Nếu \(\int {f\left( x \right){\rm{d}}x} = \frac{1}{x} + \ln \left| {2x} \right| + C\) với \(x \in \left( {0; + \infty } \right)\)thì hàm số \(f\left( x \right)\) là
A.\(f\left( x \right) = - \frac{1}{{{x^2}}} + \frac{1}{x}.\)
B. \(f\left( x \right) = \sqrt x + \frac{1}{{2x}}.\)
C.\(f\left( x \right) = \frac{1}{{{x^2}}} + \ln \left( {2x} \right).\)
D.\(f\left( x \right) = - \frac{1}{{{x^2}}} + \frac{1}{{2x}}.\)