Найти производную функции $f(x) = \ln^2(x) = (\ln x)^2$.

$u=\ln x$, $u'=1/x$. $$ f'(x) = 2u\cdot u' = 2\ln x\cdot\frac1x = \frac{2\ln x}{x}. $$

$$ \boxed{f'(x) = \dfrac{2\ln x}{x}} $$