Why does ln(x) differentiate to 1/x ?

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