Calculus · Quick Reference

Gradient Cheat Sheet

Multivariable Calculus · Lesson 18/20

In one line: the gradient turns a scalar function of several variables into a vector that points in the direction of steepest increase — like an arrow pointing uphill on a landscape.

Key Ideas

1Definition. For f(x,y), the gradient is grad f = (f_x, f_y) — a vector made up of all the partial derivatives.

2Direction of Steepest Ascent. At any point, grad f points in the direction where f increases fastest; the magnitude of grad f tells you how steep that increase is.

3Perpendicular to Level Curves. The gradient at a point is always perpendicular to the level curve (contour line) passing through that point.

4Three Variables. For f(x,y,z), grad f = (f_x, f_y, f_z) — the idea extends naturally to more variables.

5Zero Gradient. Where grad f = (0,0), the function has a flat point — a candidate for a local max, min, or saddle point.

Worked Examples

Find the gradient of f(x,y) = x^2 + y^2.

grad f = (2x, 2y)

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

grad f at (1,2) = (4, 13)

Where is the gradient of f(x,y) = x^2 + y^2 equal to zero?

At the point (0,0), which is the minimum of the bowl-shaped surface

Formulas

grad f = (f_x, f_y)

grad f = (f_x, f_y, f_z) for three variables

Practice Yourself

Find the gradient of f(x,y) = 3x + 4y.

(3, 4)

Find the gradient of f(x,y) = x^2 - y^2.

(2x, -2y)

Find the gradient of f(x,y) = xy at (2,3).

(3, 2)