Partial Derivatives

Authors

Date of this Version

1975

Comments

From Study Modules for Calculus-Based General Physics
Copyright © 1975 CBP Workshop, University of Nebraska–Lincoln.
Reproduction rights granted.

Abstract

If we have a function of more than one independent variable, then we can define a partial derivative with respect to one of the variables, which is simply the derivative of the function with all the other variables fixed, The notation using ∂, which we will use below, tells you it is a partial derivative, For example, suppose we have the function y that depends on the independent variables x and t:

y = A sin(kx – ωt),

where A, k, and ω are constants, Then, "the partial derivative of y with respect to x" is denoted by ∂y/∂x and is found by setting t constant and differentiating with respect to x:

∂y/∂x = kA cos(kx – ωt),

Similarly, "the partial derivative of y with respect to t" is denoted by ∂y/∂t and is found by setting x constant and differentiating with respect to t:

∂y/∂t = –ωA cos(kx – ωt).

Note that the value of either of the partial derivatives depends on both independent variables x and t as well as the constants A, k, and ω.