EXAMPLE 5 Where is the function

f(x)=|x|

differentiable?\ SOLUTION If

x>0

, then

|x|=

and we can choose

h

small enough that

x+h>0

and hence

|x+h|=

Therefore, for

x>0

we have\

f^(')(x)=\\\\lim_(h->0)(|x+h|-|x|)/(h)\ =\\\\lim_(h->0)((x+h)-x)/(h)

\

=\\\\lim_(h->0)()/(h)\ =\\\\lim_(h->0)

\ and so

f

is differentiable for any

x>

\ Similarly, for

x

we have

|x|=

and

h

can be chosen small enough that

x+h

and so

|x+h|=

- Therefore for

x

,\

f^(')(x)=\\\\lim_(h->0)(|x+h|-|x|)/(h)

EXAMPLE 5 Where is the function f(x)=x differentiable? SOLUTION If x>0, then x= and we can choose h small enough that x+h>0 and hence x+h= - Therefore, for x>0 we have f(x)=h0limhx+hx=h0limh(x+h)x=h0lim =limh0= and so f is differentiable for any x> Similarly, for x