Differentiate Logs with Proof

In order to differentiate logs we must use the chain rule. The simplest type of log to differentiate is a natural log this can be done as shown below.

Differentiate Natural Logs
A natural log is a log to the base e.
d/dx (ln x) = 1/x

However if we want to differentiate ln(f(x)) we must use the chain rule to get

d/dx (ln(f(x)) = f'(x)/f(x)

Proof of Derivative of Natural Logs
Consider

y=ln(x)

then from the definition of a log we get

e^y = x                   –(1)

Differentiate each side with respect to x (you need to use implicit differentiation for the left to get e^y dy/dx) to get

e^y dy/dx = 1

but from (1) we know that e^y = x which we can substitute to get

x dy/dx =1

giving the derivative

dy/dx = 1/x