Question

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

Answer 1

We have,

$=\frac{dy}{dx}=\frac{d}{dx}{(sin\frac{x}{2}+cos\frac{x}{2})}^2$

$=\frac{d}{dx}(si{n}^2\frac{x}{2}+co{s}^2\frac{x}{2}+2sin\frac{x}{2}.cos\frac{x}{2})$

$=\frac{d}{dx}(1+sinx)\left[\begin{array}{c}\because si{n}^2\theta +co{s}^2\theta =1\\ si{n}^2\theta =2sin\theta .cos\theta \end{array}\right]$

$=0+cosx$

$=cosx$

$\frac{dy}{dx}\text{at}x=\frac{\pi }{6}$

$=cos\frac{\pi }{6}$

$=\frac{\sqrt{3}}{2}$