A) \[\frac{1}{4\sqrt{x}}{{(\sec \sqrt{x})}^{3/2}}\sin \sqrt{x}\]
B) \[\frac{1}{4\sqrt{x}}\sec \sqrt{x}\sin \sqrt{x}\]
C) \[\frac{1}{2}\sqrt{x}{{(\sec \sqrt{x})}^{3/2}}\sin \sqrt{x}\]
D) \[\frac{1}{2}\sqrt{x}\sec \sqrt{x}\sin \sqrt{x}\]
Correct Answer: A
Solution :
\[\frac{d}{dx}\left( \sqrt{\sec \sqrt{x}} \right)=\frac{1}{2\sqrt{\sec \sqrt{x}}}.\frac{d}{dx}(\sec \sqrt{x})\] \[=\frac{1}{2{{(\sec \sqrt{x})}^{1/2}}}.\sec \sqrt{x}.\tan \sqrt{x}.\frac{1}{2\sqrt{x}}\] \[{{x}^{2}}=\cos 2\theta \].
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