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