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Please solve RD Sharma class 12 chapter Differentiation exercise 10.5 question 5 maths textbook solution

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Answer: \log x^{x-1}(1+\log x \log \log x)

Hint: Differentiate by  \log ^{x}\left ( x \right )

Given: (\log x)^{x}

Solution:  Let y=(\log x)^{x}

Taking log both sides

        \log y=x \log (\log x)

Differentiate w.r.t x,

        \frac{1}{y} \frac{d y}{d x}=x \cdot \frac{d}{d x}\{\log (\log x)\}+\log (\log x)\left(\frac{d}{d x}\right)(x)

                  \Rightarrow \frac{1}{y} \frac{d y}{d x}=x \frac{1}{\log x} \frac{d}{d x}(\log x)+\log (\log x)(1)

                 \begin{aligned} &\Rightarrow \frac{1}{y} \frac{d y}{d x}=\frac{x}{\log x}\left(\frac{1}{x}\right)+\log (\log x) \\\\ &\Rightarrow \frac{1}{y} \frac{d y}{d x}=\frac{1}{\log x}+\log (\log x) \\\\ &\Rightarrow \frac{d y}{d x}=y\left[\frac{1}{\log x}+\log (\log x)\right] \end{aligned}

   Now, put value of   y=(\log x)^{x}

                \Rightarrow \frac{d y}{d x}=(\log x)^{x}\left[\frac{1}{\log x}+\log (\log x)\right]

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