Help proving an integration formula via differentiation

I'm stuck on something simple, I'd be grateful for any help.

My text ("Forgotten Calculus", Bleau) states the following integration formula:

integral [dx / (x ln x)] = ln |ln x| + C.

In the course of a problem where I've applied the formula, I realised I can't prove it.

Here's my attempt, for what it's worth.

(Re notation:

exponents are denoted with the "^" sign, e.g x squared = x^2;

first derivatives are denoted with the " ' " symbol, e.g. the 1st derivative of f(x) = f'(x))

f(x) = ln |ln x| + C

I want to differentiate this via the product rule, so I rewrite the function as

f(x) = (ln x^0) * (ln x) + C

Using the product rule,

f'(x) = [ (ln x^0) * (ln x)' ] + [(ln x^0)' * (ln x) ]

= [ (ln 1) * (1/x)] + [ (0/1) * (ln x)]

= [ ln / x ] + [0]

= ln / x.

I don't see how ln / x = ln |ln x|.

Any advice?