From Förberedande kurs i matematik 2

We can see the expression as "ln of something",

\displaystyle \ln \bbox[#FFEEAA;,1.5pt]{\phantom{\ln x}}\,,

where "something" is \displaystyle \ln x.

Because we have a compound expression, we use the chain rule and obtain, roughly speaking, the outer derivative multiplied by the inner derivative,

\displaystyle \frac{d}{dx}\,\ln \bbox[#FFEEAA;,1.5pt]{\,\ln x\,} = \frac{1}{\bbox[#FFEEAA;,1.5pt]{\,\ln x\,}}\cdot \bigl( \bbox[#FFEEAA;,1.5pt]{\,\ln x\,} \bigr)'\,,

where the first factor on the right-hand side \displaystyle 1/\bbox[#FFEEAA;,1.5pt]{\,\ln x\,} is the outer derivative of \displaystyle \ln \bbox[#FFEEAA;,1.5pt]{\,\ln x\,} and the other factor \displaystyle \bigl( \bbox[#FFEEAA;,1.5pt]{\,\ln x\,} \bigr)' is the inner derivative. Thus, we get

\displaystyle \frac{d}{dx}\,\ln\ln x = \frac{1}{\ln x}\cdot \frac{1}{x} = \frac{1}{x\ln x}\,\textrm{.}