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Question
Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?
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Solution
\[\text{Let } y = \sin^2 \left[ \log\left( 2x + 3 \right) \right]\]
\[\Rightarrow \frac{d y}{d x} = \frac{d}{dx}\left[ \sin^2 \left\{ \log\left( 2x + 3 \right) \right\} \right]\]
\[ = 2 \sin\left\{ \log\left( 2x + 3 \right) \right\}\frac{d}{dx}\sin\left\{ \log\left( 2x + 3 \right) \right\} \left[ \text{Using chain rule} \right]\]
\[ = 2\sin\left\{ \log\left( 2x + 3 \right) \right\} \cos\left\{ \log\left( 2x + 3 \right) \right\}\frac{d}{dx}\log\left( 2x + 3 \right)\]
\[ = \sin\left\{ 2\log\left( 2x + 3 \right) \right\} \times \frac{1}{\left( 2x + 3 \right)}\frac{d}{dx}\left( 2x + 3 \right) \left[ \because 2\sin A \cos A = \sin2A \right]\]
\[ = \sin\left\{ 2\log\left( 2x + 3 \right) \right\}\left( \frac{2}{\left( 2x + 3 \right)} \right)\]
\[So, \frac{d}{dx}\left[ \sin^2 \left\{ \log\left( 2x + 3 \right) \right\} \right] = \sin\left\{ 2 \log\left( 2x + 3 \right) \right\}\left( \frac{2}{\left( 2x + 3 \right)} \right)\]
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