Question

sin x cos x

Answer 1

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