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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

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