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Directional Derivatives and the Gradient cheat sheet - grade college

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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.

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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).

Understanding Directional Derivatives and the Gradient

A directional derivative is a local measurement. It describes the first instant of change, not necessarily what happens after moving a long distance. Imagine a hill whose height is given by a function of east west position and north south position.

At one location, walking a few steps northeast may raise your height quickly. Walking farther in that same direction can lead to a flatter region or a descent.

This local nature is why directional derivatives are useful for approximations near a chosen point. For a small displacement, the predicted change is close to the directional rate multiplied by the distance traveled.

The unit vector condition matters because a direction should not contain a hidden distance. A vector such as two, two points in the same direction as one, one, but it is longer. If both were used directly, the longer vector would produce a rate twice as large even though the path is unchanged.

Dividing by a vector's length removes that problem. Students often make this error when a problem gives a direction from one point to another.

First subtract the starting coordinates from the ending coordinates. Then find the length of that result and divide each component by the length.

The angle idea explains the signs of directional derivatives. A positive value means the function rises as you move. A negative value means it falls.

A value of zero means your chosen motion is level to first order. On a contour map, a level curve joins locations with the same function value. Moving along the curve keeps the value unchanged, so the tangent direction has zero directional derivative.

The steepest direction must therefore cross contour lines rather than follow them. Closely spaced contour lines signal a larger local slope. This is useful in geography, weather maps, pressure fields, and images where brightness is modeled as a function of position.

Constrained optimization uses this geometry when movement is restricted to a curve or surface. Suppose a factory must choose two quantities that satisfy a fixed budget, or a point must remain on a circular track. At a highest or lowest value along that constraint, a small allowed move cannot increase or decrease the objective immediately.

The objective gradient is then perpendicular to the constraint path. The constraint gradient points perpendicular to that same path, so the two gradients point in parallel or opposite directions. This is the basis of Lagrange multipliers.

When solving these problems, first check whether the gradients exist, include endpoints when the constraint has them, and test any points where a gradient becomes zero. Those cases can contain valid extrema that the standard parallel gradient condition does not identify.