Formal definition of the derivative as a limit
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Formal definition of the derivative as a limit
We're first exposed to the idea of a slope of a line early on in our algebra careers, but I figure it never hurts to review it a bit. So let me draw some axes. That is my y-axis. Maybe I should call it my f of x-axis. y is equal to f of x. Let me draw my x-axis, just like that, that is my x-axis. And let me draw a line, let me draw a line like this. And what we want to do is remind ourselves, how do we find the slope of that line? And what we do is, we take two points on the line, so let's say we take this point, right here. Let's say that that is the point x is equal to a. And then what would this be? This would be the point f of a, where the function is going to be some line. We could write f of x is going to be equal to mx plus b. We don't know what m and b are, but this is all a little bit of review. So this is a. And then the y-value is what happens to the function when you evaluated it at a, so that's that point right there. And then we could take another point on this line. Let's say we take point b, right there. And then this coordinate up here is going to be the point b, f of b. Right? Because this is just the point when you evaluate the function at b. You put b in here, you're going to get that point right there. So let me just draw a little line right there. So that is f of b, right there. Actually, let me make it clear that this coordinate right is the point a, f of a. So how do we find the slope between these 2 points, or more generally, of this entire line? Because whole the slope is consistent the whole way through it. And we know that once we find the slope, that's actually going to be the value of this m. That's all a review of your algebra, but how do we do it? Well, a couple of ways to think about it. Slope is equal to rise over run. You might have seen that when you first learned algebra. Or another way of writing it, it's change in y over change in x. So let's figure out what the change in why over the change in x is for this particular case. So the change in y is equal to what? Well, let's just take, you can take this guy as being the first point, or that guy as being the first point. But since this guy has a larger x and a larger y, let's start with him. The change in y between that guy and that guy is this distance, right here. So let me draw a little triangle. That distance right there is a change in y. Or I could just transfer it to the y-axis. This is the change in y. That is your change in y, that distance. So what is that distance? It's f of b minus f of a. So it equals f of b minus f of a. That is your change in y. Now what is your change in x The slope is change in y over change in x. So what our change in x? What's this distance? Remember, we're taking this to be the first point, so we took its y minus the other point's y. So to be consistent, we're going to have to take this point x minus this point x. So this point's x-coordinate is b. So it's going to be b minus a. And just like that, if you knew the equation of this line, or if you had the coordinates of these 2 points, you would just plug them in right here and you would get your slope. That straightforward. And that comes straight out of your Algebra 1 class. And let me just, just to make sure it's concrete for you, if this was the point 2, 3, and let's say that this, up here, was the point 5, 7, then if we wanted to find the slope of this line, we would do 7 minus 3, that would be our change in y, this would be 7 and this would be 3, and then we do that over 5 minus 2. Because this would be a 5, and this would be a 2, and so this would be your change in x. 5 minus 2. So 7 minus 3 is 4, and 5 minus 2 is 3. so your slope would be 4/3. Now let's see if we can generalize this. And this is what the new concept that we're going to be learning as we delve into calculus. Let's see if we can generalize this somehow to a curve. So let's say I have a curve. We have to have a curve before we can generalize it to a curve. Let me scroll down a little. Well, actually, I want to leave this up here, show you the similarity. Let's say I have, I'll keep it pretty general right now. Let's say I have a curve. I'll make it a familiar-looking curve. Let's say it's the curve y is equal to x squared, which looks something like that. And I want to find the slope. Let's say I want to find the slope at some point. And actually, before even talking about it, let's even think about what it means to find the slope of a curve. Here, the slope was the same the whole time, right? But on a curve your slope is changing. And just to get an intuition for that means, is, what's the slope over here? Your slope over here is the slope of the tangent line. The line just barely touches it. That's the slope over there. It's a negative slope. Then over here, your slope is still negative, but it's a little bit less negative. It goes like that. I don't know if I did that, drew that. Let me do it in a different color. Let me do it in purple. So over here, your slope is slightly less negative. It's a slightly less downward-sloping line. And then when you go over here, at the 0 point, right here, your slope is pretty much flat, because the horizontal line, y equals 0, is tangent to this curve. And then as you go to more positive x's, then your slope starts increasing. I'm trying to draw a tangent line. And here it's increasing even more, it's increased even more. So your slope is changing the entire time, and this is kind of the big change that happens when you go from a line to a curve. A line, your slope is the same the entire time. You could take any two points of a line, take the change in y over the change in x, and you get the slope for the entire line. But as you can see already, it's going to be a little bit more nuanced when we do it for a curve. Because it depends what point we're talking about. We can't just say, what is the slope for this curve? The slope is different at every point along the curve. It changes. If we go up here, it's going to be even steeper. It's going to look something like that. So let's try a bit of an experiment. And I know how this experiment turns out, so it won't be too much of a risk. Let me draw better than that. So that is my y-axis, and that's my x-axis. Let's call this, we can call this y, or we can call this the f of x axis. Either way. And let me draw my curve again. And I'll just draw it in the positive coordinate, like that. That's my curve. And what if I want to find the slope right there? What can I do? Well, based on our definition of a slope, we need 2 points to find a slope, right? Here, I don't know how to find the slope with 1 point. So let's just call this point right here, that's going to be x. We're going to be general. This is going to be our point x. But to find our slope, according to our traditional algebra 1 definition of a slope, we need 2 points. So let's get another point in here. Let's just take a slightly larger version of this x. So let's say, we want to take, actually, let's do it even further out, just because it's going to get messy otherwise. So let's say we have this point right here. And the difference, it's just h bigger than x. Or actually, instead of saying h bigger, let's just, well let me just say h bigger. So this is x plus h. That's what that point is right there. So what going to be their corresponding y-coordinates on the curve? Well, this is the curve of y is equal to f of x. So this point right here is going to be f of our particular x right here. And maybe to show you that I'm taking a particular x, maybe I'll do a little 0 here. This is x naught, this is x naught plus h. This is f of x naught. And then what is this going to be up here, this point up here, that point up here? Its y-coordinate is going to be f of f of this x-coordinate, which I shifted over a little bit. It's right there. f of this x-coordinate, which is f of x naught plus h. That's its y-coordinate. So what is a slope going to be between these two points that are relatively close to each other? Remember, this isn't going to be the slope just at this point. This is the slope of the line between these two points. And if I were to actually draw it out, it would actually be a secant line between, to the curve. So it would intersect the curve twice, once at this point, once at this point. You can't see it. If I blew it up a little bit, it would look something like this. This is our coordinate x naught f of x naught, and up here is our coordinate for this point, which would be, the x-coordinate would be x naught plus h, and the y-coordinate would be f of x naught plus h. Just whatever this function is, we're evaluating it at this x-coordinate That's all it is. So these are the 2 points. So maybe a good start is to just say, hey, what is the slope of this secant line? And just like we did in the previous example, you find the change in y, and you divide that by your change in x. Let me draw it here. Your change in y would be that right here, change in y, and then your change in x would be that right there. So what is the slope going to be of the secant line? The slope is going to be equal to, let's start with this point up here, just because it seems to be larger. So we want a change in y. so this value right here, this y-value, is f of x naught plus h. I just evaluated this guy up here. Looks like a fancy term, but all it means is, look. The slightly larger x evaluate its y-coordinate. Where the curve is at that value of x. So that is going to be, so the change in y is going to be a f of x naught plus h. That's just the y-coordinate up here. Minus this y-coordinate over here. So minus f of x naught. So that equals our change in y. And you want to divide that by your change in x. So what is this? This is the larger x-value. We started with this coordinate, so we start with its x-coordinate. So it's x naught plus h, x naught plus h. Minus this x-coordinate. Well, we just picked a general number. It's x naught. So that is over your change in x. Just like that. So this is the slope of the secant line. We still haven't answered what the slope is right at that point, but maybe this will help us get there. If we simplify this, so let me write it down like this. The slope of the secant, let me write that properly. The slope of the secant line is equal to the value of the function at this point, f of x naught plus h, minus the value of the function here, mine f of x naught. So that just tells us the change in y. It's the exact same definition of slope we've always used. Over the change in x. And we can simplify this. We have x naught plus h minus x naught. So x naught minus x naught cancel out, so you have that over h. So this is equal to our change in y over change in x. Fair enough. But I started off saying, I want to find the slope of the line at that point, at this point, right here. This is the zoomed-out version of it. So what can I do? Well, I defined second point here as just the first point plus some h. And we have something in our toolkit called a limit. This h is just a general number. It could be 10, it could be 2, it could be 0.02, it could be 1 times 10 to the negative 100. It could be an arbitrarily small number. So what happens, what would happen, at least theoretically, if I were take the limit as h approaches 0? So, you know, first, maybe h is this fairly large number over here, and then if I take h a little bit smaller, then I'd be finding the slope of this secant line. If I took h to be even a little bit smaller, I'd be finding the slope of that secant line. If h is a little bit smaller, I'd be finding the slope of that line. So as h approaches 0, I'll be getting closer and closer to finding the slope of the line right at my point in question. Obviously, if h is a large number, my secant line is going to be way off from the slope at exactly that point right there. But if h is 0.0000001, if it's an infinitesimally small number, then I'm going to get pretty close. So what happens if I take the limit as h approaches zero of this? So the limit as h approaches 0 of my secant slope. Of, let me switch to green. f of x naught plus h minus f of x naught, that was my change in y, over h, which is my change in x. And now just to clarify something, and sometimes you'll see it in different calculus books, sometimes instead of an h, they'll write a delta x here. Where this second point would have been defined as x naught plus delta x, and then, this would have simplified to just delta x over there, and we'd be taking the limit as delta x approaches 0. The exact same thing. h, delta x, doesn't matter. We're taking h as the difference between one x point and then the higher x point, and then we're just going to take the limit as that approaches zero. We could have called that delta x just as easily. But I'm going to call this thing, which equals the slope of the tangent line, and it does equal the slope of the tangent line, I'm going to call this the derivative of f. Let me write that down. And I'm going to say that this is equal to f prime of x. And this is going to be another function. Because remember, the slope changes at every x-value. No matter what x-value you pick, the slope is going to be different. Doesn't have to be, but the way I drew this curve, it is different. It can be different. So now, you give me an x-value in here, I'll apply this formula over here, and then I can tell you the slope at that point. And it all seems very confusing and maybe abstract at this point. In the next video, I'll actually do an example of calculating a slope, and it'll make it everything a little bit more concrete.