# Distributional Derivative of Absolute Value Function

## Theorem

Let $H: \R \to \closedint 0 1$ be the Heaviside step function.

Let $\size x$ be the absolute value of $x$.

Let $T_{\size x}$ be the distribution associated with $\size x$.

Then the distributional derivative of $T_{\size x}$ is $T_{2 H - 1}$

## Proof

 $\ds \dfrac {\d \size x} {\d x}$ $=$ $\ds \begin{cases} 1 & : x > 0 \\ -1 & : x < 0 \end{cases}$ $\ds$ $=$ $\ds -1 + \begin{cases} 2 & : x > 0 \\ 0 & : x < 0 \end{cases}$ $\ds$ $=$ $\ds -1 + 2 \begin{cases} 1 & : x > 0 \\ 0 & : x < 0 \end{cases}$ $\ds$ $=$ $\ds 2 \map H x - 1$ Definition of Heaviside Step Function

Furthermore:

$\ds \lim_{x \mathop \to 0^+} \size x = \lim_{x \mathop \to 0^-} \size x = 0$

By the Jump Rule:

$T_{\size x}' = T_{2H - 1}$

$\blacksquare$