Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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# Cos √ X - Mathematics

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$\cos \sqrt{x}$

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#### Solution

$\text{ Let } f(x) = \cos \sqrt{x}$
$\text{ Thus, we have }:$
$f(x + h) = \cos \sqrt{x + h}$
$\frac{d}{dx}\left( f\left( x \right) \right) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
$= \lim_{h \to 0} \frac{\cos \sqrt{x + h} - \cos \sqrt{x}}{h}$
$\text{ We know }:$
$\cos C - \cos D = - 2\sin\left( \frac{C + D}{2} \right) \sin\left( \frac{C - D}{2} \right)$
$= \lim_{h \to 0} \frac{- 2\sin \left( \frac{\sqrt{x + h} + \sqrt{x}}{2} \right) \sin\left( \frac{\sqrt{x + h} - \sqrt{x}}{2} \right)}{h}$
$= \lim_{h \to 0} \frac{- 2\sin \left( \frac{\sqrt{x + h} + \sqrt{x}}{2} \right) \sin\left( \frac{\sqrt{x + h} - \sqrt{x}}{2} \right)}{x + h - x}$
$= \lim_{h \to 0} \frac{- 2\sin \left( \frac{\sqrt{x + h} + \sqrt{x}}{2} \right) \sin\left( \frac{\sqrt{x + h} - \sqrt{x}}{2} \right)}{2 \times \left( \sqrt{x + h} + \sqrt{x} \right)\frac{\left( \sqrt{x + h} - \sqrt{x} \right)}{2}}$
$= \lim_{h \to 0} \frac{\sin\left( \frac{\sqrt{x + h} - \sqrt{x}}{2} \right)}{\frac{\sqrt{x + h} - \sqrt{x}}{2}} \lim_{h \to 0} \frac{- \sin\left( \frac{\sqrt{x + h} + \sqrt{x}}{2} \right)}{\sqrt{x + h} + \sqrt{x}}$
$= 1 \times \frac{- \sin\sqrt{x}}{2\sqrt{x}} \left[ \because \lim_{h \to 0} \frac{\sin\left( \frac{\sqrt{x + h} - \sqrt{x}}{2} \right)}{\frac{\sqrt{x + h} - \sqrt{x}}{2}} = 1 \right]$
$= \frac{- \sin\sqrt{x}}{2\sqrt{x}}$


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 30.2 | Q 5.2 | Page 26

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