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

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 $y=x^{\frac{1}{x}}$

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