By definition, the derivative is a function which is derived from another function. The definition of the derivative is usually only written for one point, but the function is defined for all points. Derivative functions of many kinds of functions can be found, including derivatives of linear, power, polynomial, exponential, and logarithmic functions.
I want to talk about the derivative function let's say we're looking at a function like f of x equals x squared plus 1. I have a graph here, in a previous example we found the derivative of this function at x equals 3 and we used the definition of the derivative and we got 6. Now it turns out you could do that at pretty much any point, you can find the derivative at negative 1, turns out to be negative 2 and you can find the derivative at a half and it turns out to be 1.
This function and many functions have derivatives at every value of x in their domain and so that gives us the notion of the derivative function. This f prime of x defines a function of x. For every value of x you think of you can come up with a derivative value. So that's a new function that we call the derivative function for f of x.
We're going to be talking about derivative functions for a while and the idea is that given a function you want to find it's derivative that is it's a derivative function and usually that means finding a formula for the derivative function that doesn't involve the limit, and that's what we're going to be doing in the next few episodes.