Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Differentiate Each of the Following from First Principle: √ Sin ( 3 X + 1 ) - Mathematics

Differentiate each of the following from first principle:

$\sqrt{\sin (3x + 1)}$

#### Solution

$\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}$
$= \lim_{h \to 0} \frac{\sqrt{\sin \left( 3\left( x + h \right) + 1 \right)} - \sqrt{\sin \left( 3x + 1 \right)}}{h}$
$= \lim_{h \to 0} \frac{\sqrt{\sin \left( 3x + 3h + 1 \right)} - \sqrt{\sin \left( 3x + 1 \right)}}{h} \times \frac{\sqrt{\sin \left( 3x + 3h + 1 \right)} + \sqrt{\sin \left( 3x + 1 \right)}}{\sqrt{\sin \left( 3x + 3h + 1 \right)} + \sqrt{\sin \left( 3x + 1 \right)}}$
$= \lim_{h \to 0} \frac{\sin \left( 3x + 3h + 1 \right) - \sin \left( 3x + 1 \right)}{h \left( \sqrt{\sin \left( 3x + 3h + 1 \right)} + \sqrt{\sin \left( 3x + 1 \right)} \right)}$
$We have:$
$sin C-sin D= 2 cos\left( \frac{C + D}{2} \right)\sin\left( \frac{C - D}{2} \right)$
$= \lim_{h \to 0} \frac{2 \cos \left( \frac{3x + 3h + 1 + 3x + 1}{2} \right) \sin \left( \frac{3x + 3h + 1 - 3x - 1}{2} \right)}{h \left( \sqrt{\sin \left( 3x + 3h + 1 \right)} + \sqrt{\sin \left( 3x + 1 \right)} \right)}$
$= \lim_{h \to 0} \frac{2 \cos \left( \frac{6x + 3h + 2}{2} \right) \sin \frac{3h}{2}}{h \left( \sqrt{\sin \left( 3x + 3h + 1 \right)} + \sqrt{\sin \left( 3x + 1 \right)} \right)}$
$= \lim_{h \to 0} 2 \cos \left( \frac{6x + 3h + 2}{2} \right) \lim_{h \to 0} \frac{\sin \frac{3h}{2}}{h \times \frac{3}{2}} \times \frac{3}{2} \times \lim_{h \to 0} \frac{1}{\left( \sqrt{\sin \left( 3x + 3h + 1 \right)} + \sqrt{\sin \left( 3x + 1 \right)} \right)}$
$= 2 \cos \left( 3x + 1 \right) \times \left( \frac{3}{2} \right) \times \frac{1}{\sqrt{\sin \left( 3x + 1 \right)} + \sqrt{\sin \left( 3x + 1 \right)}}$
$= \frac{3 \cos \left( 3x + 1 \right)}{2\sqrt{\sin \left( 3x + 1 \right)}}$



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 3.05 | Page 26