Derivative of Absolute Value Function

Theorem
Let $\left \vert x \right \vert$ be the absolute value of $x$ for real $x$.

Then:

for $x \ne 0$.

Corollary
Let $u$ be (an image of) a differentiable real function of $x$.

Then:

for $u \ne 0$.

Proof of Corollary
Follows from the Chain Rule and the main result.