Question

If \( y=\log [\log (\sin x)] \), then find \( \frac{d y}{d x} \).

Answer 1

Given that:

\(y = \log[\log (\sin x)]\)

\(\therefore \frac{dy}{dx} = \frac d{dx} \log[\log(\sin x)]\)

\(= \frac 1{\log(\sin x)} . \frac d{dx} (\sin x)\)

\(= \frac 1{\log(\sin x)} . \frac 1{\sin x} . \frac d{dx} \sin x\)

\(= \frac 1{\log(\sin x)} . \frac 1{\sin x} . \frac {\cos x}1\)

\(= \frac {\cot x}{\log (\sin x)}\)