Question

Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?

Answer 1

\[\text{ Let, y  }= \tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\]

\[ \Rightarrow y = \tan^{- 1} a + \tan^{- 1} x \left[ \text{ Since }, \tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} \left( \frac{x + y}{1 - xy} \right) \right]\]

Differentiate it with respect to x,

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

\[ \Rightarrow \frac{d y}{d x} = 0 + \frac{1}{1 + x^2}\]

\[ \therefore \frac{d y}{d x} = \frac{1}{1 + x^2}\]