At first glance, this may seem quite complicated. However, it is simple once you make use of exponents. 

Let y=ln(x). 

This can be written as: e^y = e^ln(x)

e to the power of a natural log cancels out, which gives: 

e^y=x

Differentiating both sides with respect to x gives:

e^y (dy/dx)=1 

[This uses implicit differentiation.  Remember that you must multiply e^y by dy/dx as there isn't an x on that side]

Substituting in e^y=x gives:

x (dy/dx) =1

And so dy/dx = 1/x