$\frac{d}{dx} \{(x + a) (x^{2} + a^{2}) (x^{4} + a^{4}) (x^{8} + a^{8})\} =$

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$\frac{15 x^{16} - 16 x^{15} a + a^{16}}{{(x - a)}^{2}}$

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$\frac{x^{16} - a^{16}}{x - a}$

$\frac{x^{16} - x^{15} a + a^{16}}{{(x - a)}^{2}}$

answer is A.

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Detailed Solution

$G i v e n d e r i v a t i v e = \frac{d}{d x} (x + a) (x^{2} + a^{2}) (x^{4} + a^{4}) (x^{8} + a^{8}) m u l t i p l y a n d d i v i d e b y (x - a) = \frac{d}{d x} \frac{\{(x - a) (x + a) (x^{2} + a^{2}) (x^{4} + a^{4}) (x^{8} + a^{8})\}}{x - a} = \frac{d}{d x} \frac{\{(x^{2} - a^{2}) (x^{2} + a^{2}) ((x^{4} + a^{4}) (x^{8} + a^{8})\}}{x - a} = \frac{d}{d x} \frac{((x^{4} - a^{4}) (x^{4} + a^{4}) (x^{8} + a^{8}))}{x - a} = \frac{d}{d x} \frac{((x^{8} - a^{8}) (x^{8} + a^{8}))}{x - a} \frac{d}{dx} [\frac{1}{x - a} . (x^{16} - a^{16})] = \frac{(x - a) 16 x^{15} - (x^{16} - a^{16})}{{(x - a)}^{2}} = \frac{15 x^{16} - 16 x^{15} a + a^{16}}{{(x - a)}^{2}}$