If y = ( x + 1 ) ( x 2 - x ) x x + x + x + 1 15 ( 3 cos 2 x - 5 ) cos 3 x , then 96 y ' ( 6 ) is equal to _______ . [2024]

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Solution

Q.
If $y = \frac{(\sqrt{x} + 1) (x^{2} - \sqrt{x})}{x \sqrt{x} + x + \sqrt{x}} + \frac{1}{15} (3 \cos^{2} x - 5) \cos^{3} x,$ then $96 y' (\frac{π}{6})$ is equal to _______ . [2024]

Ans.
(105)

We have, $y = \frac{(\sqrt{x} + 1) (x^{2} - \sqrt{x})}{x (\sqrt{x} + 1) + \sqrt{x}} + \frac{1}{15} (3 \cos^{2} x - 5) \cos^{3} x$

$= \frac{\sqrt{x} (x - 1) (x + 1 + \sqrt{x})}{\sqrt{x} (x + \sqrt{x} + 1)} + \frac{1}{15} (3 \cos^{2} x - 5) \cos^{3} x$

$= x - 1 + \frac{1}{5} \cos^{5} x - \frac{1}{3} \cos^{3} x$

$\Rightarrow y' = 1 + \frac{1}{5} \cdot 5 \cos^{4} x \cdot (- \sin x) - \frac{1}{3} \cdot 3 \cdot \cos^{2} x \cdot (- \sin x)$

$= 1 - \sin x \cdot \cos^{4} x + \sin x \cdot \cos^{2} x$

$\Rightarrow 96 y' (\frac{π}{6}) = 96 [1 - \frac{1}{2} \cdot \frac{9}{16} + \frac{1}{2} \cdot \frac{3}{4}] = 105$

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