Explanation

(d)
We have, $y = \tan^{- 1}\left\lbrack \frac{\sin{x + \cos x}}{\cos{x - \sin x}} \right\rbrack$
$\Rightarrow y = \tan^{- 1}\left\lbrack \frac{1 + \tan x}{1 - \tan x} \right\rbrack$
$\Rightarrow y = \tan^{- 1}\left\lbrack \frac{\tan{\left( \frac{\pi}{4} \right) + \tan x}}{1 - \tan{\left( \frac{\pi}{4} \right)\tan x}} \right\rbrack$
$\Rightarrow y = \tan^{- 1}{\tan\left( \frac{\pi}{4} + x \right)}$
$\Rightarrow y = \left( \frac{\pi}{4} \right) + x$
On differentiating w.r.t. $x$, we get
$\frac{dy}{dx} = 1$