AP Calculus (AB)
Proof of the Derivative of Ln(x)

Goal: To show that .

The proof begins with the definition of the derivative

and then uses of the properties of logarithms to manipulate the expression so that the limit can be evaluated.

At this point we reach a minor setback since the limit cannot be evaluated in this form. Fortunately a "trick" was discovered where we will be able to find the limit by using a substitution. First, recognize that we can treat x as a fixed positive real number. The reason is that in the limit, h is approaching 0, but nothing is happening to x. Thus, h is our variable, not x.

Therefore, let . Now, when h approaches 0, u will approach infinity. We will thus use this substitution and find the limit as u approaches infinity. Further note, if , then and .

Therefore,

Some of the "stuff" in the limit can be moved out in front of the limit. Since we are evaluating the limit as u approaches infinity, 1/x is just a constant and can be moved out of the limit. Also, is a composition of functions, similar to f(g(x)). From the definition of the limit of compositions of functions, if looking for the limit of f(g(x)), if we can find the limit of g(x), then the limit of the composition is just f evaluated at the limit of g. Thus,

This is a limit that exists and is the special number e. [Click here for a proof of the definition of e]

Thus,

And the proof is complete and we see that . QED.