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PD.3 Partial Derivatives

In general, if is a function of two variables and , suppose we let only vary while keeping fixed, say , where is a constant. Then we are really considering a function of a single variable , namely, . if has a derivative at , then we call it the partial derivative of with respect to at and denote it by . Thus:

 

Picture of Partial Derivatives:

To understand the concept let's take a look at the one-dimensional case first:

Using the definition of a derivative, the above equation becomes:

Similarly, the partial derivative of with respect to at denoted by , is obtained by keeping x fixed and finding the ordinary derivative at of the function

If we now let the point vary in Equations 2 and 3, and become functions of two variables.

and

Notations for Partial Derivatives

If , we write:

                           

 

To compute partial derivatives, all we have to do is remember from Equation 1 that the partial derivative with respect to is just the ordinary derivative of the function of of a single variable that we get by keeping y fixed. Thus, we have the following rule:

Rule for Finding Partial Derivatives of

1. To find regard as a constant and differentiate with respect to

2. To find regard as a constant and differentiate with respect to

Partial Derivatives can be interpreted as rates of change. If , then represents the rate of change of with respect to when is fixed. Similarly, represents the rate of change of with respect to when is fixed.

Functions of More than Two Variables

Partial derivatives can also be defined for functions of three or more variables. For example, if is a function of three variables and then its partial derivative with respect to x is defined as

and it is found by regarding and as constants and differentiatingthen can be interpreted as the rate of change of with respect to when and are held fixed.

In general, if is a function of variables, its partial derivative with respect to the th variable x is:

Higher Derivatives

If is a function of two variables, then its partial derivatives and are also functions of two variables, so we can consider their partial derivatives , which are called the second partial derivatives of If , we use the following notation:

 

Therefore, means that we differentiate with respect to first and then with respect to . whereas is simply the reverse.

Clairaut's Theorem

Partial Differential Equations

Some equations using partial derivatives.

Laplace's Equation

 

Solutions of this equation are called harmonic functions.

Wave Equation

This equation can describe wave motion.

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