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

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

Ta có

\(S = f'\left( 1 \right) + f'\left( 2 \right) + f'\left( 3 \right) + ... + f'\left( {2018} \right)\)

\( = \left( {1 - \frac{1}{2}} \right) + \left( {\frac{1}{2} - \frac{1}{3}} \right) + \left( {\frac{1}{3} - \frac{1}{4}} \right) + ... + \left( {\frac{1} - \frac{1}} \right)\)

\( = 1 - \frac{1} = \frac.\)

Đáp án D