Differentiability of modulus composition for JEE

Formula

\[\frac{d}{dx}|f(x)|=\frac{f(x)}{|f(x)|}f'(x)=\operatorname{sgn}(f(x))f'(x)\quad (f(x)\neq 0)\]

Variables: \(f\) differentiable function.

Conditions: At points where \(f(x)=0\), differentiability must be checked separately.

Piecewise derivative, cusp/corner analysis and chain-rule problems.