gives ∇f dot T = 0 for any tangent velocity T. If the first partials are continuous near P and g = ∇f(P) is nonzero, the level curve has a tangent there and g is normal to it. To check a tangent vector, take its dot product with g. If g = (a, b), then (-b, a) is tangent because -ab + ab = 0.

Zero directional derivative describes first-order change at one point. It does not make a straight path stay level. For f = x² + 2y² at (0, 2), the gradient is (0, 8). Its dot product with (1, 0) is 0. But on the straight path (t, 2), f = 8 + t², which exceeds 8 whenever t is nonzero.

You have added the chain rule's two path contributions. For f differentiable at P and a straight path P + t v, df/dt at t = 0 is fx(P)v1 + fy(P)v2. This is rate per unit of t. For nonzero v, divide by its speed |v| to get rate per unit distance. For f = x + 2y on (2 + 2t, 1 + t), df/dt = 2 + 2 = 4. The speed is sqrt(5), so the rate per unit distance is 4/sqrt(5).

**Recap.** For differentiable f, dot its gradient with a unit direction for the rate. A nonzero gradient gives fastest increase and is normal to level-curve tangents.

*Lesson, questions, worked solutions and flashcards written for this dataset. External references were studied for mathematical facts and question design.*

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [The Multivariable Chain Rule](https://lightmysky.com/learn/mathematics/the-multivariable-chain-rule-mt_CHSHdNxPmJ)

## Optional connections

- [The Dot Product: Angles and Projections](https://lightmysky.com/learn/mathematics/the-dot-product-angles-and-projections-mt_yFBcyE4sFw)

## Opens up

- [Lagrange Multipliers](https://lightmysky.com/learn/mathematics/lagrange-multipliers-mt_0RPOBWZ1gs)
- [Critical Points and Optimisation in Two Variables](https://lightmysky.com/learn/mathematics/critical-points-and-optimisation-in-two-variables-mt_4qB5pAGeuR)
- [Conservative Fields and Path Independence](https://lightmysky.com/learn/mathematics/conservative-fields-and-path-independence-mt_5djKpI5F33)
- [Gradient Descent: Following the Slope Downhill](https://lightmysky.com/learn/computing/gradient-descent-following-the-slope-downhill-mt_EjfIT1WgMX)
- [Recovering the Field from the Potential Gradient](https://lightmysky.com/learn/science/recovering-the-field-from-the-potential-gradient-mt_EqCLB9bqG1)