Question

Differential coefficient of sec $\sec \left({\tan }^{-1}x\right)$ is
(a) $\frac{x}{1+x^2}$

(b) $x\sqrt{1+x^2}$

(c) $\frac{1}{\sqrt{1+x^2}}$

(d) $\frac{x}{\sqrt{1+x^2}}$

Answer 1

(d) $\frac{x}{\sqrt{1+x^2}}$

$$\begin{gathered} \mathrm{We}\mathrm{have},y=\sec \left({\tan }^{-1}x\right) \\ \Rightarrow \frac{dy}{dx}=\sec \left({\tan }^{-1}x\right)\tan \left({\tan }^{-1}x\right)\times \frac{d}{dx}\left({\tan }^{-1}x\right) \\ \Rightarrow \frac{dy}{dx}=\sec \left({\tan }^{-1}x\right)\tan \left({\tan }^{-1}x\right)\times \frac{1}{\sqrt{1+x^2}} \\ \Rightarrow \frac{dy}{dx}=y\left(\frac{x}{\sqrt{1+x^2}}\right) \\ \Rightarrow \frac{dy}{dx}=\left(\frac{x}{\sqrt{1+x^2}}\right)y \end{gathered}$$
This is the equation of differential equation which have coefficient $\frac{x}{\sqrt{1+x^2}}$.