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

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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= eln(x)

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

ey=x

Differentiating both sides with respect to x gives:

ey (dy/dx)=1 

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

Substituting in ey=x gives:

x (dy/dx) =1

And so dy/dx = 1/x

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