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Please solve RD Sharma class 12 chapter Differentiation exercise 10.5 question 1 maths textbook solution

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Answer: \chi^{\frac{1}{x}} \frac{(1-\ln x)}{x^{2}}

Hint: Differentiate by function x^{n}

Given: x^{\frac{1}{x}}

Solution: Let, y=x^{\frac{1}{x}}

Taking log on both sides,

\begin{aligned} &\log \mathrm{y}=\log x^{\frac{1}{x}} \\\\ &\Rightarrow \log y=\frac{1}{x} \log x \end{aligned}                    \left[\because \log a^{b}=b \log a\right]

Differentiate both sides,

\frac{1}{y} \frac{d y}{d x}=\frac{1}{x} \frac{d}{d x}(\log x)+\log x \frac{d}{d x}\left(x^{-1}\right)                         [using product rule]

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

\begin{aligned} &\frac{1}{y} \frac{d y}{d x}=\frac{(1-\log x)}{x^{2}} \\\\ &\frac{d y}{d x}=y \frac{(1-\log x)}{x^{2}} \end{aligned}

put value of y=x^{\frac{1}{x}}

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


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