 This video is going to talk about the difference in our sheet. We've learned about the average rate of change and we've learned that formula when we want to use any given two points. But we want to make it more broad, okay? And we want to find a formula so that we can use it over and over again. No matter what two points, I can plug it into this formula and I'll always be able to find what my average rate of change is or the difference quotient. So we're just going to take the average rate of change and substitute the function notation and we get a given function. I don't even know what this is. That's something ugly. f of x and a constant h, which can't be zero. Then we have f of x plus h, that's function notation. So we'll have to plug in something for that, minus, and I want to write it like this so that you will write it like that when you're working with it, minus the f function. All divided by that h. So let's see what we can do with this. Compute the difference quotient. So first we need to know what p of x plus h is because we already know what p of x is. So we have, wherever we see x, we're going to put x plus h. So x plus h quantity squared, minus two. And that's equivalent to, when you square this, you square the first one, then twice the product, so two x h, and then plus the last term squared, which is h squared, minus two. So that's our p of x plus h. So now we're going to do the difference quotient here. So we have x squared plus two x h plus h squared, minus two. That's my f of x plus h. And then minus my p of x, or that's p of x plus h, my fault. p of x plus h minus my p of x, which is x squared minus two, and it's all over h. There's three different pieces here. Remember, this is the part that everybody forgets. You have to distribute the negative. I can't say that enough times. So let's compute. x squared plus two x h plus h squared minus two. And then when I distribute it becomes minus x squared, and when I distribute it becomes plus two all over h. So let's simplify. x squared and negative x squared cancel each other out. And I have a two x h and no other x h's, so I need that one. And I have an h squared but no other h's, so plus h squared. And then negative two and positive two cancel each other out. So I really just have two x h plus h squared over h. I still have to simplify. So when I simplify, I'm going to factor. So I have a common factor of h between these two. And that leads me with two x because the h is on the outside plus a factor of h on the inside because one's on the outside. And then that's all over h. Finally, to get my final simplification here, the h's cancel and our difference quotient is two x plus h. Not simpler than what we started out with, right? One thing I want to point out to you here is that notice that my f of x really kind of were p of x in this case, really canceled out. My x squared cancels with the x squared of my p of x plus h. My two cancels with the negative two of my p of x plus h because we're subtracting the x. And when you multiply this x plus h, you still have that x part being applied in there. So that's why you'll always, your f of x should always cancel out completely and you'll be left with your h term. Keep going. So j of x, find the difference quotient. Okay, j of x plus h. Now I've got to put it in more than once. x plus h is going to be squared minus six times x plus h. And that's going to be equal to, this is always x squared plus two x h plus h squared. Minus, and then I distribute, so six x and minus six h. You've got to watch your signs, okay? So that's what this is. And I'm going to put parentheses around that and just go ahead and work right here with that. So I have to subtract my original function, x squared minus six x. And all of that is over h. So I kept that in purple and changed the color and everything else so that you could see that this was equivalent to that. But then I threw in my difference quotient all at the same time. So let's start simplifying. x squared plus two x h plus h squared minus six x minus six h minus x squared when I distribute. And minus, or plus six x when I distribute all over h. So let's go simplifying. x squared and negative x squared cancel. And negative six x and positive six x cancel. And I'm left with two x h plus h squared minus six h all over h. Notice they all have h terms. And that's really nice because then we can factor the h out. We have two x plus h minus six all over h. I factored that h, I factored one of those, and I factored that h out. The h's are gone and we have two x plus h minus six as our difference quotient. Okay, so now that we're figuring out what that difference quotient is, how to figure that out. This is the function that we had before. So let's go back because I forget. So the difference quotient is two x plus h minus six. Difference quotient. All right, so when we do the difference quotient here, we have to figure out what x is and we have to figure out what h is. Well, x is going to be the first value. Let's just put my x over there. Now I have to figure out what h is. This is not h. h is equal to the 2.0 minus the x value, or 2.0 minus 1.9. So h is going to be equal to 0.1. So now I know what h is, and I know what x is, and I know what my difference quotient is. So let's plug and chug. Two times my x, which is 1.9, plus my h, which is 0.1, and then minus six. And when I multiply all that out and add it all together, I get negative 2.1. So the difference, the average rate of change basically between these x's would be negative 2.1. That's what that's really saying. Well, we can use the same thing, but try these two points. So again, we had 2x minus h plus six, no, plus h minus six. That was our difference quotient. And we want to say that this thing here is x. And remember that we have h is equal to the 5.01, the second value, minus the x value, which was five. So we're just going to have 0.01. So plugging and chugging, we have 2 times our x, which is five. And then we have plus our h, which is 0.01, and then minus our six. Ten plus 0.01 minus six is going to give us 4.01.