Cho hàm số \(y = \frac{{\ln x}}{x}\). Các mệnh đề sau đúng hay sai?

\(y' = \frac{{{{\left( {\ln x} \right)}^\prime }.x - x'.\ln x}}{{{x^2}}} = \frac{{\frac{1}{x}.x - \ln x}}{{{x^2}}} = \frac{{1 - \ln x}}{{{x^2}}}\)

\(y'' = \frac{{{{\left( {1 - \ln x} \right)}^\prime }.{x^2} - {{\left( {{x^2}} \right)}^\prime }\left( {1 - \ln x} \right)}}{{{x^4}}}\)\( = \frac{{ - \frac{1}{x}.{x^2} - 2x\left( {1 - \ln x} \right)}}{{{x^4}}}\)\( = \frac{{ - x - 2x\left( {1 - \ln x} \right)}}{{{x^4}}} =  - \frac{{1 + 2\left( {1 - \ln x} \right)}}{{{x^3}}} =  - \frac{{3 - 2\ln x}}{{{x^3}}}\)

Suy ra: \(2y' + xy'' = 2.\frac{{1 - \ln x}}{{{x^2}}} - x\frac{{3 - 2\ln x}}{{{x^3}}}\)\( = \frac{{2 - 2\ln x - 3 + 2\ln x}}{{{x^2}}} =  - \frac{1}{{{x^2}}}\).