- Right, so let's put a little bit of notation to all of this. So, let's go back to the graph we had earlier of F of X, and let's just sort of zoom in to a little bit of it. Maybe between, you know, the input of two and the input of three. So, if we have X squared then two would give us. Let's say here would be four and three, the point three here. Let's say we just, you know, put here a nine. So, you know, we're making one unit change in this case X and we're getting a big difference in F of X. Of course, you know, as a side point, the thing we're dealing with here is A and changing A, and changing L. And that's sort of what we're mostly interested in, but we'll just do it in general. Call it X and F of X. Right, so, what I wanna know is. I wanna form a way of trying to answer the question: how fast is the output changing? Now, you know, give me some kind of answer to that question. So, that's what I'm gonna do. So, let's try and define this, and, so, yeah. At this point here, two in the X and four in the Y. What's that? Well, let's say that. Well, you know, that's F of two anywhere. Put F of 2 on there, all right. And three, nine, that's F of three. Okay. So, okay, there's the heights. The height is F of two, and the other one the height is F of three. And watch this. So, we know what the height is. So, the height now there and the actual height between them. This is height between them is gonna be F of three minus F of two. So, what is F of three minus of F of two tell us? Well, it's nine minus four, right? Which is five. It tells us how much we've shifted. So, that's the change. So, the change is five, right? So, we've gone up by five, but how far have we stepped? We stepped here from two to three. So, we've stepped by one unit. So, you know, so, this is two and that's three, and the difference here is one. So, if we divide that by three minus two, you know. Suppose we had only stepped 2.5. You know, if we're always dividing by how far we're moving horizontally then we get a rate, the pace at which we're going up. So, if I look at 2.5 here. So, if I put 2.5 into this F of 2.5. 2.5 here. Well, what's 2.5 squared? I'm gonna guess it's 60, six, 2.2 something, but we'll have a look. So, 2.5 times 2.5. 6.25, there we go. So, this is 6.25. And that's, obviously, four. So, four. So, 6.25 minus four is 2.25, so in this intermediate case, we've got 6.25 minus four then we divide by. Well, 2.5 minus two. And what does that give us. It gives us 2.25 minus. Divided by, sorry, not .5. And that tells us actually the rate we're moving at is 4.5. Right, so, it's important to divide. It's important to divide by the size of the step you're making because, you know, if we're trying to compare, suppose you're trying to compare rates, and I wanna know how fast we're moving up here and how fast we're moving down here. Well, if I'm stepping by different amounts, I'm just gonna go further up this line even though maybe my rate isn't changing. So, let me just show you what I mean by that. So, you know, so, if I had a straight line, right? And I go from here to here, well, obviously, I make a big jump. You know, I make a big jump up. And if I go from here to here, I make a very small jump up. And it's not, you know. Nothing miraculous has happened. This one is big and that one is small. It hasn't. It's not that we're getting steeper. It's just that I've moved horizontally less, right? I haven't actually. You know, I've climbed less terrain. Whereas, so, the rate isn't changing here. This, you know, so, this is like an artifact in the fact that, you know, and artifact, a consequence to the fact that I've made a different size step. See, if I divide this height by how much I've traveled with the width or the horizontal motion there, and I divide that through, well, it's gonna give me a rate, and what I'd find out here is that, you know, actually, the. So, we do it. So, we think of it in terms of just one step. You'd find that the actual. That this slope in terms of, you know, straight same step sizes, say, is always the same 'cause it's a straight line. So, the slope isn't changing. It's the same everywhere.