differentiate and find the domain of ff(x) = ln ln ln x

Expert Answers

An illustration of the letter 'A' in a speech bubbles

To differentiate `f(x) = ln(ln(ln(x)))` use the Chain Rule for differentiation of nested function.

The rule is that ` `if `f(x) = u(v(w(x)))` where `u, v` and `w` are functions

`f'(x) = u'(v(w(x)))v'(w(x))w'(x)`

Using the fact that `d/dx lnx = 1/x` we have that

` ``f'(x) = 1/ln(ln(x))1/ln(x)1/(x)` answer to first question


To find the domain of `f(x)` we proceed as follows

The domain of `x in (0,oo)`

So we need `ln(ln(ln(x)))` such that `ln(ln(x)) in (0,oo)` 

`implies ln(x) in (1, oo)`

`implies x in (e^1,oo)` answer to second question

 

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial