Question

Differentiate the following function with respect to x.

If $y=(\sin \frac{x}{2}+\cos \frac{x}{2})$, find $\frac{dy}{dx}$ at $x=\frac{\pi }{3}$.

• A. $\sqrt{3}-1$
• B. $\frac{\sqrt{3}-1}{4}$
• C. $\sqrt{3}+1$
• D. $1$

Answer 1

The correct option is B $\frac{\sqrt{3}-1}{4}$

Given equation

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

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

$\frac{dy}{dx}=\cos \frac{x}{2}\frac{d}{dx}\left(\frac{x}{2}\right)-\sin \frac{x}{2}\frac{d}{dx}\left(\frac{x}{2}\right)$

$\frac{dy}{dx}=\frac{1}{2}(\cos \frac{x}{2}-\sin \frac{x}{2})$

$x=\frac{\pi }{3}$

${\frac{dy}{dx}|}_{\pi /3}=\frac{1}{2}(\frac{\sqrt{3}}{2}-\frac{1}{2})$

$⟹\frac{\sqrt{3}-1}{4}$