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!