Question

Differentiate ${\tan }^{-1}\frac{x}{\sqrt{(1-x^2)}}$ w.r.t. ${\sec }^{-1}\frac{1}{2x^2-1}$

• A. $-\frac{1}{2}$
• B. $-\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $-\frac{1}{6}$

Answer 1

The correct option is A $-\frac{1}{2}$

Let $y={\tan }^{-1}\frac{x}{\sqrt{(1-x^2)}}$ and $z={\sec }^{-1}\frac{1}{2x^2-1}$
Put $x=\cos \theta$

$\therefore y={\tan }^{-1}\frac{\cos \theta }{\sin \theta }={\tan }^{-1}\tan (\pi /2-\theta )=\pi /2-\theta$
and $z={\sec }^{-1}\frac{1}{2{\cos }^2\theta -1}=se{c}^{-1}\left(\sec 2\theta \right)=2\theta$

$\therefore y=\frac{\pi }{2}-\frac{z}{2}\therefore \frac{dy}{dz}=-\frac{1}{2}$