. $y = {\sec ^{ - 1}}\left( {\cfrac{1}{{2{x^2} - 1}}} \right),0 < x < \cfrac{1}{{\sqrt 2 .}}$

Putting $x = \cos \theta ,$ we get

$y = {\sec ^{ - 1}}\left( {\cfrac{1}{{2{{\cos }^2}\theta - 1}}} \right)$

$\Rightarrow$ $y = {\cos ^{ - 1}}(2{\cos ^2}\theta - 1)$
$\Rightarrow$ $y = {\cos ^{ - 1}}(\cos 2\theta )$

$\Rightarrow$ $y = 2\theta \Rightarrow y = 2{\cos ^{ - 1}}x$
$\Rightarrow$ $\cfrac{{dy}}{{dx}} = - \cfrac{2}{{\sqrt {1 - {x^2}} }}$