derivative at 0 problem

lundi 28 octobre 2013

1. The problem statement, all variables and given/known data

f(x) = |x| + x

Does f'(0) exist? Does f'(x) exist for values of x other than 0?

This is from lang's a first course in calculus page 54 # 13

2. Relevant equations

lim (f(x+h) - f(x))/h

h->0

3. The attempt at a solution

So I'm not sure if I am doing this correctly

at first I took the derivative of f(x) = |x| + x

f(x) = |x| + x = √x^2 + x

from there

f'= lim h -> 0 (√(x+h)^2 + (x+h) - (√x^2 + x))/h

= x + h +x + h - x - x / h

= 2h/h

=2

I assumed this meant that f' = 2 but this doesn't make sense because there is no slope at x = 0

so I decided to take the right and left derivatives

right:

for x > 0, h>0

|x| = x

f'= (|x+h| + (x + h) - (|x| + x))/h = ((x+h) + (x + h) - ( x + x )) / h = 2

left:

for x < 0 , h < 0

|x| = -x

f' = (-(x+h) + (x + h) - (-x +x ))/h = -1

so my conclusion is that there is no derivative at x = 0 because the left and right derivatives do not equal.

I am not very confident in what I did here. If someone can help me understand it better I would really appreciate it!