 Hi, I'm Zor. Welcome to Unisor Education. Today we will talk about derivatives, basically about definition of what actually a derivative of a function at certain point is. This lecture is part of the course of advanced mathematics for teenagers and high school students. It's presented on Unisor.com. I suggest you to watch this lecture as well as the whole course actually from the website, which allows you to not only get involved in certain other aspects of educational process. For instance, you can just take exams for certain subjects and the site is free, so you can take this exam as many times as you want until you will reach a perfection. So, I do recommend you the whole course. It's on Unisor.com and this lecture is about derivatives. Okay, there are many different approaches to define derivatives. Obviously, there is some very abstract, very mathematically rigid and rigorous approach. I'll just give the definition and basically that's it. I don't think it's interesting. What's always interesting, at least for me, is how people derive to this new concept. Well, people primarily Newton and Leibniz, they have derived to the definition of the derivative primarily from practical implementation of certain things, certain principles related to physics. So, what I'm going to do is, I will just explain in a couple of examples actually, the same situation those people were in when they were trying to analyze the behavior of certain things. So, let me start first from the more abstract thing. And the abstract thing is that let's consider we have certain function which is defined. It's a real function which means real argument and real value. It's defined on certain interval from A to B where A or B can be actually infinite, minus infinity, plus infinity, whatever. Or a concrete from one value to another. And we are interested in certain point, let's call it x0, which belongs to this interval. And we would like to analyze the behavior of this function in this point. Well, this point can be any. So, basically, if we will know how to analyze behavior in this particular point, we will analogously be able to analyze the behavior in all other points. Now, what is behavior? What I mean behavior? Well, there are different aspects of the behavior and obviously not every aspect we will address. But one of the things, for instance, is the function defined in this. Well, I have said that this is within the interval of definition. It's in the domain. So, okay, the function is defined at x0, which is okay. It's something. Whether the function is continuous, for instance, in this particular point, which means no matter which way you are approaching point x0 from its neighborhood, you will always have the same limit which is equal to the value of the function at point x0, which is also important. Also, what's interesting is how the function approaches this value. If I will say that the function is approaching value f of x0 as x approaching x0, the question is how is it approaching? How fast? I mean, for instance, if you have a function which is straight line, y is equal to 5. So, it's always equal to 5. And this is point x0. So, this is 5. Well, obviously, it's always equal to 5, right? So, the function immediately takes the value 5. Now, if the graph of the function is something like this, it also takes the value 5 as a limit, but it goes with a certain speed, if you wish, which is slower than, for instance, in this case. In this case, function is closer to 5 on a further neighborhood than in this particular case. So, in this case, function is approaching to 5 faster, so to speak. So, this type of behavior of the function we are talking about, additional over and beyond just continuity. And let me just make a few examples where I will try to examine this way how the function approaches this particular value. And these examples will lead us to definition of the derivative. Okay, example number one. Let's consider you have a function which is dependency of gravity from elevation above the level of the Earth. So, obviously, if you are on the level of the Earth, you have certain value for the force of gravity. But if you go further from the Earth, this gravity is decreasing because we are further from the center of the Earth. Now, for instance, we actually have this particular function which means we can measure the gravity on every altitude, on every level of elevation. So, my question is how the gravity is changing with elevation? Is it fast? Is it not fast? I mean, how can it be checked, basically? Well, if you are, basically, well, charged with this particular task, I will ask you, okay, if you are at the elevation, for instance, of 10 kilometers above the Earth, that's where the planes are usually flying. At that particular moment, yes, I understand that you can measure the gravitation, but is gravitation changing significantly if I am on that particular level? If I am on the level of 10 kilometers above the level of the Earth, how fast gravity is changing if I will go a little bit up or a little bit down? Well, how can you approach this? Well, very simply. First, you have to measure the gravitation at point H1, for instance, 10 kilometers wherever I am asking you, and then you have to measure the gravity at the point H1, which is either a little below or a little above the H level. So, now you can actually compare the difference in gravitation levels and relate it to the ratio of elevation levels. This basically is an average change of the gravitation per unit of elevation, right? That's the ratio. However, this is not exactly the level of change or the speed of change of the gravitation at point H0. It's rather average change of the gravitation from H0 to H1. How can I make it a little bit more close to the elevation at point H0? Well, very simply, I just have to take these H0 and H1 closer to each other. Now, the closer they are, the better approximation of this average is to the speed of the change of gravitation at point H0. So, how can I actually make it a more mathematically defined concept of what is the speed of change of gravitation at point H0? Well, very simply, I just have to take the limit of this as H1 tends to H0. So, if this limit exists for the function g of H, and that's a separate story, these type of limits might not necessarily exist. But if it exists, then I can actually say that it really is maybe by definition, because that's actually where we're actually doing something to come up with a definition. This is the speed of the change of the gravitation at point H0. Now, when I'm saying H1 is approaching H0, I do not specify how. It can be from below, it can be from the top, doesn't really matter. But the limit, if it exists, should not depend on how I approach it. As long as my H1 approaching H0 anyway is possible. If this limit exists, which means no matter how I approach it will be the same number, then this limit can be actually called the degree of change of the elevation at point H0. So, that's my first example. My point is to come up with similar formulas in some other cases, and then I will give you a general definition. Okay, let's consider another example. Well, everybody knows that police likes us to be caught on the speed limit. So, there is some speed limit. At certain point, speed limit is such and such. Now, here you have a policeman who doesn't have contemporary gadgets to measure exactly the speed at exact point. But, nevertheless, he has to actually make some kind of a mechanism, if you wish, or procedure to measure the speed of the car at certain point and to be able to charge if it's exceeding the limits. So, what can this policeman do? Well, here is what he can do. Here is the road. Here is the point where he is supposed to measure whether the speed of the car exceeds the limit. Well, what he can do is, he can actually measure the distance from certain point, let's call it origin, and he is measuring this point, let's say it's D0, and then he can put another point somewhere else, and this measure will be D1. Okay? Now, basically for every car, he can measure the time when the car was in each particular point. So, let's consider at this point, the car is at time T0, and at this point the time is T1. What can this policeman say about the speed of the car? He can say the following, that the distance from D1 to D0 was covered by the time T1 minus T0, and that's the average speed on this interval from A to B. So, that he can measure. Now, that's not what he is interested in, because maybe the car was very fast here at this particular point, and then slow here, so the average speed would be below limit, whatever the limit is, but at this point it was really high, so it looks like this guy really exceeded the limit, but still with this type of measurement, he is not caught. What can the policeman do to basically catch these type of people? Now, why he slowed down? Well, maybe he saw the policeman here, right? And then he slowed down, and that's it. So, the policeman is thinking this way, if I will reduce this distance in distance and in time, therefore, I will make this average speed on the AB more precise, and it will be closer to the speed at point A. So, the closer B to A, the closer this average speed on the segment AB will be to the speed at point A. So, this particular driver, even if he sees the policeman, will not be able to decrease the speed sufficiently to compensate for excess of speed here. So, obviously, what necessary to be done is, I will, instead of putting D0, D1 and D0, I will put it as D of T1 minus D of T0. So, it's the distance where this particular car is at point T1 and point T0. So, we assume that there is some kind of function. For each car, there is a function. Distance covered at any moment in time, right? So, we assume that this is a known function, and now, obviously, the same thing. If we will limit this thing as T1 approaches T0 or point B is as close to A as possible, approaching tenths to A, then this particular limit is actually a speed at point A. Now, it actually can be a definition of the speed, because I really don't know how to define the speed any other way but this. I mean, I'm talking about speed at the point A, not average speed. Average speed is whatever else without the limit, right? But we are not interested in average. We are interested in the speed at point. And the speed at point can be calculated only by going to limit when the distance between these two points A and B is going to zero. And this is, as you see, an analogous formula with the previous one, where I was talking about gravitation based on the elevation. And the third example, which will obviously lead us to the same formula is... Let's consider we have a lot of rain in this particular area, and the rivers are rising, so we actually have to know the speed at which the level of the water in the river is rising. Obviously, you've heard many times when people from the weather forecast, whatever they're talking about, flood, they're saying, okay, you know what, the water is actually rising by two inches an hour, something like this. So, two inches an hour, it's the speed the water is rising. Now, how did they calculate it? How do they know that the speed of the water rising is exactly two inches per hour? They didn't really measure this within an hour. They might actually... And then during this hour, the speed is probably different anyway. Maybe in the beginning of the hour it was three inches per hour, and at the end of the two-hour period it was one inch per hour. So, that's not really precise. How can we precisely calculate the speed of the water rising at any particular time? Well, very similarly to the previous problem. We probably have to have some kind of a function. This is the level as a function of time. So, at each moment of time, we know what is the level, which means that if we will take the moment t0, and then we know everything what happens afterwards, we have... And before, actually, we took the moment of time t1. We know the levels, l0, which is l of t0, and l1, which is level at t1. And we have their distance, their difference. That's the distance. Well, the height, our level has been risen during the time t1 minus t0. So, this is the average speed of river rising during this period. Now, all we have to do is to have the limit as t1 is approaching t0 of this function. And we have exactly the same formula. So, that's the speed of the level of water rising at moment t1. So, t1 is fixed, t0 is fixed, t1 is approaching t0, and that's the moment where we are measuring, actually. So, that's the speed of the rising water at moment t0. Formula is exactly similar to the previous ones. Which makes us, actually, to come up with the following definition of the derivative of the function. So, back to the functions. Assume we have a certain function, which is defined at certain interval a, b. And we are interested in speed of the change of this function at point x0, which is inside this interval. And the limit of x1 going to x0 of the difference between values of function divided by difference in arguments is a definition of the derivative. And more traditional way is the following. I will make this type of substitution. So, that would be limit delta x goes to 0, right? If x1 goes to x0, then delta x is infinitesimal value. And here we have function of x1 now is x0 plus delta x minus effort x0 divided by delta x. Increment of function divided by increment of the argument, right? x1 minus x0 is delta x. So, delta x is basically an increment which we are allowing from that point x0 where we would like to measure the speed of the change of the function. And this increment should be infinitesimal, infinitesimal value, infinitely small value. I don't like the word infinitesimal. So, anyway, this is the definition of the derivative of function effort x at point x0 which belongs to a domain where the function is defined. However, I have to add very important if statements now. If this limit exists, which means no matter how delta x, infinitesimal value is approaching 0 from the left, from the right, this increment can be negative which means we are going to the left from the point x0 or positive when we are going to the right. It doesn't really matter. No matter how we approach delta x to 0, we still have to have this defined, existing and independent of the way how delta x approaches 0. Only in this case we can say that the function f of x is having the derivative defined as such. So, the whole thing was about the definition of the derivative. And this is a few approaches from purely practical reasons. The reasons similar to those which were in front of Newton and Leibniz when they invented actually this thing, this type of analysis of the function. It's the practical reasons which force them to come up with this particular definition of the derivative of the function. And again, it's not necessary that this actual limit exists. Because sometimes it doesn't. Very simple examples can very easily actually come up with examples of this. And I will devote probably a special lecture to differentiability of the function which means is it possible to have a derivative or not from the function. And there are examples of functions where this thing does not really exist at least in some point. Which are, which do not have this limit no matter which point I choose. For instance the function like it's equal to 1 if x0 is rational and 0 if it's irrational. You see this function is completely irregular. It does not have this particular limit because you never know what kind of rational or irrational parts this will be. So as you see there are definitely functions which do not actually have this particular limit. Now however I should say that most of the functions which we will be considering in this course they do have derivatives maybe not in every point but in most of the points. And obviously these functions represent for us the most interest. So most likely we will be dealing with these functions. Now this approach to define this derivative as a speed of change is just one approach. There are some other approaches and that would be the next lecture for instance. Where we will probably come up with the same function, the same expression for a derivative of a function. But from a completely different direction from geometrical from the consideration of the graph of the function. That would be the next lecture. But again my point would be to come up with also very natural kind of a problem which leads us to the same definition. All I'm trying to say is that this is not a purely artificial construction which Mathematician came up with. It has practical implementation. Okay, so that's it for today. Thank you very much. And I do recommend you to read the notes for this lecture on Unizor.com. Thanks and good luck.