Question

If the derivative of $x^4(5\sin x-3\cos x)$ is $x^3[mx\cos x+nx\sin x+p\sin x+q\cos x]$, then $m,n$, $p$ and $q$ respectively are

• A. $5,-3,-20,12$
• B. $5,3,20,-12$
• C. $5,3,12,-20$
• D. $5,-3,-12,20$

Answer 1

Answer: (B)

Let $y=x^4(5\sin x-3\cos x)$
Differentiating $y$ w.r.t. $x$, we get
$\frac{dy}{dx}=x^4\frac{d}{dx}(5\sin x-3\cos x)+(5\sin x-3\cos x)\frac{d}{dx}\left(x^4\right)$(by product rule)
$=x^4(5\cos x+3\sin x)+(5\sin x-3\cos x)4x^3$
$=x^3[x(5\cos x+3\sin x)+4(5\sin x-3\cos x)]$
$=x^3[5x\cos x+3x\sin x+20\sin x-12\cos x]$