Calculus: Early Transcendentals, Chapter 3, 3.6, Section 3.6, Problem 30

f(x)=ln(ln(ln(x))) . To find the domain of f(x) we need to solve the inequality ln(ln(x))>0 . Take the exponential on both sides, e^(ln(ln(x)))>e^0=>ln(x)>1=>e^(ln(x))>e^1=>x>e . The plot shows this result:

Next, we need to apply the Chain rule multiple times, keeping in mind the derivative of ln(x) which is 1/x . Thus,
d/dx f(x) = (1)/(ln(ln(x)))(ln(ln(x)))' =
(1)/(ln(ln(x)))*(1)/(ln(x))(ln(x))'=
(1)/(ln(ln(x)))*(1)/(ln(x))*1/x = (1)/(x ln(x) ln(ln(x)))