Answer
$$\frac{{dy}}{{dx}} = \frac{{{{\left( {\ln x} \right)}^{1/\ln x}}\left( {1 - \ln \left( {\ln x} \right)} \right)}}{{x{{\left( {\ln x} \right)}^2}}}$$
Work Step by Step
$$\eqalign{
& y = {\left( {\ln x} \right)^{1/\ln x}} \cr
& {\text{Take the natural logarithm on both sides}} \cr
& \ln y = \ln {\left( {\ln x} \right)^{1/\ln x}} \cr
& {\text{use power property for logarithms}} \cr
& \ln y = \frac{1}{{\ln x}}\ln \left( {\ln x} \right) \cr
& {\text{differentiate both sides with respect to }}x \cr
& \frac{1}{y}\frac{{dy}}{{dx}} = \frac{d}{{dx}}\left( {\frac{{\ln \left( {\ln x} \right)}}{{\ln x}}} \right) \cr
& {\text{use quotient rule for derivatives}} \cr
& \frac{1}{y}\frac{{dy}}{{dx}} = \frac{{\ln x\left( {\frac{1}{{x\ln x}}} \right) - \ln \left( {\ln x} \right)\left( {\frac{1}{x}} \right)}}{{{{\left( {\ln x} \right)}^2}}} \cr
& \frac{1}{y}\frac{{dy}}{{dx}} = \frac{{\frac{1}{x} - \frac{1}{x}\ln \left( {\ln x} \right)}}{{{{\left( {\ln x} \right)}^2}}} \cr
& \frac{1}{y}\frac{{dy}}{{dx}} = \frac{{1 - \ln \left( {\ln x} \right)}}{{x{{\left( {\ln x} \right)}^2}}} \cr
& {\text{solve the equation for }}\frac{{dy}}{{dx}} \cr
& \frac{{dy}}{{dx}} = \frac{{y\left( {1 - \ln \left( {\ln x} \right)} \right)}}{{x{{\left( {\ln x} \right)}^2}}} \cr
& {\text{replace }}y = {\left( {\ln x} \right)^{1/\ln x}} \cr
& \frac{{dy}}{{dx}} = \frac{{{{\left( {\ln x} \right)}^{1/\ln x}}\left( {1 - \ln \left( {\ln x} \right)} \right)}}{{x{{\left( {\ln x} \right)}^2}}} \cr} $$
