Question

Differentiate \[\log \left( \tan^{- 1} x \right)\]? 

Answer 1

\[\text{Let} y = \log\left( \tan^{- 1} x \right)\]

Differentiate it with respect to x we get,

\[\frac{d y}{d x} = \frac{d}{dx}\log\left( \tan^{- 1} x \right)\]

\[ = \frac{1}{\tan^{- 1} x} \times \frac{d}{dx}\left( \tan^{- 1} x \right) \left[ \text{Using chain rule} \right]\]

\[ = \frac{1}{\left( 1 + x^2 \right) \tan^{- 1} x}\]

\[So, \frac{d}{dx}\left\{ \log\left( \tan^{- 1} x \right) \right\} = \frac{1}{\left( 1 + x^2 \right) \tan^{- 1} x}\]