Absolute Value
The absolute value is defined as
\[\lvert x \rvert = \begin{cases} x &\qquad \text{ if } x \geq 0 \\ -x &\qquad \text{ if } x < 0 \end{cases}\]This is not a continuous function, so we have to break the derivative up into two cases as well. The derivative will be discontinuous at $x=0$, and thus does not exist. Using the power rule, we can evaluate each case.
If $x > 0$, then $\lvert x \rvert = x$, thus $\frac{d}{dx} \lvert x \rvert = 1 \cdot x^0 = 1$. If $x < 0$, then $\lvert x \rvert = -x$, thus $\frac{d}{dx} \lvert x \rvert = -1 \cdot x^0 = -1$.
Therefore, the derivative is the following. There are a few ways we can express it.
\[\frac{d}{dx} \lvert x \rvert = \frac{x}{\lvert x \rvert} = \text{sign}(x) = \begin{cases} 1 &\qquad \text{ if } x > 0 \\ -1 &\qquad \text{ if } x < 0 \end{cases}\]Note that the derivative at $x=0$ does not exist, since the left-hand limit would give a value of ${-}1$ and the right-hand limit would give a value of ${+}1$.