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need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.2 question 15

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Answer: 3^{x \log ^{x}}(1+\log x) \times \log _{e} 3

Hint: You must know the rules of solving derivative of logarithm and polynomial function.

Given: 3^{x \log x}


Let  y=3^{x \log x}

Differentiating with respect to x

\begin{aligned} &\frac{d y}{d x}=\frac{d}{d x}\left(3^{x \log x}\right) \\ &\frac{d y}{d x}=3^{x \log x} \times \log _{e} 3 \frac{d}{d x}(x \log x) \end{aligned}

\frac{d y}{d x}=3^{x \log x} \times \log _{e} 3\left[x \frac{d}{d x}(\log x)+\log x \frac{d}{d x}(x)\right]

\frac{d y}{d x}=3^{x \log x} \times \log _{e} 3\left[x \times \frac{1}{x}+\log x(1)\right]

\begin{aligned} &\frac{d y}{d x}=3^{x \log x} \times \log _{e} 3[1+\log x] \\ &\frac{d y}{d x}=3^{x \log x}(1+\log x) \times \log _{e} 3 \end{aligned}

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