 Hi, I'm Zor. Welcome to a new Zor education. Today will be a very short lecture, and it's only about notation of derivative, as it was presented by different mathematicians, so it's kind of historical traveling through time. Well, derivatives were relatively simultaneously described and invented, came up with whatever we wanted to use the term by different mathematicians, primarily in the 17th and 18th century. And they were using actually the different notations to basically to say more or less the same thing. So that's what I'm going to talk today. This lecture is part of the advanced math course for teenagers and high school students. It's presented on unison.com. So I do suggest you to watch this lecture from this website, because it has lots of detailed notes for each lecture. And for registered students, there are exams and other helpful stuff. And the site is free. Alright, so notation. I have introduced derivative as a limit as increment of the argument is decreasing of the value of the function with incremented value of argument minus the value of the function at point x as the increment of the argument is tending to zero, infinite decimal. So that's the original definition. Basically, it allows you to measure the speed of the change of the function, because the difference between the values divided by the difference in the argument of the function basically represents the average speed of change from x to x plus delta x, right? But as soon as we are applying the limit as this increment of the argument is decreasing down to zero, that actually defines a speed of the change at point x, exactly. And considering x can be anything wherever function is defined, we can say that this is actually a function of argument x, a different function. So f at x is original function. And this is a function which is called a derivative of the function. Obviously, it's defined in all points of the domain of the function f at x, where this limit exists. And that's not necessarily the entire domain. So the main of the derivative is not greater than the main of the original function. It can be the same if the function has this limit at any point x or it can be less than that if there are certain points where it's not defined. So that's basically the original definition. Now, if you look at this from a different perspective, so delta x is actually the difference between x plus delta x minus x, right? Now, the difference between the function values is actually an increment of the function, right? I mean, if you will consider the graph of the function, if this is point x, this is point x plus delta x, this is f at x, and this is f at x plus delta x, then this is difference between values of the function, right? f at x plus delta x minus f at x, and this is delta x, right? So this we can say that this is delta f at x. Again, it's just a notation, basically. It's a symbol. It's an increment of the function at point x. So we are basically talking about a slightly different notation, which can be introduced here, which means exactly the same thing. It's just different notation. It's just shorter. And we understand that delta f at x means an increment of the function as x is incremented by delta x at point x. That's exactly the same thing. It's not a theorem. It's just basically a definition. That's how I define delta f at x, right? Now, this being a relatively complex symbol for a derivative, different people derived different notation. And the notation which I was using in the previous lecture was f prime of x, basically meaning that this prime symbol means basically the derivative of the function at point x. Well, we can have x within index zero, but it doesn't really make any difference because x zero can be any x, right? So f prime of x is a derivative of f of x, and it's defined as this limit for all x where this limit exists makes sense. So that's one of the first notation which describes the derivative. And it's relatively common notation. Now, based on this, oh, by the way, this notation belongs to, who does it belong to? Lagrange. Yeah, Lagrange. Italian mathematician Lagrange. I think that's how it's spelled. Lagrange. I think he invented this at the end of 17th century or beginning of the 18th century, something like this. By the way, in the notes, I have exact dates at theunisor.com. I have exact dates when these people which I will be quoting when they lived. All right, so now Lagrange had used this type of notation for derivative. Now, another mathematician, Leibniz, German, by the way, also somewhere in the same end of 17th beginning of 18th century, he was actually more inclined to this and to get rid of this cumbersome limit and basically having a good notation, he decided that he will do this. D instead of delta means we have come to D as a limit with delta x tending to 0. Again, this is just a notation. It doesn't mean so far anything more than that. So this is increment of the function. This is increment of the argument. Both are infinitesimals, right? Because as soon as delta x goes to 0, delta f at x also should go to 0. Otherwise, the limit will not exist, right? Otherwise, it will be infinity. So we have assumed that we're talking about the point x where this limit or this limit exists, right? So it makes sense to talk about these two as two infinitesimals. And that's what basically this means. This particular notation represents the fact that we are talking about a ratio of two infinitesimals. One infinitesimal is an increment of the argument and another infinitesimal is the increment of the function. And again, that was introduced by Leibniz to German mathematician. Sometimes this can be written as D by dx of f at x, which basically means again the same thing, just slightly different notation. Now, what's next? Well, next is Newton, Isaac Newton. Sir Isaac Newton was actually using a dot on the top. So instead of the prime symbol, it was a dot. We will not probably be using this, although in physics, in classical mechanics, for instance, dots do have their own usage, just because traditionally it's like a respect to Sir Isaac Newton who basically invented classical mechanics. So the dot, we will not probably be using it. And another notation is this. Now, this notation was used by Euler. So Lagrange, Italy, Leibniz, Germany, Newton, England, and Euler, he was actually a Swiss mathematician, but he lived almost his entire life in Russia. And he basically created the Russian School of Mathematics. So that's what he was using. Most likely, we will be using one of these two. And what's interesting is the following. You see, this is a ratio of two infinitesimals, right? So let's just consider them not to be as values, but as infinitesimal, which is a sequence, right? It's a process in as much as limit is basically a process and something is changing. But we can actually talk about infinitesimal variables. And in this case, for an x, for a chosen x, this is basically a value. This is the value of the derivative of the function f of x at point x, right? These are two infinitesimals. And we assume that there is a limit. But if these are infinitesimals, then we can say that from this follows. So we are actually considering this not to be as an entire symbol, but as a real ratio between two infinitesimals. And that's why we can put this particular equality, that this infinitesimal, which is increment of the function, is equal to the derivative at that point, f of x, multiplied by the increment of the argument. Now, geometrically, again, it's kind of obvious because our derivative is tangent of this. Now, this is delta x and this is delta f of x, right? And from this right triangle, this catechus, which is delta f of x, is equal to this catechus, multiplied by the tangent of this angle, right? Now, as soon as we are going to limit as delta x going to zero, then we can have this particular equation. Now, again, that's about notation, but there is also some kind of a terminology which I wanted to introduce. So far, we were talking about one particular new word, which is a derivative. We have defined it. Now, there is one more word, actually, a couple of related words, which I would like to introduce immediately related to derivative. One word is differentiation. Differentiation is a process of determining the derivative. It's called differentiation. And what's interesting, this is called differential, and this is called differential. So, this is differential of argument, this is differential of the function. So, differentiation and differential, these are two new kind of words, which you have to basically relate to the concept of derivative. And what's important, the differential are actually not a particular value. These are sequences, which is infinitesimal sequences of values, which are going to zero, this one as well as this one. That's basically it. That's all I wanted to talk about as syntax, notation, language, words, whatever else, related to derivatives. So, differentiation is the process of taking, determining, finding out the value of derivative, and then you have different differentials, differential of the function, differential of the argument. And notations, we will be using mostly these two notations. Well, that's it for this very short lecture, just kind of an introduction into the language of mathematics, of calculus, if you wish. And probably I will spend some more time in the next lectures to basically derive the derivatives from certain basic functions, which we were doing before with, just as an example how derivative is actually derived. Okay, so that's it for today. Thank you very much and good luck.