Question

If $\mathrm{y}={\sec }^{-1}\left(\frac{\left(\sqrt{\mathrm{x}+1}\right)}{\left(\sqrt{\mathrm{x}-1}\right)}\right)+{\sin }^{-1}\left(\frac{\left(\sqrt{\mathrm{x}-1}\right)}{\left(\sqrt{\mathrm{x}+1}\right)}\right)$, then $\frac{dy}{dx}=$

• A. $0$
• B. $\frac{1}{\sqrt{x+1}}$
• C. $1$
• D. None of these

Answer 1

The correct option is A

$0$

Find the value of $\frac{dy}{dx}$:

Given,

$$\begin{gathered} \mathrm{y}={\sec }^{-1}\left(\frac{\left(\sqrt{\mathrm{x}+1}\right)}{\left(\sqrt{\mathrm{x}-1}\right)}\right)+{\sin }^{-1}\left(\frac{\left(\sqrt{\mathrm{x}-1}\right)}{\left(\sqrt{\mathrm{x}+1}\right)}\right) \\ \Rightarrow \mathrm{y}={\cos }^{-1}\left(\frac{\left(\sqrt{\mathrm{x}-1}\right)}{\left(\sqrt{\mathrm{x}+1}\right)}\right)+{\sin }^{-1}\left(\frac{\left(\sqrt{\mathrm{x}-1}\right)}{\left(\sqrt{\mathrm{x}+1}\right)}\right)\mathbf{}\mathbf{}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{s}\mathbf{e}{\mathbf{c}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{c}\mathbf{o}{\mathbf{s}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{a}}\mathbf{\right)}\mathbf{]} \\ \Rightarrow y=\frac{\mathrm{\pi }}{2}\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{c}\mathbf{o}{\mathbf{s}}^{\mathbf{-}\mathbf{1}}\mathbf{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{-}\mathbf{1}}\mathbf{x}\mathbf{}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{]} \end{gathered}$$

Differentiate $y$ with respect to $x$,

$\frac{dy}{dx}=0\left[\because \frac{d\left(\mathrm{constant}\right)}{\mathrm{dx}}=0\right]$

Hence, the correct option is A.