Question

lf $y_k$ is the $k^{th}$ derivative of $y$ w.r.t. $x$ and $y=\cos \left(\sin x\right)$ then $y^{\prime }\sin x+y^{\prime \prime }\cos x=$

• A. $y{\sin }^{3}x$
• B. $-y{\sin }^{3}x$
• C. $-y{\cos }^{2}x$
• D. $-y{\cos }^{3}x$

Answer 1

The correct option is C $-y{\cos }^{2}x$

$y^{\prime }=-\sin \left(\sin x\right)\cos x$
$y^{\prime \prime }=-\cos \left(\sin x\right){\cos }^{2}x+\left(\sin x\right)\left(\sin \right(\sin x\left)\right)$
$y^{\prime \prime }\cos x+y^{\prime }\sin x=-\cos \left(\sin x\right){\cos }^{2}x$

$y^{\prime \prime }\cos x+y^{\prime }\sin x=-y{\cos }^{2}x$