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Question
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Solution
\[\text{Let} y = \log\left( \tan^{- 1} x \right)\]
Differentiate it with respect to x we get,
\[\frac{d y}{d x} = \frac{d}{dx}\log\left( \tan^{- 1} x \right)\]
\[ = \frac{1}{\tan^{- 1} x} \times \frac{d}{dx}\left( \tan^{- 1} x \right) \left[ \text{Using chain rule} \right]\]
\[ = \frac{1}{\left( 1 + x^2 \right) \tan^{- 1} x}\]
\[So, \frac{d}{dx}\left\{ \log\left( \tan^{- 1} x \right) \right\} = \frac{1}{\left( 1 + x^2 \right) \tan^{- 1} x}\]
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