If we didn't already know the derivative of ln x, we could figure it out using the chain rule.
We know that
eln x = x.
Take the derivative of each side of this equation.
The derivative of x is 1.
To find the derivative of eln x we need to use the chain rule. The outside function is e{□} and its derivative is also e{□}.
The inside function is ln x. Since we don't yet know the derivative of ln x (at least, we're pretending we don't) we'll write its derivative as (ln x)'.
The chain rule says
(eln x)' = eln x · (ln x)'
Since eln x = x, we can simplify this to
(eln x)' = x · (ln x)'
Now return to the equation
eln x = x.
The derivative of the right-hand side is 1, and the derivative of the left-hand side is x · (ln x)', therefore
x · (ln x)' = 1.
Dividing both sides by x, we find