 Hi, I'm Zor. Welcome to a new Zor education. Today we will continue talking about derivatives and primarily about the geometric meaning of the derivative to a function. And I'm talking about tangential line to a curve, which represents the graph of this function. Now, this lecture is part of the advanced course of mathematics for teenagers and high school students. I suggest you to watch this lecture from Unizord.com because every lecture has very detailed notes. Plus you can have test exercises and even take exams. The site is free and you can take as many exams as you want until you will reach the perfection. So, back to derivatives. In the previous lecture, we were talking about derivative as the speed of change of the function at certain point. So, if you have a function which is defined at some interval, now a or b can be infinite as well, minus infinity, plus infinity or concrete values. And then you have some kind of a point inside this interval. Then the limit of this, which is increment of the function from point x0 to next point, so to speak, x0 plus delta x, divided by increment of argument as this increment goes to 0. I have actually called a derivative of this function. Obviously, if this particular limit exists, which means it exists in any way, delta x, infinitesimal, infinitesimal goes to 0. So, if it exists, then it's called a derivative. And it basically measures the speed of the function change at point x0, exactly at that particular point. Because if you have two points, for instance x0 and x1, you can always measure the average speed of change by subtracting from f of x1 minus f of x0 divided by x1 minus x0. But now we have to make this interval between x0 and x1 as small as possible. This is delta x is actually the difference. So, this is well, kind of algebraic sense, if you wish, of the derivative as a speed of change of the function. And now I'm going to talk about a different approach to analysis of the function, which basically leads to the same expression. And again, my purpose is to put some kind of a base for this definition, base of real things which are standing behind this pretty formal definition, limit of something. I mean, it should have some meaning. So, that's the meaning which we're talking about. So, that was speed of change was the meaning of this derivative in algebraic sense. And now we're talking about geometric sense. And we're talking about graph of the function f of x. Okay, this is function f of x. Oops. And we're talking about one particular point x0. And what I'm interested right now is a tangential line to function to the curve which represents the graph of this function at this particular line. So, the tangential line looks something like this. Now, obviously, we have to talk about what is a tangential line. We know from the course of geometry what is tangential line to a circle. Well, it's a line which has only one common point, right? So, that's kind of an easy definition. In case of just any function, it's not really exactly like that because, for instance, this line also has only one point in common with the graph of the function. But it's certainly not a tangential line, right? Now, in cases like this, in cases like this, for instance, this is function, which, this is the graph of the function. Now, at this particular point, I can have many lines which have only one common point with the curve of the function. Because if this, in this case, you can say, okay, we have to really define this tangential line as, number one, having only one point in common, but number two, the curve should lie on one side of this tangential line because this doesn't really go by this definition because the curve goes on both lines. But in this case, the graph goes still on one side and we have multiple straight lines which have only one point in common. So, we have to really talk about definition of the tangential line first. And I can only suggest you something, that tangential line is not only line which has the one common point with this particular curve. But there are a couple other properties. Number one, this line actually has another intersection. So, we're not talking about one common point as being just an overall requirement. It's only local requirement, which means there is some kind of a neighborhood of point x0 where this line has only one common point. And you can add actually that the line should, that the curve in this neighborhood should lie on one side of the tangential line. That's also important. But how can I prevent these points? Well, these points can be prevented by, that exist at this particular point only one such line. So, if there is only one such line which in the neighborhood of point x0 has one point, which is basically the line, the point which has coordinates x0 and function of x0, right? This is coordinates of this point, common point. So, the tangential line should go through this point. In the neighborhood of x0, there is a small neighborhood. So, the research on epsilon, where whatever I'm saying is true from x0 minus epsilon to x0 plus epsilon if you wish. So, there is some neighborhood where this line has only one common point, which is this one, that the curve goes on one side of this line and it's only one and only one such line, which has this common point. So, this point, this situation is excluded. But then, I have a very interesting point and this is the point of the function y is equal to x to the power of 3. Look at this graph. Graph looks like this. Now, what is a tangential line in this particular case when x is equal to 0? Well, actually the only reasonable answer is that the line which is actually coinciding with an x-axis, right? So, it looks like our requirement of having the curve even locally on one side of the tangential line is not really necessarily true in some cases. So, we really should approach this differently. Well, the way how I suggest you to approach the tangential line is exactly the way which I am going to use to come up with a formula which will help us to express the equation for this line. Now, line is usually defined by two points, right? One point for the tangential line we always have. This is the point where it's supposed to touch our curve. Now, the other point, well, it can be this or it can be this or whatever else. It's kind of difficult to define. What's easier is if you have one line, what's easier is to define an angle this tangential line makes with some known line like for instance the x-axis, right? And that is actually the way we will proceed. How can I find out what is the angle? In this case, it's better to talk about tangent of the angle of this tangential line with the x-axis. So, and that actually will lead us to a definition as well. So, let's approach it this way. What if I will take another point which is somewhere in the neighborhood of x0, but I will make it a little bit further. Here is x1, okay? So, this is the point x0f of x0 and this is the point x1f of x1. This is A, this is B. And let's consider the chord between them. And now, what I am going to do is I'm going to move x1 towards x0, basically the same thing as I was measuring the speed if you remember in the previous lecture. What happens? Well, the next position would be here, right? Next position would be here. And as x1 moves closer and closer to x0, my line, my chord if you wish, would be closer and closer to the line which we will probably be right to call a tangential line. What's important is that no matter how I move x1 to x0, maybe I will move it from here. So, I will take this chord and this chord and this chord, etc. So, if no matter how I move x1 to x0, my result, my limit, I'm not afraid to say the limit right now. My limit would be the same line, then that line actually should be called a tangential line. Obviously, if this limit exists, because in this particular case, this limit does not exist at all. Because if we move from the left, our line would always be like this. If we move from the light, from the right, the line would be always like this. And they will never go to the same limit, obviously, which means that there is no limit. But in this case, when the line, when the curve is really smooth, then my limit exists and it can rightfully be called the tangential line. Now, back to how can we define it algebraically. We have one point. So, all we need is an angle between tangential line and let's say x axis, right? And this angle can very well be measured by the tangent because the tangent we actually do know. And here is how we can calculate it. So, let's consider the first version of my x1. So, what is the tangent of the angle between this line and the x axis, which is this angle? Well, obviously, this is an increment of the function from x0 to x1, which is f of x1 minus f of x0, right? This is the piece. Divided by, this is the calculus, and this is the calculus of this angle, right? So, divided by x1 minus x0. And we are talking about the limit of this as x1 goes to x0. Now, if this limit exists, that means that we will have the tangent of this tangential line, which is passing through point x0 and is tangential to curve. So, as in the case of the derivative as a speed of change of the function, I prefer to call instead of x1 and x0, I prefer to use x0 and delta x. Delta x is the difference between. So, instead of writing this, I will have limit with delta x infinitesimal. And the ratio is f of x0 plus delta x minus f of x0 divided by delta x. So, this limit is a tangent of a tangential line to the graph, actually, to the curve which represents the graph of the function f of x at point x0. And this is just another view to derivatives, how important derivatives are. Now, how important tangential lines? Well, we will have some problems which will illustrate importance of the tangential line. Like, you know, some kind of a preview, just think about this. If the function has some kind of a maximum, then basically in the maximum tangential line is supposed to be horizontal, right? In the simplest case. So, that's why it's very important to have this tangent of the tangential line to be equal to zero in this case, because the line actually has a zero angle with an x-axis. So, tangent is equal to zero. And this is an indication of some kind of a maximum or maybe minimum or something else. We'll talk about this separately. My point right now is to come up with exactly the same formula for the tangent of the tangential line as I did for the speed of the change of the function. Seems to be completely unrelated things, but they go to the same formula for something which we call a derivative of the function. So, that's why basically the basis of the whole calculus is to be able to understand how the function behaves. And its derivative is very, very important characteristic of this function. Now obviously we can take this derivative at any point, not only point x zero, but in any other point, which means that derivative actually might be a function of that any point, right? And this indicates that this is a derivative. It's just some kind of notation. So, using this notation, this is a derivative of the function f at point x zero. But again, since we can take a derivative at any point, we can talk about a function, a new function, which is a derivative of the function f. All right. So, I do suggest you to read the notes for this lecture on unizor.com. And it's probably very important if you take the whole course actually, because the course has many interrelated topics. Like for instance, right now I'm talking about my previous lecture, where I was talking about speed of the function change, right? So, I always reference to some other things. So, I do suggest to take the whole course. It's actually a very good foundation for just the general knowledge of mathematics. And also my point is that mathematics actually helps you to develop your creativity, your logic, your analytical thinking, which basically is helpful in any profession. So, if you think that mathematics is basically the only purpose is to pass some exams, you're terribly wrong. Mathematics helps you, even if you will be a doctor or a lawyer or a plumber or a architect or whatever else. In any of these professions, mathematics, even if it's not used exactly as it is specified in the math, like formulas, etc., it's still important because it develops your thinking, your creativity, your logic, etc. All right, that's it. Thank you very much and good luck.