Derivative of Absolute Value Function/Corollary

Corollary to Derivative of Absolute Value Function
Let $u$ be a differentiable real function of $x$.

Then:


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

for $u \ne 0$.

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

Proof
From Derivative of Absolute Value Function:
 * $\dfrac {\mathrm d}{\mathrm d u} \left \vert {u} \right \vert = \dfrac u {\left \vert {u} \right \vert}$

Hence by the Chain Rule:
 * $\dfrac {\mathrm d}{\mathrm d x} \left \vert {u} \right \vert = \dfrac u {\left \vert {u} \right \vert} \dfrac {\mathrm d u}{\mathrm d x}$