I am not sure if I am getting this or not. Would you please check a couple of problems before I go any further.
f(x) = ln2x
f'(x) =1/u (du/dx 1/2x)(d/dx2x)
f'(x) = 2/2x
and
f(x) = x ln x
f'(x) = x lnx^3/2
f'(x) = x(1/2lnx)
Just for clarity, I would write the first problem this way:
f(x) = ln(2x)
Let u = 2x <--> u' = 2
f(x) = ln(u)
f'(x) = (ln(u))'*(u)'
f'(x) = (1/u)*2
f'(x) = 2/(2x) = 1/x
However, you don't always need to do a substitution for a problem like this. Try to get used to doing this problem without doing substitutions because they can be hard to keep track of in harder problems (and they take more work to write).