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Question
Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?
Solution
\[\text{ Let, y } = \sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\]
\[\text{ put x } = cot \theta\]
\[ \therefore y = \sin^{- 1} \left( \frac{1}{\sqrt{1 + co t^2 \theta}} \right)\]
\[ \Rightarrow y = \sin^{- 1} \left( \frac{1}{\sqrt{\cos e c^2 \theta}} \right)\]
\[ \Rightarrow y = \sin^{- 1} \left( \sin\theta \right) \]
\[ \Rightarrow y = \theta\]
\[ \Rightarrow y = co t^{- 1} x .............\left[ \text{since, }cot\theta = x \right]\]
Differentiate it with respect to x,
\[\frac{d y}{d x} = - \frac{1}{\left( 1 + x^2 \right)}\]
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