Sign in to save

Bookmark this page so you can find it later.

Sign in to save

Bookmark this page so you can find it later.

Directional Derivatives and the Gradient cheat sheet - grade college

Click image to open full size

Calculus Grade college

Directional Derivatives and the Gradient Cheat Sheet

A printable reference covering directional derivatives, gradients, unit vectors, maximum increase, level curves, and constrained optimization for college.

TopicsDerivativesVectorsFunctionsGraphsCalculusStudy Guide
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

Directional derivatives measure how a multivariable function changes as you move from a point in a chosen direction. The gradient collects the partial derivatives into a vector that points toward the steepest increase. This cheat sheet helps students connect formulas, computations, and geometric meaning in one quick reference.

It is useful for studying partial derivatives, tangent planes, optimization, and vector calculus applications.

The central formula is Duf(a,b)=∇f(a,b)⋅uD_{\mathbf{u}}f(a,b)=\nabla f(a,b)\cdot \mathbf{u}, where u\mathbf{u} must be a unit vector. The gradient is ∇f=⟨fx,fy⟩\nabla f=\langle f_x,f_y\rangle in two variables and ∇f=⟨fx,fy,fz⟩\nabla f=\langle f_x,f_y,f_z\rangle in three variables. The largest directional derivative is ∥∇f∥\lVert \nabla f \rVert, and it occurs in the direction of ∇f\nabla f.

On level curves and surfaces, the gradient is perpendicular to the level set when ∇f≠0\nabla f\neq \mathbf{0}.

Key Facts

  • For f(x,y)f(x,y), the gradient is ∇f(x,y)=⟨fx(x,y),fy(x,y)⟩\nabla f(x,y)=\langle f_x(x,y),f_y(x,y)\rangle.
  • For f(x,y,z)f(x,y,z), the gradient is ∇f(x,y,z)=⟨fx(x,y,z),fy(x,y,z),fz(x,y,z)⟩\nabla f(x,y,z)=\langle f_x(x,y,z),f_y(x,y,z),f_z(x,y,z)\rangle.
  • The directional derivative of ff at (a,b)(a,b) in unit direction u\mathbf{u} is Duf(a,b)=∇f(a,b)⋅uD_{\mathbf{u}}f(a,b)=\nabla f(a,b)\cdot \mathbf{u}.
  • A direction vector v\mathbf{v} must be normalized before use: u=v∥v∥\mathbf{u}=\frac{\mathbf{v}}{\lVert \mathbf{v}\rVert}.
  • The maximum rate of increase at a point is ∥∇f(a,b)∥\lVert \nabla f(a,b)\rVert, and it occurs in the direction ∇f(a,b)∥∇f(a,b)∥\frac{\nabla f(a,b)}{\lVert \nabla f(a,b)\rVert} when ∇f(a,b)≠0\nabla f(a,b)\neq \mathbf{0}.
  • The minimum directional derivative is −∥∇f(a,b)∥-\lVert \nabla f(a,b)\rVert, and it occurs in the direction −∇f(a,b)∥∇f(a,b)∥-\frac{\nabla f(a,b)}{\lVert \nabla f(a,b)\rVert} when ∇f(a,b)≠0\nabla f(a,b)\neq \mathbf{0}.
  • If θ\theta is the angle between ∇f\nabla f and u\mathbf{u}, then Duf=∥∇f∥cos⁡θD_{\mathbf{u}}f=\lVert \nabla f\rVert \cos \theta.
  • For a level curve f(x,y)=cf(x,y)=c, the gradient ∇f(a,b)\nabla f(a,b) is normal to the curve at (a,b)(a,b) if ∇f(a,b)≠0\nabla f(a,b)\neq \mathbf{0}.

Vocabulary

Directional derivative
The directional derivative Duf(a,b)D_{\mathbf{u}}f(a,b) is the instantaneous rate of change of ff at (a,b)(a,b) in the unit direction u\mathbf{u}.
Gradient
The gradient ∇f\nabla f is the vector of partial derivatives of a scalar function.
Unit vector
A unit vector is a vector with length 11, so it gives direction without changing the scale of a directional derivative.
Level curve
A level curve is a set of points satisfying f(x,y)=cf(x,y)=c, where the function has the same value everywhere on the curve.
Level surface
A level surface is a set of points satisfying f(x,y,z)=cf(x,y,z)=c in three-dimensional space.
Steepest ascent
Steepest ascent is the direction of greatest increase of a function, given by the direction of ∇f\nabla f when the gradient is nonzero.

Common Mistakes to Avoid

  • Using a non-unit direction vector in Duf=∇f⋅uD_{\mathbf{u}}f=\nabla f\cdot \mathbf{u} is wrong because the result is scaled by the vector length instead of representing rate per unit distance.
  • Forgetting to evaluate ∇f\nabla f at the given point is wrong because the gradient usually changes from point to point.
  • Confusing the gradient with the directional derivative is wrong because ∇f\nabla f is a vector, while DufD_{\mathbf{u}}f is a scalar rate of change.
  • Assuming the gradient points along a level curve is wrong because ∇f\nabla f is perpendicular to the level curve when ∇f≠0\nabla f\neq \mathbf{0}.
  • Calling ∥∇f∥\lVert \nabla f\rVert the rate in every direction is wrong because it is only the maximum possible directional derivative at that point.

Practice Questions

  1. 1 Find ∇f(1,2)\nabla f(1,2) for f(x,y)=x2y+3y2f(x,y)=x^2y+3y^2.
  2. 2 Compute Duf(1,1)D_{\mathbf{u}}f(1,1) for f(x,y)=x2+xyf(x,y)=x^2+xy in the direction of v=⟨3,4⟩\mathbf{v}=\langle 3,4\rangle.
  3. 3 For f(x,y,z)=xy+z2f(x,y,z)=xy+z^2, find the maximum directional derivative at (2,1,−1)(2,1,-1) and the unit direction where it occurs.
  4. 4 If ∇f(a,b)=⟨0,0⟩\nabla f(a,b)=\langle 0,0\rangle, explain why there is no unique direction of steepest increase at (a,b)(a,b).