 Hi, I'm Zor. Welcome to Unizor Education. Today we will talk about certain relatively simple theorems, mini theorems I can say, about derivatives. And this is part of the advanced course of mathematics for teenagers and high school students. You can find this course on Unizor.com if you watch this from some YouTube website. I do suggest you to go to Unizor.com because the same lectures actually are there. They're just reference to YouTube. But Unizor.com contains lots of very detailed notes for each lecture. And also there is certain educational functionality which you can basically benefit from. You can enroll into specific topic or entire course. You can take exams and everything is free. So basically I do suggest you to watch this from Unizor.com. Okay, so today we will talk about the few theorems related to derivatives. So my first one is, well, very short, if the function is differentiable, then it's continuous. Now, differentiable graphically means that if you have certain function, then there is a tangential line at every point wherever this function is defined. Now, the fact that there is a tangential line actually is sufficient for the fact that the line should be smooth. Because if line is not smooth, for instance, if this is something like this, there is no tangential line at this point. And if the line even has some kind of a breakage here, it's also obvious that there is no one particular tangential line in this point. So basically having a derivative is a stronger property. It dictates more smooth behavior of the function than just continuity. So from existence of the derivative follows the continuity. And here is how we can explain it and prove, basically. So first of all, we have a function, f at x. And then we are thinking that this is defined at certain interval, which might be actually infinite on either side or on both sides. Now, let's check one concrete point in this interval. So I'm saying that at this point, if the function is differentiable, then it's continuous at this particular point. Now, differentiability means that exists this limit and it's equal to derivative at point x0, right? That's what existence of the derivative means. Now, what is the continuity? Continuity is that the limit of f at x is equal to f of x0 as x goes to x0. That's what continuity means. That the function, the value of the function goes to the value of the function at the limit point if the argument goes to that limit point. So what I'm saying is this is stronger than this, which means from this follows this. Now, how can we basically define it, prove it? Well, first of all, it's kind of obvious because if x goes to x0, then this is infinitesimal. Now, this is some concrete constant value, right? So if the limit of something divided by infinitesimal is some kind of a constant value, that means that this thing is supposed to be infinitesimal, right? And if this is infinitesimal, that's what it means basically, right? Which means f of x is getting closer and closer to f of x, f of x0. So that's just, you know, very brief kind of almost obvious proof of this. If you want certain, you know, algebraically rigorous proof, if you wish, you can say the following way. Well, if this, if the limit of this is this, it means that the difference between them is infinitesimal, right? So I know that f of x minus f of x0 divided by x minus x0 minus f of x0 is infinitesimal as x goes to x0, right? Now, I will use the common denominator. So f of x minus f of x0 minus f of x0 times x minus x0 equals epsilon x minus x0. Now, now, this is even simpler. So this is infinitesimal as x goes to x0. Now, this is a constant, which means constant multiplied by infinitesimal is also infinitesimal. So if we go to a limit as x goes to 0, this thing goes to 0, and therefore, this thing must go to 0. Or f of x goes to f of x0, which is the same thing, right? That's it. Very simple proof. So at any point where there is a derivative of the function, it's continuous. Is reverse theorem correct? No. From continuity, differentiability does not really follow. An example is very simple. Let's take, for instance, function y is equal to absolute value of x. You see this point? There is no tangential line at this point, because if you approach 0 from the right, tangential line will be this one. If you approach from the left, tangential will be this, which means there is no limit at this point. If you jump from left to right or left to right, you will get completely different results. So obviously, there is no derivative at point 0. But the function is continuous, obviously, right? Okay. So that's my first theorem. From differentiability follows continuity. Next. Let's consider that you have a differentiable function at each point of certain interval where it is defined. Let's assume that the function is monotonically increasing. If the function is monotonically increasing something like this, then my statement is that its derivative, and I assume the derivative exists. I said that the function is differentiable and monotonic increasing. Then its derivative must be positive. Now, what is derivative? Graphically, as you remember, this is tangent of this angle. So if the function is increasing, what I'm saying is that we always have some kind of a tangential line with this angle being from 0 to 90 degree. If you have decreasing, see, the angle is greater than 90 degree. But that would be the next theorem. But so far, my statement is that the derivative will be positive. And therefore, the angle will be from 0 to 90. Okay. How can I prove it? Well, very simple actually. Let's take again the definition of the derivative. And this is the limit. Think about this. If x is approaching x0 from the right, then f of x is greater than f of x0. And x is greater than x of x0. So this thing would be positive. If x is on the left, if x is less than x0, then this is negative. And this is negative, right? If the function is monotonically increasing, if x is less than x0, then f of x would be less than f of x0. And the difference will be negative. So it's negative divided by negative. And again, you have the positive sign of this particular ratio. So this ratio is always positive, which means its limit would be, well, positive or maybe 0, but definitely not negative. So that's why this is non-negative. Let's put it this way. So my statement is from the monotonicity, I think that's the proper word, of the function f of x. When it's monotonically increasing, the derivative of this function is positive or 0, non-negative, at any point wherever this derivative exists. So I assume it's an entire interval. So that's my first theorem. Now, obviously, the next one is if the function is monotonically decreasing. And it's as obvious that if the function is decreasing, then if x is greater than x0, that would be positive. But function is decreasing. So this one would be negative, right? So f of x would be less than f of x0. And the ratio would be negative. Vice versa. If x is on the left from the x0, so this is negative now. Now, if x is less, but the function is decreasing, it means that f is greater than f of x0. So this is positive. Again, the ratio of positive divided by negative is again negative. So in any case, we will have a negative value. And the limit would be in this particular case, obviously negative or equal to 0. So now we have for monotonically decreasing function, derivative would be negative or 0, non-positive. Now, question is, is the reverse true? For instance, I know that my derivative of the function is positive. Is it sufficient condition for the function to be increasing? Well, yes. Okay, let's consider this. So I know that my f of x is greater than 0. Question is, if this is given, can I prove that? Well, let's just consider any two points, right? x and x0. Now, let's consider the limit again, the limit, the definition of derivative. So let's consider this is positive as x goes to x0. What does it mean, actually? Well, it means that for all those x which are on the right from the x0, this is supposed to be, this is greater than this. So this is positive, which means that if the limit is greater than 0, then there will be certain neighborhood of x0 whenever f of x will be greater than f of x0. So in the immediate neighborhood of x0, my function must be increasing. Right? So for any point, for any point x0 in the immediate point, in immediate neighborhood, so this is my function, this is my x0. So if this is positive, it means that if I approach sufficiently close to my x0, then this for any x within this particular neighborhood, this is supposed to be positive. Because on the right, if x on the right, this is positive, now this must be positive. If on the left, this is negative, so this must be negative. So for any point x0, I have a certain neighborhood where the function is monotonically increasing. Now, if that is true for any point, which means that the function must be increasing always, right? Because if there is some moment where it's not, let's say, where it's not monotonically increasing, I will actually make all these considerations around this point and I will see that this is contradicting my initial assumption that the derivative is positive. So basically, we can just prove it from the opposite. There is no point where it can decrease, because for any point in the immediate neighborhood of that point, the function must increase. And obviously, you understand that the symmetrical, so to speak, theorem, so whenever my derivative is negative at every point of AB, my function must be decreasing. Because again, for every point where I assume maybe there is a point where it's not decreasing, then I can just find that this is not true, basically, exactly the same way. So there is always some kind of immediate neighborhood where the function must decrease. Because if this is less than 0, in this case, if this is greater than 0, then this must be less than 0. If this is less than 0, this must be greater. So always in the immediate neighborhood of x0, I will have monotonically decreasing function, which means that should be always for the whole interval AB. So these are very simple theorems, which are kind of relating graphically behavior of the function, which is monotonicity, and its derivative. So derivative actually kind of points what happens with the function. For the positive derivative, the function must increase. For the negative derivative, the function should decrease. So this is the property of the differentiable functions. Very simple, because as you see, the proof was actually completely trivial. I do suggest you to try to do it yourself using maybe epsilon-delta language. I think it's very easy, because whatever I was just talking about, these immediate neighborhood, etc., this is assumed the language epsilon-delta. So for every epsilon, there is some kind of a delta whenever you are within delta neighborhood of x0, then this is supposed to be within at least epsilon neighborhood from f at x0. All right, that's it for today. Review again these particular theorems on the website, on Unisor.com. That's it. Thank you very much, and good luck.