Explanation

(c)
$y = \tan^{- 1}{\frac{1}{1 + x + x^{2}} + \tan^{- 1}{\frac{1}{x^{2} + 3x + 3} +}}$
…+upto $n$ terms
$= \tan^{- 1}{\frac{(x + 1) - x}{1 + x(x + 1)} + \tan^{- 1}{\frac{(x + 2) - (x + 1)}{1 + (x + 1)(x + 2)} +}}$
…+ upto $n$ terms
$= \lbrack\tan^{- 1}{(x + 1) - \tan^{- 1}{x +}}\tan^{- 1}{(x + 2) - \tan^{- 1}(x + 1)}$
$+ \ldots + \tan^{- 1}{(x + n) - \tan^{- 1}{\{ x + (n - 1)\}\rbrack}}$
$= \tan^{- 1}{(x + n) - \tan^{- 1}x}$
$\therefore\ y^{'}(x) = \frac{1}{1 + (x + n)^{2}} - \frac{1}{1 + x^{2}}$
$\Rightarrow \ y^{'}(0) = \frac{1}{1 + n^{2}} - 1 = \frac{n^{2}}{1 + n^{2}}$