Explanation

(d)
Given, $r = \left\lbrack 2\phi + \cos^{2}\left( 2\phi + \frac{\pi}{4} \right) \right\rbrack^{1/2}$
$\frac{dr}{d\phi} = \ \frac{\left\lbrack 2 - 2\cos{\left( 2\phi + \frac{\pi}{4} \right)\sin{\left( 2\phi + \frac{\pi}{4} \right).2}} \right\rbrack}{2\sqrt{2\phi + \cos^{2}\left( 2\phi + \frac{\pi}{4} \right)}}$
$= \frac{\left\lbrack 1 - \sin{\ \left( 4\phi + \frac{\pi}{2} \right)} \right\rbrack}{\sqrt{2\phi + \cos^{2}{\left( 2\phi + \frac{\pi}{4} \right)\ }}}$
$\Rightarrow \ \ \left( \frac{dr}{d\phi} \right)_{\phi = \pi/4} = \ \frac{\left\lbrack 1 - \sin\left( \pi + \frac{\pi}{2} \right) \right\rbrack}{\sqrt{2.\frac{\pi}{4} + \cos^{2}\left( \frac{\pi}{2} + \frac{\pi}{4} \right)}}$
$= \frac{1 + 1}{\sqrt{\frac{\pi}{2} + \frac{1}{2}}} = 2\sqrt{\frac{2}{1 + \pi}}$