Question

Find the derivative of $x^4(5\sin x-3\cos x).$

Answer 1

Let $f\left(x\right)=x^4(5\sin x-3\cos x)$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}\left(x^4\right(5\sin x-3\cos x\left)\right)$
$\Rightarrow f^{\prime }\left(x\right)=(5\sin x-3\cos x)\frac{d}{dx}\left(x^4\right)+\left(x^4\right)\frac{d}{dx}(5\sin x-3\cos x)$
$\Rightarrow f^{\prime }\left(x\right)=4x^3(5\sin x-3\cos x)+\left(x^4\right)\left(\frac{d}{dx}\right(5\sin x)-\frac{d}{dx}(3\cos x\left)\right)$
$\Rightarrow f^{\prime }\left(x\right)=4x^3(5\sin x-3\cos x)+(5\cos x+3\sin x)\left(x^4\right)$
$\Rightarrow f^{\prime }\left(x\right)=x^3\left[4\right(5\sin x-3\cos x)+x(5\cos x+3\sin x\left)\right]$
$\Rightarrow f^{\prime }\left(x\right)=x^3(20\sin x-12\cos x+5x\cos x+3x\sin x)$
$\Rightarrow f^{\prime }\left(x\right)=x^3(5x\cos x+3x\sin x+20\sin x-12\cos x)$