 Welcome, let's quickly recap the main ideas from section 1.4 and active calculus on the derivative function. So previously in the course, we've defined the derivative of a function f at a specific point a to be f prime of a, and that's equal to the limit as h approaches zero of f of a plus h minus f of a all divided by h. In this section, we're not really doing anything new to the formula. We're simply observing that if we let a be a number whose value we don't specify until later, rather than a specific number that we specify right now, then we can go through the same limit process to arrive at a formula for f prime of a. We can then specify the value of a and very quickly calculate a lot of derivatives all at once at different points without having to retake a limit every time. For example, we saw in the book that if f of x is 4x minus x squared, then f prime of a is 4 minus 2a no matter what the value of a is. So again, this makes it very easy to compute derivative values because instead of having to calculate a limit every time we needed a value of f prime of a, we just calculate the limit once and for all using an unspecified value of a. And then we get a formula for f prime of a and use that resulting formula to calculate the derivative at specific points. The trade off here is that the algebra involved in coming up with f prime of a in this way can be a little more complicated. But if we're willing to pay that price, then the pay off is pretty good. So since we can calculate f prime of a for any value of a, that makes the derivative of f a brand new function unto itself. We call this the derivative function, f prime of x. And we're now using x to make it clear that this derivative is a function with a variable input and not just a fixed point like specifically one or specifically negative five or so on. And we're going to define the derivative function f prime of x. No surprise to be the limit as h goes to zero of f of x plus h minus f of x divided by h provided that the limit exists. That's the one main concept of this section. There are two applications of this concept in this section. They just have to do with treating f prime of x, the derivative function, like any other function. First, we need to be able to find a formula for f prime of x if we're given a formula for f of x. And second, if we're given a graph of f of x, we'd like to be able to sketch the graph of f prime of x. So both of those tasks will have examples in the videos that are coming up right now in addition to the ones in the textbook.