Question

x⁴ (5 sin x − 3 cos x)

Answer 1

$$\begin{gathered} \mathrm{Let}u=x^4;v=5\sin x-3\cos x \\ \mathrm{Then},u\text{'}=4x^3;v\text{'}=5\cos x-3(-\sin x)=5\cos x+3\sin x \\ \text{According to the product rule:} \\ \frac{d}{dx}\left(uv\right)=uv\text{'}+vu\text{'} \\ \frac{d}{dx}\left(x^4\left(5\sin x-3\cos x\right)\right)=x^4\left(5\cos x+3\sin x\right)+\left(5\sin x-3\cos x\right)4x^3 \\ =x^3\left(5x\cos x+3x\sin x+20\sin x-12\cos x\right) \\ =x^3\left(\left(3x+20\right)\sin x+\left(5x-12\right)\cos x\right) \\ =3x^4\sin x+20x^3\sin x+5x\cos x-12\cos x \end{gathered}$$