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Provide solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 18 sub question (i)

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Answer: x^{x} \sqrt{x}\left(\log x+1+\frac{1}{2 x}\right)

Hint:  \text { Diff by } x^{x} \sqrt{x}

Given: x^{x} \sqrt{x}


Let, y=x^{x} \sqrt{x}

Taking log both side  

\log y=\log x^{x} \sqrt{x} \; \; \; \; \; \; \; \; \; \; \; \quad[\log m n=\log m+\log n]

\log y=\log x^{x}+\log \sqrt{x}

\log y=x \log x+\log (x)^{\frac{1}{2}}

\log y=\left(x+\frac{1}{2}\right) \cdot \log x

\frac{d}{d x} \log y=\frac{d}{d x}\left(x+\frac{1}{2}\right) \cdot \log x

\begin{aligned} &\frac{1}{y} \frac{d y}{d x}=\frac{d}{d x}\left(x+\frac{1}{2}\right) \cdot \log x+\left(x+\frac{1}{2}\right) \cdot \frac{d}{d x} \log x \\\\ &\frac{d y}{d x}=x^{x} \sqrt{x}\left(\log x+1+\frac{1}{2 x}\right) \end{aligned}

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