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Question
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
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Solution
\[\text{Let } y = \log \sqrt{\frac{x - 1}{x + 1}}\]
\[ \Rightarrow y = \log \left( \frac{x - 1}{x + 1} \right)^\frac{1}{2} \]
\[ \Rightarrow y = \frac{1}{2}\log \left( \frac{x - 1}{x + 1} \right)\]
\[ \Rightarrow y = \frac{1}{2}\left[ \log\left( x - 1 \right) - \log\left( x + 1 \right) \right]\]
Differentiate it with respect to x
\[\frac{d y}{d x} = \frac{1}{2}\left[ \frac{d}{dx}\left\{ \log\left( x - 1 \right) \right\} - \frac{d}{dx}\left\{ \log\left( x + 1 \right) \right\} \right]\]
\[ = \frac{1}{2}\left( \frac{1}{x - 1} - \frac{1}{x + 1} \right)\]
\[ = \frac{1}{2}\left( \frac{2}{x^2 - 1} \right)\]
\[ = \frac{1}{x^2 - 1}\]
\[So, \frac{d y}{d x} = \frac{1}{x^2 - 1}\]
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