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).

Figure 2.1.4
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