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