Derivative of Absolute Value Function/Corollary

Corollary to Derivative of Absolute Value Function

Let $u$ be a differentiable real function of $x$.

Then:

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

for $u \ne 0$.

At $u = 0$, $\size u$ is not differentiable.

Proof

From Derivative of Absolute Value Function:

$\dfrac \d {\d u} \size u = \dfrac u {\size u}$

Hence by the Chain Rule for Derivatives:

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

$\blacksquare$