Question

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

Answer 1

$y=(\sin \frac{x}{2}+\cos \frac{x}{2}{)}^2$

$⟹\frac{dy}{dx}=2\left[{\cos }^2\right(x/2)-{\sin }^2(x/2\left)\right]\left(1/2\right)]$

$⟹\frac{dy}{dx}=\cos x$

Now $\frac{dy}{dx}$ at $x=\pi /6$ is$=\cos \left(\pi /6\right)=\frac{\sqrt{3}}{2}$