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Derivative Functions - Overview

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity, and the derivative of velocity with respect to time is acceleration. The concept of a derivative has been generalized to other contexts than just functions of a single real variable. For instance, the derivative of a function of several real variables is a function of several real variables, and the derivative of a function from Rn to Rm is a function from Rn to Rm.

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Okay, so we have defined the derivative of a function at a point f prime A b. So for a function ffx assuming the slum it exists at X equals A. This gives the instantaneous rate of change. Today, it gives the slope of the line tangent to the graph of F of X at X equals A Yeah, etcetera, etcetera. Okay, so what we're gonna do now is defined, actually a new function. So define a function f prime of X. Now here, X is a variable, but the function is just going to be this limit of this difference question. Half of explosive. Agh! Minus F of X all over H Now this is called Okay, the derivative function. Okay. Now, of course, dysfunction has a domain. It has a natural domain, and it's natural domain is going to be related to the domain of f Fact is going to be contained in the domain of death. So the domain is going to be the following set. So first of all, we're on Lee going to include X values that air in the domain of F. So it's the set of X values in the domain of F for which this limit F prime of X exists. So this function well, what are the outputs? Well, the outputs are the derivatives f prime of whatever number I plug in. But I'm only going to get an output if the function is differential at that point. And, of course, it doesn't make sense to try toe, see if a function is differential at a point. If that point is not in the domain of the function. So that's why we're really restricting to just looking at X values in the domain of the function and from this definition is clear that the domain of S Brian is going to be a subset of the domain of F, so we're not gonna have any new points. But in general, they're gonna be points in the domain of F for which f Prime of X does not exist. So, really, this is gonna be in general a strict containment. They're gonna be points in the domain of Beth. They're not in the domain of s prime. So let's make some of this terminology a little bit more clear. So f prime of X is defined as a function at X equals a exactly when cool F of X is differential Bill today. Okay, so we have another way to just say the same thing. The function is differential to derivative exists. F Prime of X is defined at X equals a again. All of these things mean exactly the same thing. But what will say is that ffx is differential. And so now notice. I'm not saying at any particular point differential. Yes, F Prime of X is defined at every point in the domain of death. So this is very similar to our conversation about a continuous function, so a function could be continuous at a point. But we also said a function was just a continuous function if it was continuous over his whole domain. Similarly, here a function is differential. If it's derivative, function is defined at every point in the domain of death. Or maybe if we restrict just some open interval, we say f is differential over that open interval. If the derivative is defined at every point over that open interval. So again there's gonna be a lot of terminology, and there's it seems to be that there's a lot of abuse of terminology around derivatives because you're already saying derivative at a point differential at a point derivative function to find it a point. But then you also have differential over an interval f prime of X to find over an interval. So it's just a lot kind of muddied up in the waters now. But for now, we're really going to be talking a lot about the derivative at a point. So f of X being differential at a point. The derivative existing in a point, the derivative function being defined it a point, etcetera. So just, you know, take some time to think about all these these terms and these definitions relax and understand that there really saying the same thing. They're just used in slightly different context, just to mean different things. Okay, so we want to talk a little bit about what the derivative is telling us. And what does it tell us specifically about the function ffx. And now we're going to spend a lot of time talking about this question. But we just want to introduce it right now. So here just a couple things f prime tells us where, eh? FedEx is increasing and decreasing. And now Why is that? Well, when the derivative function is positive? What that means is that all of those X values correspond two points on the function that have positive slope. In other words, the derivative is increasing our site. The function is increasing, the derivative is positive, and then also where the derivative is negative. That tells us that the function is decreasing. It also tells us where ffx has horizontal tangents. And what's the significance of a horizontal change tangent. But we'll see later the horizontal tangents. Now what I mean by that are points on the graph of F of X, for which the tangent line is horizontal. And those air actually points where the function is potentially changing from being increasing or decreasing. And this has a lot of physical significance. So, for instance, physically horizontal tangents can signify when a ball reaches its highest point because the bullet in that point is changing from having a positive derivative, so a positive velocity to a negative velocity. So it tells us about changing directions again. It's going back to understanding how functions air changing. Here's an economics example. The horizontal candidates will tell us when a profit function has a has an optimal profit. So for how many units of something should I sell to maximize the profit? Well, that's going to occur when the function has a horizontal tangent. So it's telling us a lot of information and horizontal tangents will take center stage when we discuss applications of the derivative and we're going to discuss a bunch. And well, as you'll see, these horizontal tangents are extremely important. So this is just something to keep in the back of your mind. We're going to hit on all of this later, but the derivative function is telling us a lot of information about ffx. So I want to go back and talk about the relationship between continuity. So where function is continuous and then different ability, so function being continuous function being differential. And so what we know is that being differential is stronger than being continuous. Okay, so in other words, differential ability tells us that a function iss smooth at a point. So it's burying smoothly, whereas continuity just says that there's a connection so you can see that a function can connect like at a corner and not be smooth, but it's necessary for a function to be smooth. That at least the function be connecting a to that point. Okay, so precisely what do I mean? What I mean is, if f prime of a exists, okay? And now notice. There's lots of different ways to think about this. I'm saying that the derivative function evaluated at a is defined or this limit exists. It doesn't really matter. How do you think about it? The tangent line exists, etcetera. Then what can I conclude? I can conclude that ffx is continuous. Hey, So this is why what we call a property being stronger than another property differential ability implies continuity. So let's actually see why this is the case, so we can actually prove the statement. Now, we're not going to do very many proofs in this class, but we do want to highlight the simple ones. The ones that don't really require a lot of depth toe, understand? And so the proof here is pretty simple. We're going to assume that this exists, um, in his H purchase. Zero of f of vehicles, age minus f of a all over h exists. So this is what we know. So what we want to show Is that the limit? His ex approaches a above affects is equal to F today. Okay, but what is the limit is X approaches A of earth affects. Well, what I can dio is a magic. So the first thing I'm going to do is I'm going to take ffx, subtract f of a and then add of the thing. Okay, so all I did is subtract FAA and then out of today, So these were still equal. And now I'm going to do one more thing. So I'm gonna add Ecevit back. But what I'm going to do first is I'm going to divide by X minus a and then multiply by X minus a and now notice because I'm taking the limit. Is x a purchase? A. I'm not multiplying and dividing by zero because I don't care about what happens when X is equal today. But now the lemon is X approaches. A of this difference question is exactly the same thing as the lemon is a cheaper to zero of this difference question. The only difference is I just need toe let x equal. What? Well, I need to let x equal a plus h and then we see is H goes to zero X is approaching a and then you just plug in a plus h A plus h you get exactly this, but this exists as a limit. This is what we're assuming. So we can actually just apply the limit loss so I can take the limit of this factor times the limit of this factor. So this is f prime of a times. Well, the limit is expert is a of X minus A is zero. So time zero and then the limit of the constant F Abe is just f obey. But this is a number times zero, which is zero plus f of a. So all in all we have today, so the limit is expected A of ffx. It's just that the big but that's exactly what we need to show to show the function is continuous. So if a function is differential below the point, it must be continuous. And while this seems like a really simple fact, it wouldn't be that useful. It is extremely useful when you start talking about the differentiations and some of the rules that we want to use to make finding derivative functions easier. So what about the converse now? Converse? I mean the opposite statement. And so we just said and concluded that if a function is differential, it has to be continuous, but our all continuous functions differential. Now what you think about that for a second? So of course, the answer is no. And we've actually already seen an example. Absolute value of X is continuous answer, but not differential. And this is kind of the go to example. If you ever asked, given example of a function that's continuous but not differential, absolute value of X, it's continuous zero because the limit from the left of zero limit from the right of zero. But those two piece wise functions are meeting at a corner. They're not meeting smoothly at that point. Where they meet is not differential. So just a couple of notes just to kind of generalized this idea. If F of X has a corner, then the derivative function will have a jump. Dis continuity, jump this continuity. This isn't too hard to see because meaning in a corner just means that the two slopes are not lining up. So if I think about the derivative function, telling me the slope everywhere, like for absolute value of X. At zero, the slope is negative one, but then, at zero, it jumps upto one. So there's a there's a jump dis continuity in the derivative function. Okay, And then also, if ffx has a vertical tangent or a cusp, then what can you say about the derivative function than F Prime of X? Has a vertical ascent up. And why is that? Because remember that a vertical tangent or a cusp had the slope of the function going either to plus or minus infinity in one direction. But that means that the derivative function is getting bigger and bigger and bigger and bigger or further away from zero. I should say whether it's positive or negative doesn't matter. So F Prime of X is gonna have vertical ascent it. So these were just some fax thio. Keep in the back of your mind and they'll come up as we look at the examples. So the very last thing we want to talk about with derivative functions is something that's kind of obvious, but I think it needs to be very explicitly set, and that's the concept of higher derivatives. Second derivatives, third derivatives, fourth derivatives. But the idea is really simple. Let's start with the second derivative. So the second derivative, this is just the silliest definition to me, The second derivative of a function f x denoted So the derivative was f prime of X. So the second derivative will be Yeah, double prime of ax. What do you think it iss? Well, it's the derivative of the first derivative Easy is that so? If you find the first derivative function, then you just take the derivative again. Now you may have to exclude some points where the first derivative is defined, where the second isn't. But that's fine. But we have the second derivative function and of course, similar for higher derivatives. And so notation Aly for higher derivatives. We have the third derivative, which is the derivative of thesis and riveted. But at some point, you know, if you take the 10th derivative, you don't want to put 10 primes. So what you sometimes do is right in parentheses, the derivative that you want. So this is like the third derivative, so that you take the derivative and then the derivative and then the derivative again, The fourth derivative. You take the derivative four times, five times, etcetera five. And then So sometimes you see something kind of weird. If you ever see F and then zero of X So what is the zeroth derivative of X? Well, that doesn't really seem to make sense. But what you mean is, you just mean the function itself. And so you can sort of think about this chain of derivatives. The zeroth derivative is just death. And then the first derivative is the derivative second derivative, 3rd, 4th, etcetera. And so, you know, it kind of brings up the question. What's the point of higher derivatives? Okay, so the first derivative tells us the slope of the function, the instantaneous rate of change, slope of the tangent line. And that's just telling us how the function is changing. But that's kind of a first order, uh, description of how the function is changing what we'll see later. So we'll talk a lot more about the second derivative later on in the course, the second derivative is going to tell us how the first derivative is changing now on applications. This is really important think back to velocity. So the velocity tells how a position function is changing. But what is the derivative of Ah, velocity function? What should be telling us how the velocity is changing? That's called Acceleration. So acceleration is the second derivative of position, and it tells us how the velocity is changing. So again we'll get into these ideas more later. But we do just want to explicitly say what the second derivative it's. So if I want to find a second derivative, I just have to take the derivative, find the first derivative and then take that function, take the derivative again to get the second derivative.

Georgia Southern University
Top Calculus 1 / AB Educators
Anna Marie V.

Campbell University

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