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MathidiaMathematics Academy
Multivariable calculus
Lesson 33 of 75
Standard practice

Gradient and directional derivative

Objective. Use gradients to find steepest change and tangent planes.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

The gradient of a scalar function f(x, y, z), written ∇f, is the vector of its partial derivatives (fₓ, fᵧ, f_z), and it points in the direction of steepest increase of f at a given point, with magnitude equal to the maximum rate of change. The directional derivative of f at a point in the direction of a unit vector u is D_uf = ∇f · u, giving the rate of change along any chosen direction. At a point where f = c, the gradient ∇f is normal to the level surface, which is exactly why ∇f at a point gives the normal vector needed for the tangent plane there. For f(x, y) = x² + y² at (1, 2), ∇f = (2, 4), so the steepest ascent direction there is (2, 4).

Worked example

Find the gradient of f(x, y) = x²y + 3y² at the point (2, 1).

y) = x²y + 3y² at the point (2
Coordinate grid with the line y) = x²y + 3y² at the point (2
  1. 1Compute the partial derivatives: fₓ = 2xy and fᵧ = x² + 6y.
  2. 2Evaluate fₓ at (2, 1): fₓ = 2(2)(1) = 4.
  3. 3Evaluate fᵧ at (2, 1): fᵧ = (2)² + 6(1) = 10.
  4. 4Write the gradient vector using these two components.

Answer: ∇f(2, 1) = (4, 10)

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Common mistake. Students often normalise the direction vector incorrectly or forget to do so at all before computing a directional derivative; D_uf = ∇f · u only gives the true rate of change when u is a unit vector.

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