Derivative of Absolute Value Function

Theorem
Let $\size x$ be the absolute value of $x$ for real $x$.

Then:


 * $\dfrac \d {\d x} \size x = \dfrac x {\size x}$

for $x \ne 0$.

At $x = 0$, $\size x$ 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.

Also see

 * Distributional Derivative of Absolute Value Function