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Question
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
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Solution
\[\frac{dy}{dx} = \frac{d}{dx} \left( \sin \frac{x}{2} + \cos \frac{x}{2} \right)^2 \]
\[ = \frac{d}{dx}\left( \sin^2 \frac{x}{2} + \cos^2 \frac{x}{2} + 2 \sin \frac{x}{2}\cos \frac{x}{2} \right)\]
\[ = \frac{d}{dx}\left( 1 + \sin x \right)\]
\[ = \frac{d}{dx}\left( 1 \right) + \frac{d}{dx}\left( \sin x \right)\]
\[ = 0 + \cos x\]
\[ = \cos x\]
\[\frac{dy}{dx} at x =\frac{\pi}{6}= cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\]
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