Calculus · Quick Reference

Partial Derivatives Cheat Sheet

Multivariable Calculus · Lesson 15/20

In one line: when a function depends on more than one variable, partial derivatives measure how it changes with respect to just one variable at a time, holding the others fixed.

Key Ideas

1Functions of Several Variables. A function like f(x,y) depends on two (or more) inputs, producing a surface rather than a curve.

2Notation. Partial derivative with respect to x is written as (partial f / partial x) or f_x. Same idea applies for y, giving f_y.

3How to Compute. To find f_x, differentiate f(x,y) with respect to x while treating y as a constant. To find f_y, do the reverse.

4Geometric Meaning. f_x gives the slope of the surface in the x-direction at a point; f_y gives the slope in the y-direction.

5Higher-Order Partials. You can take partial derivatives of partial derivatives, e.g., f_xx, f_yy, or the mixed partial f_xy.

Worked Examples

Find the partial derivatives of f(x,y) = x^2*y + y^3.

f_x = 2xy, f_y = x^2 + 3y^2

Find f_x and f_y for f(x,y) = 3x^2 + 4xy + y^2.

f_x = 6x + 4y, f_y = 4x + 2y

Find f_xy (mixed partial) for f(x,y) = x^3*y^2.

f_xy = 6x^2*y

Practice Yourself

Find f_x for f(x,y) = x^2 + y^2.

Find f_y for f(x,y) = x^2 + y^2.

Find f_x for f(x,y) = 3xy + y^2.

Find f_y for f(x,y) = 3xy + y^2.

3x + 2y