Question

Differentiate

$x^{1/x}$

Answer 1

$\mathrm{Let}y=x^{\frac{1}{x}}...\left(\mathrm{i}\right)$
Taking log on both sides,
$$\begin{gathered} \log y=\log x^{\frac{1}{x}} \\ \Rightarrow \log y=\frac{1}{x}\log x\left[\because \log a^b=b\log a\right] \end{gathered}$$
Differentiating with respect to x,
$$\begin{gathered} \Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{1}{x}\frac{d}{dx}\left(\log x\right)+\log x\frac{d}{dx}\left(x^{-1}\right)\left[\mathrm{Using}\mathrm{product}\mathrm{rule}\right] \\ \Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{1}{x}\times \frac{1}{x}+\left(\log x\right)\times \left(-\frac{1}{x^2}\right) \\ \Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{1}{x^2}-\frac{\log x}{x^2} \\ \Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{\left(1-\log x\right)}{x^2} \end{gathered}$$
$\Rightarrow \frac{dy}{dx}=x^{\frac{1}{x}}\left[\frac{1-\log x}{x}\right]$ [From (i)]