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  • Need solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Multiple choice question 3

Need solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Multiple choice question 3

Answers (1)

Answer:

        \left(\frac{2}{3}\right)^{\frac{1}{2}}

Hint:

        Differentiate the given function and replace x by \frac{\pi }{6} and solve

Given:

        \cot ^{-1}\left\{(\cos 2 x)^{\frac{1}{2}}\right\} \text { at } x=\frac{\pi}{6}

Solution:  

        \begin{aligned} &y=\cot ^{-1}(\sqrt{\cos 2 x}) \\\\ &\frac{d y}{d x}=\frac{-1}{1+\cos 2 x} \frac{d}{d x} \sqrt{\cos 2 x} \end{aligned}

        \frac{d y}{d x}=\left[\frac{-1}{1+\cos 2 x}\right]\left[\frac{1}{2}(\sqrt{\cos 2 x})\right](-2 \sin 2 x)

               =\frac{\sin 2 x}{(1+\cos 2 x) \sqrt{\cos 2 x}}

\text { At } x=\frac{\pi}{6}

        \frac{d y}{d x}=\frac{\sin \frac{\pi}{3}}{\left(1+\cos \frac{\pi}{3}\right) \sqrt{\cos \frac{\pi}{3}}}

               =\frac{\left(\frac{\sqrt{3}}{2}\right)}{\left(1+\frac{1}{2}\right) \sqrt{\frac{1}{2}}}

               \begin{aligned} &=\sqrt{\frac{2}{3}} \\\\ &=\left(\frac{2}{3}\right)^{\frac{1}{2}} \end{aligned}

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