A calculus problem by Trung Đặng Đoàn Đức

$\frac{d}{dx}\sqrt[x]{x}=\frac{d}{dx}e^{\frac{\ln x}{x}}$

$=e^{\frac{\ln x}{x}}\times\frac{d}{dx}\frac{\ln x}{x}$ , by the Chain Rule

$=e^{\frac{\ln x}{x}}\times\frac{1-\ln{x}}{x^2}$ , by the Quotient Rule.

Setting this to $0$ , we have $1-\ln{x}=0$ , or $x=e$ . It is easy to check that this is indeed the maximum, using a second derivative test. Thus, our answer is $\boxed{\sqrt[e]{e}}$ .