Найти производную функции $$ f(x) = \frac{1+\sqrt{x}}{x}. $$

$u=1+\sqrt x$, $u' = \dfrac{1}{2\sqrt x}$; $v=x$, $v'=1$.

$$ f'(x) = \frac{\dfrac{1}{2\sqrt x}\cdot x - (1+\sqrt x)}{x^2} = \frac{\dfrac{\sqrt x}{2} - 1 - \sqrt x}{x^2} = \frac{-\dfrac{\sqrt x}{2} - 1}{x^2}. $$

$$ \boxed{f'(x) = -\dfrac{\sqrt{x}+2}{2x^2}} $$