Question

Помогите найти производную y=x^(1/x)

Answer 1

$y=x^{\frac{1}{x}}\\\\lny=ln(x^{\frac{1}{x}})\\\\lny=\frac{1}{x}\cdot lnx\\\\(lny)'=(\frac{1}{x}\cdot lnx)'\\\\\frac{y'}{y}=-\frac{1}{x^2}\cdot lnx+\frac{1}{x}\cdot \frac{1}{x}\\\\y'=y\cdot (-\frac{1}{x^2}\cdot lnx+\frac{1}{x^2})\\\\y'=x^{\frac{1}{x}}\cdot \frac{1}{x^2}\cdot (-lnx+1)\\\\y'=x^{\frac{1}{x}-2}\cdot (1-lnx)\\\\y'=x^{\frac{1-2x}{x}}\cdot (1-lnx)$