Answer
$f'(x) = \frac{1}{x~\cdot~ln~x~\cdot ~ln~ln~x}$
The domain is $~~(e, \infty)$
Work Step by Step
$f(x) = ln~ln~ln~x$
We can differentiate $f(x)$:
$f'(x) = \frac{1}{ln~ln~x}\cdot \frac{d}{dx}(ln~ln~x)$
$f'(x) = \frac{1}{ln~ln~x}\cdot \frac{1}{ln~x} \cdot\frac{d}{dx}(ln~x)$
$f'(x) = \frac{1}{ln~ln~x}\cdot \frac{1}{ln~x} \cdot\frac{1}{x}$
$f'(x) = \frac{1}{x~\cdot~ln~x~\cdot ~ln~ln~x}$
When we consider the function $~~ln~ln~ln~x~~$ it is required that $~~ln~ln~x \gt 0$
Then $~~ln~x \gt 1$
Then $~~x \gt e$
The domain is $~~(e, \infty)$
