Question

(x sin x + cos x) (x cos x − sin x)

Answer 1

$$\begin{gathered} u=x\sin x+\cos x;v=x\cos x-\sin x \\ u\text{'}=x\cos x+\sin x-\sin x=x\cos x;v\text{'}=-x\sin x+\cos x-\cos x=-x\sin x \\ \text{Using the product rule:} \\ \frac{d}{dx}\left(uv\right)=uv\text{'}+vu\text{'} \\ \frac{d}{dx}\left[\left(x\sin x+\cos x\right)\left(x\cos x-\sin x\right)\right]=\left(x\sin x+\cos x\right)\left(-x\sin x\right)+\left(x\cos x-\sin x\right)\left(x\cos x\right) \\ =-x^2{\sin }^{2}x-x\cos x\sin x+x^2{\cos }^{2}x-x\cos x\sin x \\ =x^2\left({\cos }^{2}x-{\sin }^{2}x\right)-x\left(2\sin x\cos x\right) \\ =x^2\cos \left(2x\right)-x\left(\sin \left(2x\right)\right) \\ =x\left[x\cos \left(2x\right)-\sin \left(2x\right)\right] \end{gathered}$$