Derivative of Absolute Value Function

Theorem
Let $\left \vert x \right \vert$ be the absolute value of $x$ for real $x$.

Then:


 * $\dfrac \d {\d x} \left \vert x \right \vert = \dfrac x {\left \vert {x} \right \vert}$

for $x \ne 0$.

At $x = 0$, $\left \vert x \right \vert$ is not differentiable.

Proof
Now consider $x = 0$.

From the definition of derivative:

From Limit iff Limits from Left and Right, the limit does not exist.