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If f(x) = cot–1 (cos2x)1/2, then f^{\prime}\left(\frac{\pi}{6}\right) \text { is : }

Option: 1

\frac{1}{\sqrt{3}}


Option: 2

\frac{2}{\sqrt{3}}


Option: 3

\sqrt{\frac{2}{3}}


Option: 4

\frac{-2}{\sqrt{3}}


Answers (1)

best_answer

\begin{aligned} f^{\prime}(x) & =\frac{-1}{1+\cos 2 x} \times \frac{1}{2}(\cos 2 x)^{-1 / 2}(-\sin 2 x \times 2) \\ & =\frac{\sin 2 x}{\sqrt{\cos 2 x(1+\cos 2 x)}} \\ \Rightarrow f^{\prime}\left(\frac{\pi}{6}\right) & =\frac{\sin \frac{\pi}{3}}{\sqrt{\cos \frac{\pi}{3}}\left(1+\cos \frac{\pi}{3}\right)} \\ & =\frac{\frac{\sqrt{3}}{2}}{\sqrt{\frac{1}{2}\left(1+\frac{1}{2}\right)}}=\sqrt{\frac{2}{3}} \end{aligned}

Hence (C) is correct answer

Posted by

manish painkra

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