Example 2: Derivative of the Root-Function

Find f'(x) given f(x) = √x.

Case 1

x < 0. Since √x is not defined, f'(x) does not exist

Case 2

x = 0. When Δx is a negative infinitesimal, the term

is not defined because is undefined. When Δx is a positive infinitesimal,

the term

is defined but its value is infinite. Thus for two reasons, f'(x) does not exist.

Case 3

x > 0. Let y = √x. Then

We then make the computation

Taking standard parts,

Therefore, when x > 0,

So the derivative of 

f(x) = √x

is the function

and the set of all x > 0 is its domain (see Figure 2.1.4).