Question

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 

Options

•  cos 9     

• sin 9   

•  0     

• 1

Answer 1

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

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).