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Explain solution RD Sharma class 12 chapter Differentiation exercise 10.5 question 8 maths

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

Hint: Differentiate by applying e^{x}

Given: e^{x \log x}

Solution:  Let y=e^{x \log x}

        y=x^{x}                \left[\because a^{b}=e^{b \log a}\right]

Taking log on both sides

        \begin{aligned} &\log y=\log e^{x \log x} \\\\ &\log y=x \log x \end{aligned}

Differentiate w.r.t x,

        \frac{1}{y}=x \frac{d}{d x}(\log x)+\log x \frac{d}{d x} x                   [ use multiplication rule]

        \begin{aligned} &\frac{1}{y} \frac{d y}{d x}=x \cdot \frac{1}{x}+\log x .1 \\\\ &\frac{d y}{d x}=y+y \log x \end{aligned}

        \begin{aligned} &\frac{d y}{d x}=y(1+\log x) \\\\ &=e^{x \log x}(1+\log x) \end{aligned}\begin{aligned} &\frac{d y}{d x}=y(1+\log x) \end{aligned}

                =e^{x \log x}(1+\log x)

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