People say that |x| is not differentiable, but isn't the derivative of |x|, |x|/x? I have heard that $|x|$ is not differentiable, but I found a derivative that meets all the requirements necessary.
Here is my proof that the derivative of $|x|$ is $|x|/x$. First, note that $|x|$ has a slope of $1$ when x is positive. $|x|/x$ also equals $1$ when $x$ is positive. $|x|$ has a slope of $-1$ when $x$ is negative. $|x|/x$ equals $-1$ because $|x|$ is positive and $x$ is negative. $|x|$ has an undefined slope at $x=0$ and |x|/x is also undefined at $x=0$. Therefore, $|x|$'s derivative is $|x|/x$.
Anything wrong with this?

I think what the OP is trying to say is that there isn't generally a single, unified formula given for the derivative of $|x|$, and he has provided one.  I think the formula looks correct, and has the same domain as the piecewise derivative as well.
