How do you find f'(x) using the definition of a derivative for f(x)=absx?

1 Answer
Oct 17, 2015

See the explanation.

Explanation:

The function f(x)=|x| is continuous function, defined for AAx in R but it's not differentiable for AAx in R (e.g. derivative doesn't exist in every point).

By definition, function is differentiable at a point if its derivative exists at that point.

f(x)=x for x>=0
f(x)=-x for x<0

For x>0:
f'(x)=lim_(h->0) (x+h-x)/h = lim_(h->0) h/h =lim_(h->0) 1 = 1

For x<0:
f'(x)=lim_(h->0) (-(x+h)-(-x))/h = lim_(h->0) (-x-h+x)/h

f'(x) =lim_(h->0) (-h)/h = lim_(h->0) (-1) = -1

From the previous:

f'(0+epsilon) = 1
f'(0-epsilon) = -1

f'(0+epsilon) != f'(0-epsilon)

Function is not differentiable at x=0.