Question

If $f\left(x\right)=\left(4\sin x{\cos }^{2}x\right)$ the find the $7^{th}$ derivative of $f\left(x\right).$

• A. $3^7\cos 3x+\cos x$
• B. $-3^7\cos 3x+\cos x$
• C. $-3^7(\cos 3x+\cos x)$
• D. $3^7(\cos 3x+\cos x)$

Answer 1

The correct option is B $-3^7\cos 3x+\cos x$

$f\left(x\right)=4{\cos }^{2}x\sin x$

$=2\sin 2xcosx$

$2\sin a\cos b=\sin (a+b)+\sin (a-b)$

$f\left(x\right)=\sin 3x+\sin x$
$f^1\left(x\right)=3\cos 3x-\cos x$
$f^{11}\left(x\right)=-3\sin 3x-\sin x$
$f^{111}\left(x\right)=-3^3\cos 3x+\sin x$
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$f^7\left(x\right)=-3^7\cos 3x+\cos x$