Cho hàm số \(f\left( x \right) = \ln \frac.\) Tính tổng \(S = f'\left( 1 \right) + f'\left( 2 \right) + \ldots + f'\left( {2018} \right)\)?

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

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

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