Question

how would you differentiate ln ln ln x, and what would be the domain??

Answer 1

Let u = ln x
d/dx ln ln x = d/dx (ln u)
= dln u/du * du/dx = (1/u)* (1/x)
= 1/(x ln x)
Now add another step. Let v = ln (ln x)
d/dx ln ln ln x = d/dx ln v
= d(ln v)/dv * dv/dx
= (1/v)/(x ln x)
= [1/(ln ln x)]*1/(x ln x)
= 1/[x ln x]^2
In order for the function to be defined, x must be positive. So must ln x and ln ln x. Use these facts to determine the domain. For ln x to be positive, x must exceed 1. For ln ln x to be positive, ln x must exceed 1, which means x > e.