Logarithmic differentiation for variable powers for JEE

Formula

\[y=[u(x)]^{v(x)}\Rightarrow \ln y=v\ln u\Rightarrow \frac{y'}{y}=v'\ln u+v\frac{u'}{u}\]

Variables: \(u=u(x)>0,\ v=v(x)\).

Conditions: For real-valued \(u^v\), typically take \(u>0\).

Functions of the type \(f(x)^{g(x)}\), products of many factors, powers with variable exponents.