Question

$\frac{d^{20}y}{d{x}^{20}}\left(2\cos x\cos 3x\right)$ is equal to

• A. $2^{20}(\cos 2x-2^{20}\cos 3x)$
• B. $2^{20}(\cos 2x+2^{20}\cos 4x)$
• C. $2^{20}(\sin 2x+2^{20}\sin 4x)$
• D. $2^{20}(\sin 2x-2^{20}\sin 4x)$

Answer 1

The correct option is B $2^{20}(\cos 2x+2^{20}\cos 4x)$

We know that,
$2\cos A\cos B=\cos (A+B)+\cos (A-B)$
$\therefore y=2\cos x\cos 3x=\cos 4x+\cos 2x$
$\frac{dy}{dx}=-4\sin 4x-2\sin 2x$
$\frac{d^{2}y}{d{x}^2}=-{\left(4\right)}^2\cos 4x-{\left(2\right)}^2\cos 2x$
$\frac{d^{3}y}{d{x}^3}=4^3\sin 4x+2^3\sin 2x$
Similarly,
$\therefore \frac{d^{20}y}{d{x}^{20}}=4^{20}\cos 4x+2^{20}\cos 2x$