Cho hàm số f(x) = \ln {{2018x} / {x + 1

Ta có \(f'\left( x \right) = {\left( {\ln \frac{{2018x}}{{x + 1}}} \right)^\prime } = \frac{1}{{\frac{{2018x}}{{x + 1}}}} \cdot {\left( {\frac{{2018x}}{{x + 1}}} \right)^\prime } = \frac{{x + 1}}{{2018x}} \cdot \frac{{2018}}{{{{\left( {x + 1} \right)}^2}}} = \frac{1}{{x\left( {x + 1} \right)}}\).

Vậy \[S = f'\left( 1 \right) + f'\left( 2 \right) + \ldots + f'\left( {2018} \right)\]\( = \frac{1}{{1 \cdot 2}} + \frac{1}{{2 \cdot 3}} + .. + \frac{1}{{2018 \cdot 2019}}\)

\( = \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + .. + \frac{1}{{2018}} - \frac{1}{{2019}}\)\( = 1 - \frac{1}{{2019}} = \frac{{2018}}{{2019}}.\) Chọn D.