# Mark the Correct Alternative in of the Following: If Y = Sin ( X + 9 ) Cos X Then D Y D X at X = 0 is - Mathematics

MCQ

Mark the correct alternative in  of the following:
If $y = \frac{\sin\left( x + 9 \right)}{\cos x}$ then $\frac{dy}{dx}$ at x = 0 is

•  cos 9

• sin 9

•  0

• 1

#### Solution

$y = \frac{\sin\left( x + 9 \right)}{\cos x}$

Differentiating both sides with respect to x, we get

$\frac{dy}{dx} = \frac{\cos x \times \frac{d}{dx}\sin\left( x + 9 \right) - \sin\left( x + 9 \right) \times \frac{d}{dx}\cos x}{\cos^2 x} \left( \text{ Quotient rule } \right)$
$= \frac{\cos x \times \cos\left( x + 9 \right) - \sin\left( x + 9 \right) \times \left( - \sin x \right)}{\cos^2 x}$
$= \frac{\cos\left( x + 9 \right)\cos x + \sin\left( x + 9 \right)\sin x}{\cos^2 x}$
$= \frac{\cos\left( x + 9 - x \right)}{\cos^2 x}$
$= \frac{\cos9}{\cos^2 x}$
Putting x = 0, we get

$\left( \frac{dy}{dx} \right)_{x = 0} = \frac{\cos9}{\cos^2 0} = \cos9$

Thus, $\frac{dy}{dx}$  at x = 0 is cos 9.

Hence, the correct answer is option (a).

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
Q 10 | Page 48