Aus Online Mathematik Brückenkurs 2

The quotient rule gives

\displaystyle \begin{align} & \left( \frac{x}{\ln x} \right)^{\prime }=\frac{\left( x \right)^{\prime }\centerdot \ln x-x\centerdot \left( \ln x \right)^{\prime }}{\left( \ln x \right)^{2}} \\ & \\ & =\frac{1\centerdot \ln x-x\centerdot \frac{1}{x}}{\left( \ln x \right)^{2}}=\frac{\ln x-1}{\left( \ln x \right)^{2}}=\frac{1}{\ln x}-\frac{1}{\left( \ln x \right)^{2}} \\ \end{align}