A) \[0\]
B) \[1\]
C) \[\frac{\pi }{4}\]
D) \[\sqrt{\pi }\]
Correct Answer: D
Solution :
Given, \[y=\log \,(\sin \,({{x}^{2}}))\] \[\frac{dy}{dx}=\frac{1}{\sin \,{{x}^{2}}}.\,\cos \,{{x}^{2}}.2x\] \[=2x\,cot\,{{x}^{2}}\] At \[x=\frac{\sqrt{\pi }}{2},\frac{dy}{dx}=\frac{2\sqrt{\pi }}{2}\,\cot \,{{\left( \frac{\sqrt{\pi }}{2} \right)}^{2}}\] \[=\sqrt{\pi }\,\,\cot \,\,\left( \frac{\pi }{4} \right)=\sqrt{\pi }\]
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