Example 3: Derivative of the Rational Function

Find the deritative of f(x) = 1/x.

Case 1

x = 0. Then 1/x is undefined so f'(x) is undefined.

Case 2

x ≠ 0.

Simplifying,

Taking the standard part,

Thus

The deritative of the function f(x) = 1/x is the function f'(x) = - 1/x² whose domain is the set of all x ≠ 0. Both functions are graphed in Figure 2.1.5.