Derivatives of absolute values

[fk378]

Homework Statement

Where is the function f(x) = |x| differentiable?

Homework Equations

[f(x+h) - f(x)] / h

The Attempt at a Solution

I know that the graph of f(x)=|x| shows a corner at the origin from which 2 lines project at opposite slopes, as in they are symmetric about the y-axis.
I've seen the solution but I don't understand why f(x) is differentiable for every number except 0.

 

[Dick]

|x|=x for x>0 and |x|=(-x) for x<0. x and -x are both differentiable. Remember the derivative expression is a limit. It's only correct in the limit where h->0. Think of h as REALLY SMALL.

 

[Feldoh]

You can re-write |x| as a piecewise function

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

Find the derivative and I think you'll see your answer -- Oh and you might want to read up on what a "cusp" is.