Proof of the derivative of ln x

Why d/dx (lnx) = 1/x? Here's the proof.
This video shows the proof of the derivative of lnx by using the chain rule.

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Proof of the limit of (1+1/n)^n = e using:

Proof of the derivative of ln x using:

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If have to prove more than one case at once. Here are some combinations.

limit of (1+1/n)^n = e and d/dx(lnx) = x
or