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Question
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Solution
\[\text{ Let, y }= \tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\]
\[ \Rightarrow y = \tan^{- 1} a + \tan^{- 1} x \left[ \text{ Since }, \tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} \left( \frac{x + y}{1 - xy} \right) \right]\]
Differentiate it with respect to x,
\[\frac{d y}{d x} = \frac{d}{dx}\left( \tan^{- 1} a \right) + \frac{d}{dx}\left( \tan^{- 1} x \right)\]
\[ \Rightarrow \frac{d y}{d x} = 0 + \frac{1}{1 + x^2}\]
\[ \therefore \frac{d y}{d x} = \frac{1}{1 + x^2}\]
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