I would use the limit definition of a derivative.
where h is a very small change in x
y' = [ln(x+h) - ln(x)]/h
Means the same as the:
Limit: (ln(x+h) - ln(x))/h
as h --> 0
Using the properties of logarithms, combine them into the logarithm function.
Limit: ln((x+h)/x)/h
ln(1+h/x)*(1/h)
change of variable
h/x = n ; h = nx
as h--> 0 so does n-->0
So now we have ...
Limit: (1/nx)*ln(1+n)
as n--> 0
now the definition of Euler's number "e"
Limit: (1+n)^(1/n) as n-->0
So we change the limit to
Limit: (1/x)*ln((1+n)^(1/n))
as n-->0
"limit of a log is the log of a limit"
(1/x)* ln( Limit: (1+n)^(1/n) as n-->0 )
= (1/x)* ln(e)
= (1/x)*(1)
= 1/x
