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Derivative at a Point - Overview

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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Okay, So before we define the derivative, let's go back to one of the first concepts we discussed in the last topic. And that was the average rate of change of a function over an interval. So recall that the average rate of change of the function ffx over some interval. And now I'm going to write the interval in a little bit different way. I'm not just going to write a to B. I'm actually going to write a thio A plus h where h here. I'm just thinking about is being a positive number. Okay, so I'm just kind of parameter rising the interval by its length. Okay, So h is the length of the interval and of course, the interval ISS starting at. And if I just let h equal b minus a which is the length of the interval. You see that this really is just a to B, Okay, so don't think too much about that. But over this interval, the average rate of change of the function is just f of April's H minus half a day all over h okay. And notice again that I'm dividing by the length of the interval which is B minus a, but I'm saying B minus a may be able to put that down. So be mind to say I'm thinking about is h based on how we defined average rate of change before and so just a note. This is the slope of the secret line between two points on the graph of dysfunction. So between the points A and that's a day and then a plus age, an F of April's h okay. And of course we have this nice picture of the Secret Line. So we had the function of X and then the secret line. So this is a and over here is April's age than the Secret Line just connects these two points. Okay, But of course, we're also interested in the instantaneous rate of change, and in fact, that's going to be a little bit more useful for our applications. So we're really looking for the tangent line at X equals that So the average rate of change is over and honorable. So here's our interval. The instantaneous rate of change or what we'll call the derivative is going to be at a point, and so what we're going to do is we're really gonna let this h good zero and watch these secret lines level off to the tangent Mark, assuming that our function has a derivative at that point. Okay, so now that we've sort of reviewed and I keep saying this, we're derivative. Let me actually tell you specifically what the derivative iss. All right. So the derivative of a function at X equals a So the notice the difference between average rate of change and what we're defining here, the average rate of change is over an interval, and the derivative is at a specific point. It's an instantaneous rate of change. So the derivative at X equals a is given by, Well, it's basically this same difference question that we had for average rate of change. But again, we won't let that h go to zero. And the whole reason we went through this process of talking about limits is to have a way to precisely say H is getting closer and closer to zero. So we're taking the limit is h goes to zero, and now we introduce some notation. That's what this kind of colon equal sign means. So I'm going to define this notation to mean the derivative instead of saying f obey. Well, that's the function value. I'm going to say f prime of a. So this is the derivative right here. So when you see f prime of a all it is is it's the limit of this difference question. So you're just taking those seeking lines in their slope the average rate of change squeezing them in around some point X equals a and that's giving you the derivative. Now, one important note is that this definition Onley makes sense. If this limit exists so very important kind of caveat is that this is only defined if the limit exists. And for now, we'll just assume that this limit exists when we talk about its properties. So here we go, the derivative What is the driven? Okay, so I sort of just made up a vocab word derivative. You know, it doesn't mean anything if I just say it, I need to tell you what it really is. The derivative is the instantaneous rate of change as well as the slope of the tangent line. Okay, So, again, if you want to go back and think about our velocity example so the average rate of change was the ridge velocity of a position function over a time interval. But if you look on the speedometer in your car, that tells you the instantaneous rate of change. So actually, the derivative is that instantaneous speed that you're traveling or instantaneous velocity. If you are thinking about well, I'm not going forward or backward, but we'll talk about that later when we talk about applications. But this is what you need to know. For now, the derivative it has, this technical definition is a limit. That's how you actually compute it. You if you want to figure out what it is, so if it's two or five or negative three or whatever it is, that's the slope of that tangent line geometrically. But its interpretation is that it's the instantaneous rate of change of the function. It's the slope, if you will, the function, which of course, is changing. So let me give you a quick sketch. So again, if we have a function and we'll just kind of sketch it like this, you get sort of arbitrary, and then we have a point. X equals a somewhere right here. Then our tangent line, assuming it exists. Of course, assuming that limit exists, that limit exists. Then the tangent line exists. Okay, so here's ffx or function, and then we have a tangent line and we'll be a little bit more specific about what the tangent line is in a second. But the derivative so f prime of a is equal to this number. Yeah, the slope of the tangent line. Okay, so geometrically. It's very clear what the derivative is telling us. And actually, if we want the equation of the tangent line, notice that for the tangent line we have the slope. And if we want a point on the line, well, we know a point on the line. It's a common F obey member that this point is the point of Tange insee for the tangent line. It's the point where the the tangent line just barely touches the function. So a f obey is a point on the line. So the tangent line really is why, minus f of s I'm just using point slope form equals m times X minus A or if you prefer, I can add s obey over. So this is kind of the general form for the tangent line right there. That's how you find it. So in order to find the tangent line, you need the slope. By taking that limit. Finding the derivative at X equals a and then just using this point of tange Insee, a common method to find the equation of the tangent line. So the next thing we need to do is sort of go back to this question. Does the derivative always exist? And of course, the answer is going to be No, because it's defined is a limit. And as we've already seen in our discussion on limits, limits don't always exist. So the derivative F prime of a does not always exist, but just to introduce um, or notation or some more terminology if the derivative exists. So if f prime of a exists or, in other words, if the tangent line exists of the function, or we'll see later if the function has what's called a linear approximation at that point. So if F prime of exists, then F of X is what's called Differential at X equals a and now this is going to be more related to when we talk about derivative functions. But for now, we just want to introduce this terminology because it'll just kind of get thrown out. It's just in our mathematical language. The derivative exist a functions differential. The tangent line exists at that point, although there's just a lot of different ways to say the same thing. And probably that's why math is confusing. That's why anything is confusing. If there was only one way to say something, then it would probably be easier toe learn. But it wouldn't be as broad and kind of rich in content because there's all these different ways to think about the derivative or think about differential bility. But the key thing when you're learning this for the first time is to realize that all of these concepts are exactly the same thing. The instantaneous rate of change, the slope of the tangent line, the derivative differential ability. They're all just saying the same thing. So let's get into the really interesting question of talking about when a function is not differential. We're gonna spline spend plenty of time talking about functions that do have derivatives. That's going to be an essential part of this class, but we're defining this concept. And so it's gonna be worth while to think about cases where the derivative does not exist. Okay, so when is a function not differential will recall that the derivative is defined as a limit. So this question when is epics not differential is the same as this question. When does f prime of a not exist? And when I say not exist, I mean as a limit because this is a limit. When ordered to answer this question, we got to think back. So how can a limit not exist? Well, there's some really, really complicated cases that we're not going to consider. We're really just going to consider the simplest case. Okay, when As a limit, the left hand limit doesn't equal the right hand. So we're gonna introduce some work terminology, but it's really nothing new. So we're going to define what's called the left hand derivative. And if you're really tracking, you're going to know exactly what this is. It's just the left hand limit of F prime event. So instead of just taking the limit is H goes to zero. We're just going to take the limit is H goes to zero from the left of this difference question and then similarly, will define the right hand derivative. So this is really nothing more than looking at the slope from the left and from the right. What is the slope approaching? If they're approaching the same thing, then this limit will exist and the function will be differential. Okay, so we need these two things to be equal. That's the key point for ethics to be differential. Well, the left hand drive it. It has to equal the right hand derivative. And in the first example, we'll see if a function that's not differential, that's exactly what we'll do. We'll look at the left hand derivative, and I look at the right hand directive and show that they're not equal. Therefore, the function will not be differential. Okay, so let's look at everybody's favorite function now. Of course, I'm being sarcastic. I doubt this is everybody's favorite function, but the function we're going to look at is absolute value of X, and I'm just going to explicitly write down absolute value of X as a piece wise function to be very clear. What it iss so recall. The absolute value of X is just going to be X when X is greater than or equal to zero. But it's actually going to be minus X when X is less than zero. Okay, so here's the question. What is F prime of zero? What's the derivative of the function at zero? Or, in other words, what's the slope of the tangent line passing through zero? Well, let's just sketch what this function looks like, first of all, just to kind of illustrate what the problem is going to be. So this function looks like this. It looks like Y equals eggs going this way, but it looks like why equals negative X going this way. So if you look at all of these points, so like if you look at negative one, then the slope is just going to be the slope of this line. Negative one. Or if I look on the other side like it one, the slope or the derivative is just going to be one right, because these air just lines. They're just kind of put together in pieces. But what about zero? What is this slope of this function at zero. Well, let's look at the two limits. Let's look at the left hand derivative in the right hand derivative. So the limit is h approaches zero from the left of absolute value of zero plus H minus absolute value of zero all over. H. Well, this is just the limit is h goes to zero from the left of absolute value of H over H. Okay, but kind of saving, You know, all of the steps. This is something we saw with limits. This is a one sided limit and because H is approaching zero from the left absolute value of age, we want to use this definition. So this is really minus age over age or minus one. And that really makes a lot of sense. Because if you just think about the slope is as you purchase your from the left, it's negative one, right? It's just a slope. It's just a line passing through the origin of slope negative went and then the right handed derivative. It's going to be the same thing. Except this time when we look at absolute value of age were to the right of zero. So we're just going to use H. So this is really gonna be H over H, which is one. So the limit is just one. And again, that makes sense, because to the right, our function is just y equals X, which has slope one. So here we say the absolute value of X is not differential at Syria, but notice that actually f is continuous and zero, and this is something that will mention later. It actually being differential is stronger than being continuous. So, in other words, if you're differential than you will be continuous at that point in another way, to think about different ability is what we'll talk about next. The different ability is really talking about smoothness in some sense. So again, let's look at just kind of the geometric meaning of differential bility. So if a function is differential, then F is sometimes said to be smell at X equals day. And this really goes back to the idea of looking at how function is changing. If a function is differential, then somehow it's changing nicely there. It's just kind of accelerating or decelerating. In some sense, there's not any sort of jerking going on, and so let's just look at a few examples of when a function is not differential and see the types of situations where a function cannot be smooth in some sense. So let's look at some examples. So one example we've already seen. That's when a function has what's called a corner. And that's the case, like for absolute value of X. And so you see that a corner is not smooth, right? So if you kind of trace your hand along the corner, it doesn't roll Canada from one. You know, it doesn't change and roll around the corner. It's sharp as a sharp corner, so it's not smooth. Okay, so sharp means not smooth. Okay? And again, we saw this by showing that the left hand derivative doesn't equal the right hand derivative. So if we kind of approached the corner from the left, we have a slope going like this. Like if you look at the top of the house like if I go up one side, I have a slope this way and I go up the other way. I have a slope this way, and they needed a corner like that. Then it's not Smith. Okay, so another example is what's called a cusp in a cusp more or less is the same thing as the corner but a cusp instead of just kind of meeting at two different angles. Cusp really is, um, coming down in smoothing out and then coming back the other way. So really, the left and right hand derivatives air becoming undefined so the limits aren't existing. But somehow there they're infinitely. So 11 is approaching infinity and one's approaching minus infinity. Not just being left and right. So, you know, from the left here, you get a number from the right here. You get a number. They're not equal here. You're really getting that. The limit is an infinite limit, right? And so from the left, the slope is approaching minus infinity, and from the right, the slope was approaching. Infinity is the idea. So this is really thinking about infinite slopes, whereas here and kind of ah limit. Since here we're just thinking about those limits not being equal. And then the final example that will look at is what's called a vertical tangent. And this is similar to a cust. Except that actually there's some consistency in the left and right hand limits that they're actually both either approaching infinity or approaching minus infinity. And so, if our function looks something like this, so we go up and we see that our tangent line, the slope is actually becoming undefined. So the tangent line exists here, but the slope is undefined. So Tangent line has undefined slip. Or, in other words, the tangent line is vertical. I mean, hence the name vertical tangent. So actually the Tangela exists, but it just it's burke. It has an undefined slope. So if you look at the left hand derivative in the right hand derivative in this case, they would actually both be approaching infinity, whereas with a cust in one direction, you're approaching infinity in another direction. You're approaching minus infinity. So there's really not a well defined tangent line in that case, just like with a corner. So you see these air just three examples of cases where functions can fail to be differential, okay? Or they could fail to be smooth in some sense. So these air just important to keep in the back of your mind and in the examples will actually see specific cases where these come up