Just differentiate it piece-wise as well.
You should have f(x) = |x| = x if x>=0 or f(x) = |x| = -x if x < 0.
For x < 0 what will the derivative be? For x > 0 what will the derivative be? For x = 0, take a look at the graph of |x|. Does it look like the derivative will exist there? Also for x = 0, look at the one-sided limits for
f(0 + h)/h