Derivatives of Logarithm Functions

Now that we have learned the derivatives of the exponential functions, we are ready to move on to the logarithm functions.  We will use a different approach here, and in a later note, we will look at extending this approach to other functions.

In effort to keep the notation from getting in the way, we will let  and  as we develop the derivative of the latter function.

We know that, for a fixed base, the exponential function and the logarithm function are inverses.  In particular, we have .  We will differentiate the resulting equation  and solve it for .  Remember that for , we have ; we will use that in our derivation.  (Note the application of the Chain Rule in the first step!)

We thus have .  We can extend our work to arbitrary bases by using the Chain Rule and the Change of Base Formula, .

So our final result is .