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Question
Differentiate (log sin x)2 ?
Solution
\[\text{Let } y = \left( \log \sin x \right)^2 \]
\[\text{ Differentiate with respect to x we get }, \]
\[\frac{d y}{d x} = \frac{d}{dx} \left( \log \sin x \right)^2 \]
\[ = 2\left( \log \sin x \right)\frac{d}{dx}\left( \log \sin x \right)\]
\[ = 2\left( \log \sin x \right) \times \frac{1}{\sin x}\frac{d}{dx}\left( \sin x \right)\]
\[ = 2\left( \log \sin x \right) \times \frac{1}{\sin x} \times \cos x \]
\[ = 2\left( \log \sin x \right)\cot x\]
\[So, \frac{d}{dx} \left( \log \sin x \right)^2 = 2\left( \log \sin x \right)\cot x\]
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