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

# 4 X + 5 Sin X 3 X + 7 Cos X - Mathematics

$\frac{4x + 5 \sin x}{3x + 7 \cos x}$

#### Solution

$\text{ Let } u = 4x + 5 \sin x; v = 3x + 7 \cos x$
$\text{ Then }, u' = 4 + 5 \cos x; v' = 3 - 7 \sin x$
$\text{ Using the quotient rule }:$
$\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}$
$\frac{d}{dx}\left( \frac{4x + 5 \sin x}{3x + 7 \cos x} \right) = \frac{\left( 3x + 7 \cos x \right)\left( 4 + 5 \cos x \right) - \left( 4x + 5 \sin x \right)\left( 3 - 7 \sin x \right)}{\left( 3x + 7 \cos x \right)^2}$
$= \frac{12x + 15 x \cos x + 28 \cos x + 35 \cos^2 x - 12x + 28 x \sin x - 15 \sin x + 35 \sin^2 x}{\left( 3x + 7 \cos x \right)^2}$
$= \frac{15 x \cos x + 28 x \sin x + 28 \cos x15 \sin x + 35\left( \sin^2 x + \cos^2 x \right)}{\left( 3x + 7 \cos x \right)^2}$
$= \frac{15 x \cos x + 28 x \sin x + 28 \cos x15 \sin x + 35}{\left( 3x + 7 \cos x \right)^2}$

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 30.5 | Q 21 | Page 44