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?

If $A\subset\Bbb R$ and $f$ is a function from $A$ into $\Bbb R$, we say that $f$ is differentiable if $f$ is differentiable at every $a\in A$. So, the absolute value function is not differentiable, since it is not differentiable at $0$. Otherwise, you are right, but I would simply say that if we differentiate it at a point $a\ne0$, then we get $1$ if $a>0$ and $-1$ if $a<0$.
