Answer
$$\frac{{dy}}{{dx}} = \frac{1}{{x\ln \left( {\ln x} \right)\ln x}}$$
Work Step by Step
$$\eqalign{
& y = \ln \left( {\ln \left( {\ln x} \right)} \right) \cr
& {\text{Find the derivative of }}y{\text{ with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\ln \left( {\ln x} \right)} \right] \cr
& {\text{use the chain rule}}{\text{,}} \cr
& \frac{{dy}}{{dx}} = \frac{1}{{\ln \left( {\ln x} \right)}}\frac{d}{{dx}}\left[ {\ln \left( {\ln x} \right)} \right] \cr
& \frac{{dy}}{{dx}} = \frac{1}{{\ln \left( {\ln x} \right)}}\left( {\frac{1}{{\ln x}}} \right)\frac{d}{{dx}}\left[ {\ln x} \right] \cr
& {\text{solve the derivative}} \cr
& \frac{{dy}}{{dx}} = \frac{1}{{\ln \left( {\ln x} \right)}}\left( {\frac{1}{{\ln x}}} \right)\left( {\frac{1}{x}} \right) \cr
& {\text{simplify}} \cr
& \frac{{dy}}{{dx}} = \frac{1}{{x\ln \left( {\ln x} \right)\ln x}} \cr} $$
