Question

Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?

Answer 1

\[\text{ Let, y } = \sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\]

\[\text{ put x } = cot \theta\]

\[ \therefore y = \sin^{- 1} \left( \frac{1}{\sqrt{1 + co t^2 \theta}} \right)\]

\[ \Rightarrow y = \sin^{- 1} \left( \frac{1}{\sqrt{\cos e c^2 \theta}} \right)\]

\[ \Rightarrow y = \sin^{- 1} \left( \sin\theta \right) \]

\[ \Rightarrow y = \theta\]

\[ \Rightarrow y = co t^{- 1} x .............\left[ \text{since, }cot\theta = x \right]\]

Differentiate it with respect to x,

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