Derivative of ln x

If we didn't already know the derivative of ln x, we could figure it out using the chain rule.

We know that

e^{ln x} = x.

Take the derivative of each side of this equation.

The derivative of x is 1.

To find the derivative of e^{ln 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

(e^{ln x})' = e^{ln x} · (ln x)'

Since e^{ln x} = x, we can simplify this to

(e^{ln x})' = x · (ln x)'

Now return to the equation

e^{ln 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