Question

Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] with respect to x.

Answer 1

\[\text{ Let y } = \sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\]
\[\text{ Putting x } = \cot \theta \Rightarrow \theta = \cot^{- 1} x\]
\[ \therefore y = \sin^{- 1} \left( \frac{1}{\sqrt{1 + \left( \cot \theta \right)^2}} \right)\]
\[ = \sin^{- 1} \left( \frac{1}{\sqrt{1 + \cot^2 \theta}} \right)\]
\[ = \sin^{- 1} \left( \frac{1}{cosec \theta} \right)\]
\[ = \sin^{- 1} \left( \sin \theta \right)\]
\[ = \theta\]
\[ \therefore y = \cot^{- 1} x\]
\[\text{ Diff w.r.t to x, we get } \]
\[\frac{dy}{dx} = \frac{- 1}{1 + x^2}\]