$y = {\cos ^{ - 1}}\left( {\cfrac{{2x}}{{1 + {x^2}}}} \right), - 1 < x < 1.$

Putting $x = \tan \theta ,$we get
$y = {\cos ^{ - 1}}\left( {\cfrac{{2\tan \theta }}{{1 + {{\tan }^2}\theta }}} \right) \Rightarrow y = {\cos ^{ - 1}}(\sin 2\theta )$

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

$\Rightarrow$ $y = \cfrac{\pi }{2} - 2{\tan ^{ - 1}}x \Rightarrow \cfrac{{dy}}{{dx}} = - \cfrac{2}{{1 + {x^2}}}$