[tex]y=\left(\ln^{2}\left(x\right)\right)\Rightarrow y'=2\cdot \ln\left(x\right)\cdot \left(\ln\left(x\right)\right)'=\dfrac{2\,\ln\left(x\right)}{x}[/tex]

[tex]y''=\left(\dfrac{2\,\ln\left(x\right)}{x}\right)'=2\cdot \frac{\left(\ln\left(x\right)\right)'\cdot x-\left(x\right)'\cdot \ln\left(x\right)}{{x}^{2}}=2\cdot \frac{\dfrac{1}{x}\cdot x-1\cdot \ln\left(x\right)}{{x}^{2}}=\dfrac{2\,\left(1-\ln\left(x\right)\right)}{{x}^{2}}[/tex]