A) \[\frac{1}{4\sqrt{x}}\sec \sqrt{x}\sin \sqrt{x}\]
B) \[\frac{1}{4\sqrt{x}}{{(\sec \sqrt{x})}^{3/2}}\cdot \sin \sqrt{x}\]
C) \[\frac{1}{2}\sqrt{x}\sec \sqrt{x}\sin \sqrt{x}\]
D) \[\frac{1}{2}\sqrt{x}{{(\sec \sqrt{x})}^{3/2}}\cdot \sin \sqrt{x}\]
Correct Answer: B
Solution :
Let\[y=\sqrt{\sec \sqrt{x}}\] On differentiating w.r.t.\[x,\] we get \[\frac{dy}{dx}=\frac{1}{2}{{(\sec \sqrt{x})}^{-1/2}}\cdot \frac{d}{dx}(\sec \sqrt{x})\] \[=\frac{1}{2\sqrt{\sec \sqrt{x}}}\cdot \sec \sqrt{x}\cdot \tan \sqrt{x}\cdot \frac{1}{2\sqrt{x}}\] \[=\frac{1}{4\sqrt{x}}{{(\sec \sqrt{x})}^{1/2}}\frac{\sin \sqrt{x}}{\cos \sqrt{x}}\] \[=\frac{1}{4\sqrt{x}}{{(\sec \sqrt{x})}^{1/2}}\cdot \sin \sqrt{x}\cdot \sec \sqrt{x}\]
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