Question

Differentiate (log sin x)² ?

Answer 1

\[\text{Let } y = \left( \log \sin x \right)^2 \]

\[\text{ Differentiate with respect to x we get }, \]

\[\frac{d y}{d x} = \frac{d}{dx} \left( \log \sin x \right)^2 \]

\[ = 2\left( \log \sin x \right)\frac{d}{dx}\left( \log \sin x \right)\]

\[ = 2\left( \log \sin x \right) \times \frac{1}{\sin x}\frac{d}{dx}\left( \sin x \right)\]

\[ = 2\left( \log \sin x \right) \times \frac{1}{\sin x} \times \cos x \]

\[ = 2\left( \log \sin x \right)\cot x\]

\[So, \frac{d}{dx} \left( \log \sin x \right)^2 = 2\left( \log \sin x \right)\cot x\]