Chapter 6 - Applications of Integration - 6.8 Logarithmic and Exponential - 6.8 Exercises - Page 480: 8

$$\frac{1}{{x\ln x}}$$

$$\eqalign{ & \frac{d}{{dx}}\left( {\ln \left( {\ln x} \right)} \right) \cr & {\text{use the rule }}\frac{d}{{dx}}\left( {\ln u} \right) = \frac{1}{u}\frac{{du}}{{dx}},{\text{ letting }}u = \ln x \cr & = \frac{1}{{\ln x}}\frac{d}{{dx}}\left( {\ln x} \right) \cr & = \frac{1}{{\ln x}}\left( {\frac{1}{x}} \right) \cr & {\text{multiply}} \cr & = \frac{1}{{x\ln x}} \cr} $$