Question

The derivative of $se{c}^{-1}\left(\frac{1}{2x^2-1}\right)$ with respect to $\sqrt{1-x^2}$ at $x=1$, is

• A. $2$
• B. $-2$
• C. $1$
• D. none of these

Answer 1

The correct option is C $1$

Let $y={\sec }^{-1}\left(\frac{1}{2x^2-1}\right)$
Put $x=\cos \theta$
$\Rightarrow y={\sec }^{-1}\left(\frac{1}{2{\cos }^2\theta -1}\right)$
$\Rightarrow y=se{c}^{-1}\left(\sec 2\theta \right)$
$\Rightarrow y=2\theta$
$\Rightarrow y=2{\cos }^{-1}x$
$\Rightarrow \frac{dy}{dx}=\frac{-2}{\sqrt{1-x^2}}$
Let $z=\sqrt{1-x^2}$
$\Rightarrow \frac{dz}{dx}=\frac{-x}{\sqrt{1-x^2}}$
So, $\frac{dy}{dz}=\frac{dy}{dx}.\frac{dx}{dz}$
$=1$
$\Rightarrow \frac{dy}{dz}=1$