 Hi, I'm Zor. Welcome to Unisor Education. Today we will talk about differential, not differentiating, which is basically finding the derivative. But what is differential? Differential of the function. Well, this lecture is part of the course of advanced mathematics, which is presented on Unisor.com for teenagers and for high school students. It's basically a very comprehensive and advanced course, which contains many problems, because I consider actually solving problems as the main point, actually, of studying mathematics. All right, so, and by the way, the site is free and it has certain functionality, like taking exams for instance. Okay, okay, differential. Well, let me start first from the definition of the derivative, which we are familiar with. So we take a point x0, make some kind of an increment of the argument and take the difference between value of the function at incremented value of the argument and original value. And divide by increment and the limit as this increment goes to zero. So that's the definition of the function and we assume that this limit exists. So the function is smooth, differentiable, and whatever else. Now, there is a very important notation, which is attributed to Leibniz. It looks like some kind of a division. And at the end of this lecture, I will tell you why this is justified, basically. Okay, so let's examine basically what this is all about. Well, first of all, this limit makes the whole formula a little bit more, I would say, involved, too bulky, if you wish. And mathematicians prefer to have as short notation as possible. So in this particular case, delta x is basically implied as some kind of a variable, which is converging to zero. Now, if we do not use the limit here, then it would be just a particular fixed value of delta x. And that would be a particular value of difference of the function divided by difference of the argument. And this is not a derivative, right? It's some kind of a concrete value, which obviously depends on the point x zero, our base point, which we started from. And it also depends on delta x. So the first step towards simplification of this formula, so to speak, is to imply this limit. So it's not like we are saying, okay, we don't have limit, but we kind of assume this is a limit. But we would like to have another notation, which implies that some kind of an increment of the argument is not actually a concrete number, but a variable which is converges to zero. And the invention was, instead of using delta x, use dx. Now, in some way, it's the same thing. But at the same time, you have to understand that if you are using this particular notation for increment, you actually assume that it's not just any increment, it's a process of increment converging to zero. You can consider it as a sequence which is going to zero. You can consider it as a function of time. And as the time goes by, this is becoming smaller and smaller. So this is an infinitesimal variable. Now, this is a concrete increment. And without the limit, it's really just a particularly fixed number. This is an infinitesimal variable. That's very, very important. And if we are using this, we actually can skip the notation limit here, because it's assumed in this notation that this is what it is. So that's number one. Now, number two, let's just write it down a little bit differently. f of x, zero plus, well, I will use dx right now if you don't mind. Minus f of x divided by dx. Now, we are assuming that dx is an infinitesimal variable. Right? So basically, we are talking about this limit. So what is this? Is this a derivative at point x, zero? No, because this is an infinitesimal variable. And this is a difference between two infinitesimal variables. So it's some kind of a variable. This is a number. So we cannot put equal sign among them, right? So what I have to do, I have to add an infinitesimal variable. So we assume that this limit exists, right? Which means that this thing is actually equal to its limit plus infinitesimal variable, because that's actually the meaning of the limit, right? If this is a limit of this, so this is equal to limit plus infinitesimal. So the difference between them is supposed to be infinitesimal variable, right? Okay. Now, let me rewrite it slightly differently. f of x, zero plus dx minus f of x, zero, it's supposed to be zero here, equals to f of x, zero times dx plus epsilon times dx. Now, what is interesting here is that this is a difference between two values of the function at point x, zero and an incremented point dx, x, zero plus dx. Now, and this difference is, oops, supposed to be derivative. Now, this difference is actually, bulk of it is proportional to the increment of the argument, right? And the coefficient of proportionality is the derivative at point x, zero. Now, then there is a little bit extra. And what is this a little bit extra? Now, we know that dx is an infinitesimal variable. Derivative is a constant. So this is of an order of dx. How about this? Well, don't forget that dx is infinitesimal, but epsilon is also infinitesimal, which means that the whole member is of the order little o of dx. I hope you remember this notation. It means it's an infinitesimal of a higher order of decreasing. It goes to zero faster than just dx. Why? Because dx goes to zero and epsilon goes to zero. So they are independent, two independent variables. And that's why this is, this can be considered as a major component of this difference. And this is considered to be an infinitesimal of a higher order than this one. This is of an order dx. This is of an order infinitesimal epsilon times dx, which is a higher order of decreasing of infinitesimal variable. So this is the major component. This is a minor component. Okay, we've done that. Now let me put some kind of a graphic of the function and illustrate this graphically. So let's assume that this is your function. Now this is x zero. This is x zero plus dx. And again, we're assuming that there is certain process when this point is getting closer and closer to this point because dx is infinitesimal variable. Now what is derivative at point x zero? Well, that's actually the tangent of the tangential line, right? Now this is f of x zero. This is f of x zero plus dx, right? Okay, this is dx. Let's put points A, B, C, D, E, okay? Now, so the difference between this point and this point, A, B is actually dx, right? This is a dx, our increment. A, C is function of x zero. And B, E is function of x zero plus dx. Now, this is f. Now what is EF? EF is the difference between an incremented value of the function and its original value at this point. So this is actually, you know what, let me just change my pan of x zero plus dx minus function value of x zero, right? That's what EF is. Now let's look at this. So this is EF. Now what is this? It's dx times the derivative at this point. dx times tangent of this angle between tangential line and horizontal line of this angle, if you wish. So this is basically df, right? So what's interesting is that on the left of this is EF. On the right, there is a df plus something. And this is something Ed, which is epsilon times dx. Now what's interesting is as this point B is getting closer and closer to A, my df segment and EF segment are actually getting closer and closer to each other. And the difference between them, Ed, is an infinitesimal of a higher order than each of these guys. So that's what it means. Now this is something which I would say delta f of x, point x zero. df is, by definition, is a differential of function f at point x zero. So this is a definition. So this piece, which is almost the same as this one, when our points are getting closer and closer. And this is a major component of increment of the function. So this major component is called differential. So this is a differential of function at point x zero. So what is the differential? It's not a function because it contains, obviously, x zero, x zero can be anywhere. So it's kind of a function of x zero, but it's also a function of dx. And dx is a variable which is a completely different thing. So it's basically a function of two things, initial point where the whole thing is basically considered at x zero and an infinitesimal increment of this. So that's what differential of the function is. It's a product of differential of the argument which is an infinitesimal variable with coefficient of proportionality equals to the derivative of this particular point. So that's how we are approaching this concept of differential. Differential of the function is a major component of its increment. So EF is an increment. DF is a major component of the increment. And that's what differential is. Now, why is it better to deal with differential in many cases than with the increment of the function? Well, increment of the function is kind of a difficult thing because you have to really compute the value of the function in two things. So it depends explicitly on how big your increment actually is. The differential depends only on the value of x zero because dx is any variable which goes to zero actually, any infinitesimal variable. So what's most important actually is in the differential is obviously this coefficient of proportionality which is a derivative. And this component is just any infinitesimal variable which goes to zero. It can be, for instance, a sequence one second, one third, one fourth, one fifth, etc., one nth. It can be. It doesn't really make any difference from the differential standpoint because any variable which is converging to zero can play the role of dx. The coefficient of proportionality is obviously the most important things. So this thing is called a differential at point x zero. Well, now you can probably skip the zero because it doesn't really matter. Zero can be anywhere. So we can just skip this thing. And now it's quite justifiable to write, this is also zero, to write something like this that f derivative of x is equal to differential of the function divided by differential of the x. So we divide this by this. So this actually justifies this Leibniz notation. It can be considered as a division if you understand what differential of the function actually is because differential of the function is always proportional to the increment of the argument. And the coefficient of proportionality is exactly this type of a thing. So whatever you choose, you can define derivative first and differential from there or you can start from differential as a linear component, linear component of the increment of the function and then go to the derivative. It doesn't really matter. But anyway, these are conceptual ideas behind a differential of the function. Now, let me do a little bit. Well, it's a recollection of something which we have already covered before. Let me just remind you that we have covered something which is called a Taylor's series. Now, let me just remind you that function, let's say x is incremented value from x zero. So it can actually be represented as a value of x zero plus first derivative at x zero times x minus x zero divided by one plus second derivative at point x zero x minus x zero square divided by one point two plus etc. And common member would be f nth derivative at point x zero x minus x zero to the power of n divided by n factorial etc. It's an infinite series. So this is a Taylor's series. We have covered this at one particular lecture before. And I would like you to look at this. Well, obviously x minus x zero can be changed to delta x in which case you would have x plus delta x is equal to x f at x zero plus f at x zero delta x divided by one plus etc. So what's important here that let me make another transformation. This minus this is equal to, all right? So this is a Taylor's series basically or a little bit transformed Taylor's series. It basically tells you exactly the same thing as I was just talking about that increment of the function has certain major component where the increment of the argument is proportional to this increment about increment of the function is majorly proportional to increment of the argument with the coefficient of proportionality being a derivative plus something which is delta x square delta x cube etc which are little o of delta x, right? Because they're all starting from delta x square. Obviously they're all infinitesimals of higher degree of decreasing, right? So all of these are decreasing much faster to zero than this one. So this is actually a major component and this is just another justification for considering this to be a major component of this and that's why again it's called differential of the function at point x zero. Okay, so that's just another kind of a considerations which basically gives us exactly the same thing. So let's just talk again about the difference between increment and the differential. Increment is a concrete number differential like this one where delta x is a concrete number, right? By the way it's supposed to be zero. Now when we're talking about differential, differential is an infinitesimal variable always. So it's a process of converging to zero. Question is how is it converging to zero? Okay, the differential of the function is converging to zero proportionally to infinitesimal variable of increment of the argument. So it's differential, it's not an increment, it's basically a process. It's a limit if you wish is implied and the coefficient proportionality is first derivative. So what you can actually derive from this is that increment of the function divided by differential of the function at this point if limit of the argument goes to zero. This is what? This is basically this, right? Now obviously you can just reduce this and you will get one. In limit this is negligible relative to this. So whenever you divide by delta x as it goes to zero is the same as dx, right? dx is implied actually that delta x is going to zero. And this thing is an infinitesimal of the higher order than dx. So it gives the infinitesimal as a result. So we have one plus infinitesimal and since it's a limit it's just one. So this is very important that the, on the graph again, let me just do a little graph here. So if this is delta f but if you have a tangential line on this side you have df because it goes proportionally to dx. So the ratio between this and this is getting closer and closer to one as we approach as this point is converging to this point. Because the difference between them which is basically encompassed here is a higher order infinitesimal which we can just disregard. So this is basically about what is differential. For whatever reason I don't know why I myself had certain problems in the beginning to understand the conceptual meaning of that differential. So what actually differential of the function is. Well again you should consider it in a limit kind of a reasoning approach. It's a variable which is converging to zero and this converging is implied in the word differential. So you don't have to use the word limit etc. So basically that's it for today. I do suggest you to read everything whatever I said about differentials on the website on Unisor.com. You just go to calculus and derivatives and then there is a differential topic. It might just you know put a little bit more light into whatever you've heard right now. Other than that that's it for today. Thank you very much and good luck.