ddx{(x+1)(x2+1)(x4+1)(x8+1)}=(15xp−16xq+1)(x−1)−2⇒(p,q)=

(x+1)(x2+1)(x4+1)(x8+1).(x−1)x−1=x16−1x−1 ddx{x16−1x−1}=(x−1)(16.x5)−(x16−1)(1)(x−1)2 =(15.x16−16x15+1)(x−1)−2Comparing with given problem(p,q)=(16,15).

ddx{(x+1)(x2+1)(x4+1)(x8+1)}=(15xp−16xq+1)(x−1)−2⇒(p,q)=

(x+1)(x2+1)(x4+1)(x8+1).(x−1)x−1=x16−1x−1 ddx{x16−1x−1}=(x−1)(16.x5)−(x16−1)(1)(x−1)2 =(15.x16−16x15+1)(x−1)−2Comparing with given problem(p,q)=(16,15).

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$\frac{d}{d x} {(x + 1) (x^{2} + 1) (x^{4} + 1) (x^{8} + 1)} =$ $(15 x^{p} - 16 x^{q} + 1) {(x - 1)}^{- 2}$
$\Rightarrow (p, q) =$

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(12,11)

(15,14)

(16,14)

(16,15)

answer is D.

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Detailed Solution

$\frac{(x + 1) (x^{2} + 1) (x^{4} + 1) (x^{8} + 1) . (x - 1)}{x - 1} = \frac{x^{16} - 1}{x - 1} \frac{d}{d x} {\frac{x^{16} - 1}{x - 1}} = \frac{(x - 1) (16. x^{5}) - (x^{16} - 1) (1)}{{(x - 1)}^{2}} = (15. x^{16} - 16 x^{15} + 1) {(x - 1)}^{- 2}$
Comparing with given problem
(p,q)=(16,15).