Question

Find the derivative of $(a{x}^2+\sin x)(p+q\cos x)$ where $a,p,q$ are fixed non-zero constants.

Answer 1

Let $f\left(x\right)=(a{x}^2+\sin x)(p+q\cos x)$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}\left(\right(a{x}^2+\sin x\left)\right(p+q\cos x\left)\right)$
$\Rightarrow f^{\prime }\left(x\right)=(p+q\cos x)\frac{d}{dx}(a{x}^2+\sin x)+(a{x}^2+\sin x)\frac{d}{dx}(p+q\cos x)$
$\Rightarrow f^{\prime }\left(x\right)=(p+q\cos x)(\frac{d}{dx}a{x}^2+\frac{d}{dx}\sin x)+(a{x}^2+\sin x)(\frac{d}{dx}p+\frac{d}{dx}q\cos x)$
$\Rightarrow f^{\prime }\left(x\right)=(p+q\cos x)(2ax+\cos x)+(a{x}^2+\sin x)(-q\sin x)$
$\therefore f^{\prime }\left(x\right)=-q\sin x(a{x}^2+\sin x)+(p+q\cos x)(2ax+\cos x)$