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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
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Solution
\[y = \frac{\sin\left( x + 9 \right)}{\cos x}\]
Differentiating both sides with respect to x, we get
\[ = \frac{\cos x \times \cos\left( x + 9 \right) - \sin\left( x + 9 \right) \times \left( - \sin x \right)}{\cos^2 x}\]
\[ = \frac{\cos\left( x + 9 \right)\cos x + \sin\left( x + 9 \right)\sin x}{\cos^2 x}\]
\[ = \frac{\cos\left( x + 9 - x \right)}{\cos^2 x}\]
\[ = \frac{\cos9}{\cos^2 x}\]
\[\left( \frac{dy}{dx} \right)_{x = 0} = \frac{\cos9}{\cos^2 0} = \cos9\]
Thus, \[\frac{dy}{dx}\] at x = 0 is cos 9.
Hence, the correct answer is option (a).
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