ddxx+1x2+1x4+1x8+1=15xp-16xq+1x-1-2⇒p,q=

y=(x+1) (x2+1) (x4+1) (x8+1) y=(x-1)(x-1) (x+1) (x2+1) (x4+1) (x8+1) =(x2-1) (x2+1) (x4+1) (x8+1)(x+1) y=x16-1x-1 dydx=(x-1) 16x15-x16-1 (1)(x-1)2 =15 x16-16 x15+1 (x-1)-2 p=16, q=15

ddxx+1x2+1x4+1x8+1=15xp-16xq+1x-1-2⇒p,q=

y=(x+1) (x2+1) (x4+1) (x8+1) y=(x-1)(x-1) (x+1) (x2+1) (x4+1) (x8+1) =(x2-1) (x2+1) (x4+1) (x8+1)(x+1) y=x16-1x-1 dydx=(x-1) 16x15-x16-1 (1)(x-1)2 =15 x16-16 x15+1 (x-1)-2 p=16, q=15

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

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(16, 15)

(12, 11)

(15, 14)

(16, 14)

answer is D.

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Detailed Solution

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