MathematicsIf y=cos 6x+6cos 4x+15cos 2x+10cos 5x+5cos 3x+10cos x then dydx=

If $y = \frac{\cos 6 x + 6 \cos 4 x + 15 \cos 2 x + 10}{\cos 5 x + 5 \cos 3 x + 10 \cos x}$ then $\frac{dy}{dx} =$

A
$2 \sin x + \cos x$
B
$- 2 \sin x$
C
$\cos 2 x$
D
$\sin 2 x$

Solution:

$y = \frac{(\cos 6 x + \cos 4 x) + 5 (\cos 4 x + \cos 2 x) + 10 (\cos 2 x + 1)}{\cos 5 x + 5 \cos 3 x + 10 \cos x}$ $= \frac{2 \cos 5 xcos x + 5 (2 \cos 3 xcos x) + 10 (2 \cos^{2} x)}{\cos 5 x + 5 \cos 3 x + 10 \cos x} = 2 \cos x$
$∴ \frac{dy}{dx} = - 2 \sin x$

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