Question

$\mathrm{If}y={\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)}^2,\mathrm{find}\frac{dy}{dx}\mathrm{at}x=\frac{\mathrm{\pi }}{6}.$

Answer 1

$$\begin{gathered} \frac{dy}{dx}=\frac{d}{dx}{\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)}^2 \\ =\frac{d}{dx}\left({\sin }^2\frac{x}{2}+{\cos }^2\frac{x}{2}+2\sin \frac{x}{2}\cos \frac{x}{2}\right) \\ =\frac{d}{dx}\left(1+\sin x\right) \\ =\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(\sin x\right) \\ =0+\cos x \\ =\cos x \\ \frac{dy}{dx}\text{at x =}\frac{\mathrm{\pi }}{6}\text{= cos}\frac{\mathrm{\pi }}{6}\text{=}\frac{\sqrt{3}}{2} \end{gathered}$$