Derivative of Absolute Value Function/Corollary
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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:
- $\dfrac \d {\d x} \size u = \dfrac u {\size u} \dfrac {\d u} {\d x}$
$\blacksquare$