Explanation

(c)
Since, $y = (1 + x)\left( 1 + x^{2} \right)\left( 1 + x^{4} \right)\ldots(1 + x^{2^{n}})$
$\Rightarrow (1 - x)y = \left( 1 - x^{2} \right)\left( 1 + x^{2} \right)\left( 1 + x^{4} \right)\ldots(1 + x^{2^{n}})$
$= \left( 1 - x^{4} \right)\left( 1 + x^{4} \right)\ldots\left( 1 + x^{2^{n}} \right)$
$\cdots\ \cdots\ \cdots\ \cdots\ \cdots$
$\cdots\ \cdots\ \cdots\ \cdots\ \cdots$
$= \left( 1 - x^{2^{n}} \right)\left( 1 + x^{2^{n}} \right) = 1 - x^{2^{n + 1}}$
$\therefore\ y = \frac{1 - x^{2^{n + 1}}}{(1 - x)}$
$\frac{dy}{dx} = \frac{(1 - x)\left( - 2^{n + 1} \right).{x^{2}}^{n + 1 - 1} - \left( 1 - {x^{2}}^{n + 1} \right)( - 1)}{(1 - x)^{2}}$
$\therefore\left. \ \frac{dy}{dx} \right|_{x = 0} = \frac{(1 - 0)\left( - 2^{n + 1}.0 \right) - (1 - 0)( - 1)}{1} = 1$