So far, the only functions we know how to find the derivative of are polynomials, rational functions, the sine and cosine and combinations of those functions. For the next several sections we're going to look at how to find the derivatives of other common functions starting with logarithms and exponentials.
Video Lectures
Fair warning: This is the most theoretical of all of our lectures. Our goal is to come up with derivatives of the exponential and logarithmic functions but, to do that, we need a technical definition of e. If you’re following along with the theoretical side of the lectures then you’re in the right place. If you’re just interested in practical matters and you’re willing to accept this limit at face value then you can skip ahead to the next presentation.
From here out our goal is going to be to add to the "base functions" that we can differentiate starting with the natural exponential function.
To find the derivative of our next base function, the natural logarithm, we're going to start by using the Chain Rule to get a relationship between the derivative of a function and the derivative of its inverse.
By playing on the derivatives for the natural logarithmic and exponential functions, we can get formulas for the corresponding functions with any valid base.
