 Hi, I'm Zor. Welcome to Indizor Education. We are talking about derivatives, and I would like to spend some time physically deriving a concrete derivative from a concrete function. This is just one of the examples, which I believe very useful, and I will spend some time basically deriving derivatives for different elementary functions we were dealing before. Now, this lecture is part of the advanced mathematics course for teenagers and high school students. It's presented on unizor.com. I suggest you to watch this lecture from this website, because every lecture has very nice notes and exams, and the site is free, so you can gain a lot of new information from the site itself. All right, so what kind of examples this particular lecture will be devoted to? These are related to the power function. Primarily, I have two examples. One function is f at x is equal to x to the power of n, where n is some integer number, and another f at x is equal to square root of x. These two functions, I would like to basically address today in this lecture, and I'll just take derivative from this function exactly as it is written in the definition of the derivative. So what's the definition of derivative? It's limit as increment of the argument goes to zero of increment of the function divided by increment of the argument. Right? Now, fine, so let's just use this particular definition and find the derivative of the function x to the power of n. So I have to take x plus delta x to the power of n delta x minus x to the power of n divided by delta x, right? And find the limit as delta x goes to zero. All right, now in this course, I believe it's in the subject when I was talking about induction, I actually introduced the Newton's binomial formula, which is basically formula which gives the value of this a plus b to the power of n. Now, let me just very briefly put it a to the power of n plus n a to the power of n minus 1 b plus n n minus 1 divided by 2 n minus 2 b square, et cetera, et cetera, et cetera. And the last, so you see these power of a is diminishing and n minus 1 and minus 2. And the power of b is increasing from zero. This is zero, basically b to the power of zero, then b to the first, the second, et cetera. And these are binomial coefficients which participate in this expression. Now, based on this, based on this, where this is my x and this is my delta x, how my x plus delta x would look in this particular representation? Well, it's x to the nth of power of n plus n x n minus 1 times delta x, right? Plus n n minus 1 divided by 2 x to the power n minus 2 times delta square x plus, et cetera. So next one will contain delta cube, delta to the fourth, delta x to the fifth, et cetera, up to delta x to the nth with certain other co-efficient, certain coefficients. Now, x is fixed, right? x is not changing, x is a fixed parameter. Delta x is going to zero. So if I will subtract minus x to the power of n, what happens? Well, that's what happens. If I will divide it by delta x, that's what happens. This will be this, this will be to the first power, and then something will be to the delta square x, et cetera, et cetera, right? And if I am going with delta x to zero, all these members, and there is only finite number of them, it's basically n minus 1 number, or n minus 2 numbers, whatever, they are all actually going to zero, right? Delta x, delta x square, delta x cube, up to delta x to the power of n. If delta x goes to zero, all these will disappear in the limit, and the only thing which will be left is this one. So this is equal to n x n minus 1. And that's the answer. So the f prime of x is equal to n times, n times x to the power n minus 1. That's the derivative of x to the power of n. Well, well, that's it. Let's talk about square root. Again, we will use exactly the same definition. So what do I have to do? I have to find limit as delta x goes to zero, square root of x plus delta x minus square root of x, right? That's the increment of the function, square root of x, divided by delta x. Okay. Now, obviously, again, this is 2 infinitesimals. It's kind of difficult to find this limit without any kind of additional work. So additional work is, I will multiply numerator and denominator by square root of x plus delta x plus square root of x. Now, what happens? Well, this is a minus b times a plus b equals a square minus b square, right? Where a is this and b is this. So on the top, I have a square minus b square, which is x plus delta x minus x. That's what I have on the top. On the bottom, I have delta x times square root of this plus this. Now, what's next? x plus delta x minus x is delta x and here is delta x. So here, we have one and if delta x goes to zero, delta x goes to zero, this thing will be equal to 1 over 2 square root of x, right? This is square root of x. If delta x goes to zero and this is square root of x. So that's my derivative. And what's interesting is Well, first of all, what's the domain of this function? x greater or equal to zero, right? What's the main of this function? x greater than zero equal is not good because equal will give me zero in the denominator. So this is one of the examples that my domain of the of the derivative is slightly smaller by one point than the domain of the original function. Which, by the way, is not the case in case of the x to the power of n because x to the power of n is defined for all real values x as well as this one. So everything is fine. But here, we have this domain is slightly less by one point less. And let me just explain it graphically. So basically this is a derivative, right? So f star of x is equal to 1 over 2 square root of x. And let me talk about geometrical interpretation of this. Square root of x has such a graph. Now you remember that derivative is a tangent of tangential line, right? Tangential line. And tangent of this angle is actually the derivative, right? Now, what's the tangential line here, for instance? It's this one. How about it point zero? Well point zero is basically a vertical line. So it's 90 degrees and the tangent of 90 degrees actually infinity, right? So it doesn't exist. Tangent is not defined at angle of 90 degree. And that's why this thing is not defined at x is equal to zero. The same thing. It goes to infinity is infinitely increasing value. And that's why this tangent is infinitely increasing from relatively small value in this case because the angle between this and the x axis is relatively small. When the tangent goes to this particular direction, almost 90 degrees, the tangential line, the tangent goes to infinity. So that's the meaning of this and that's why it's not defined at x point zero. Here we do have tangent, obviously, to the curve. However, this geometrical interpretation as a tangent of the angle is not really valid in this particular point. Basically, that's it. These are two examples I wanted to talk today. In some other lecture, I will address some other functions like trigonometric function or exponential functions. And well, little by little, we will build the repertoire of the function. We know how to take derivatives off. So that's it for today. Thank you very much. I do suggest you to read the notes for this lecture as well as other lectures on Unisor.com. Thanks very much and good luck.