## Instructional Materials in Physics and Astronomy

#### Title

#### Date of this Version

1975

#### 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(k*x* – ω*t*),

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(k*x* – ω*t*),

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(k*x* – ω*t*).

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.

## Comments

From

Study Modules for Calculus-Based General Physics

Copyright © 1975 CBP Workshop, University of Nebraska–Lincoln.Reproduction rights granted.