ln(4x) is a composite function, composed of the functions lnx and 4x. Because of that, we should use the chain rule:
dy/(dx) = (dy)/(du) (du)/dx
We already know that (lnx)' = 1/x. So, we want what's inside of the natural logarithm to be a single variable, and we can do this by setting u = 4x. Now we could say that (lnu)' = 1/u, with respect to u. Essentially, the chain rule states that the derivative of y with respect to x, is equal to the derivative of y with respect to u, where u is a function of x, times the derivative of u with respect to x. In our case, y = ln(4x). Differentiating u with respect to x is simple, since u = 4x: u' = 4, with respect to x. So, we see that:
dy/(dx) = 1/u * 4 = 4/u
Now we can change u back into 4x, and get 4/(4x) = 1/x.
Interestingly enough, [ln(cx)]' where c is a non-zero constant, where it is defined, is equal to 1/x, just like (lnx)', even though we are using the chain rule.
