Question

Differentiate

${\sin }^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)$

Answer 1

$$\begin{gathered} \mathrm{Let},y={\sin }^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right) \\ \mathrm{put}x=cot\theta \\ \therefore y={\sin }^{-1}\left(\frac{1}{\sqrt{1+co{t}^2\theta }}\right) \\ \Rightarrow y={\sin }^{-1}\left(\frac{1}{\sqrt{\cos e{c}^2\theta }}\right) \\ \Rightarrow y={\sin }^{-1}\left(\sin \theta \right) \\ \Rightarrow y=\theta \\ \Rightarrow y=co{t}^{-1}x\left[\sin ce,cot\theta =x\right] \end{gathered}$$

Differentiate it with respect to x,

$\frac{dy}{dx}=-\frac{1}{\left(1+x^2\right)}$