Question

If $f\left(x\right)=x\cos x$, find $f^{\prime \prime }\left(x\right)$.

• A. $-\cos x-2\sin x$
• B. $-x\cos x-2\sin x$
• C. $-2x\sin x-2\cos x$
• D. $-x\sin x-2\cos x$

Answer 1

The correct option is B $-x\cos x-2\sin x$

Using the Product Rule, we have
$f^{{}^{\prime }}\left(x\right)=x\frac{d}{dx}\left(\cos x\right)+\cos x\frac{d}{dx}\left(x\right)$
$=-x\sin x+\cos x$
To find $f^{\prime \prime }\left(x\right)$, we differentiate $f^{{}^{\prime }}\left(x\right)$ w.r.t $x$ :
$f^{\prime \prime }\left(x\right)=\frac{d}{dx}(-x\sin x+\cos x)$
$=-x\frac{d}{dx}\left(\sin x\right)+\sin x\frac{d}{dx}(-x)+\frac{d}{dx}\left(\cos x\right)$
$=-x\cos x-\sin x-\sin x=-x\cos x-2\sin x$