First slide

Derivatives

Question

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

Moderate

A

$(12, 11)$
B

$(15, 14)$
C

$(16, 14)$
D

$(16, 15)$

Solution

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

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