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If Y = ( Sin X 2 + Cos X 2 ) 2 , Find D Y D X a T X = π 6 . - Mathematics

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

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Solution

\[\frac{dy}{dx} = \frac{d}{dx} \left( \sin \frac{x}{2} + \cos \frac{x}{2} \right)^2 \]
\[ = \frac{d}{dx}\left( \sin^2 \frac{x}{2} + \cos^2 \frac{x}{2} + 2 \sin \frac{x}{2}\cos \frac{x}{2} \right)\]
\[ = \frac{d}{dx}\left( 1 + \sin x \right)\]
\[ = \frac{d}{dx}\left( 1 \right) + \frac{d}{dx}\left( \sin x \right)\]
\[ = 0 + \cos x\]
\[ = \cos x\]
\[\frac{dy}{dx} at x =\frac{\pi}{6}= cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\]

Concept: The Concept of Derivative - Algebra of Derivative of Functions
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 30 Derivatives
Exercise 30.3 | Q 19 | Page 34

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