Explanation

(d)
$y = \tan^{- 1}\left( \frac{\sqrt{x -}x}{1 + x^{3/2}} \right)$
$= \tan^{- 1}\left( \frac{\sqrt{x} - x}{1 + \sqrt{x}.x} \right)$
$= \tan^{- 1}{(\sqrt{x})} - \tan^{- 1}{(x)}$
On differentiating w.r.t. $x$, we get
$y^{'} = \frac{1}{1 + x}.\frac{1}{2\sqrt{x}} - \frac{1}{1 + x^{2}}$
$\Rightarrow y^{'}(1) = \frac{1}{2}.\frac{1}{2} - \frac{1}{2} = - \frac{1}{4}$