The absolute value function is defined as
[tex]|x| = \begin{cases}x & \text{for }x \ge 0 \\ -x & \text{for }x < 0\end{cases}[/tex]
If x is strictly positive (x > 0), then |x| = x, and d|x|/dx = dx/dx = 1.
If x is strictly negative (x < 0), then |x| = -x, and d|x|/dx = d(-x)/dx = -1.
But if x = 0, the derivative doesn't exist!
In order for the derivative of a function f(x) to exist at x = c, the limit
[tex]\displaystyle \lim_{x\to c}\frac{f(x) - f(c)}{x-c}[/tex]
must exist. This limit does not exist for f(x) = |x| and c = 0 because the value of the limit depends on which way x approaches 0.
If x approaches 0 from below (so x < 0), we have
[tex]\displaystyle \lim_{x\to 0^-}\frac{|x|}x = \lim_{x\to0^-}-\frac xx = -1[/tex]
whereas if x approaches 0 from above (so x > 0), we have
[tex]\displaystyle \lim_{x\to 0^+}\frac{|x|}x = \lim_{x\to0^+}\frac xx = 1[/tex]
But 1 ≠ -1, so the limit and hence derivative doesn't exist at x = 0.
Putting everything together, you can define the derivative of |x| as
[tex]\dfrac{d|x|}{dx} = \begin{cases}1 & \text{for } x > 0 \\ \text{unde fined} & \text{for }x = 0 \\ -1 & \text{for }x < 0 \end{cases}[/tex]
