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  • Please solve RD Sharma class 12 chapter 10 Differentiation exercise Multiple choice question 1 maths textbook solution

Please solve RD Sharma class 12 chapter 10 Differentiation exercise Multiple choice question 1 maths textbook solution

Answers (1)

Answer:

        \frac{1}{2e}

Hint:

        Use quotient rule to differentiate the function

Given:

        f(x)=\log _{x^{2}}\left(\log _{e}\right) x

Solution:  

        f(x)=\log _{x^{2}}\left(\log _{e}\right) x

                  =\frac{\log (\log x)}{(\log (x))^{2}}

        f(x)=\frac{\log (\log x)}{2 \log x}

Use quotient rule

        f^{\prime}(x)=\frac{\frac{1}{2}\left[\log \left(\frac{1}{\log x}\right)\left(\frac{1}{x}\right)-\log (\log (x))\left(\frac{1}{x}\right)\right]}{(\log x)^{2}}

                   =\frac{1}{2} \frac{\left[\frac{(1-\log (\log x))}{x}\right]}{(\log x)^{2}}

Put x=e

        f^{\prime}(e)=\left(\frac{1}{2}\right) \frac{\left[\frac{(1-\log (\log e))}{e}\right]}{(\log e)^{2}}

        \begin{aligned} &=\left(\frac{1}{2}\right) \frac{(1-0)}{e} \quad[\text { Since } \log e=1] \\\\ &f^{\prime}(e)=\frac{1}{2 e} \end{aligned}

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