 Hi, I'm Zor. Welcome to a new Zor education. Well, we continue talking about calculus. We have basically completed, as far as theories concerned, differentiation and integration of simple functions. And I would like to continue this with functions of more than one argument. And the first thing to talk about is there is a thing which is called a partial derivative. So basically partial derivative is some kind of an operation similar to regular derivative. But the partial derivative is for functions of more than one argument. You can take partial derivative of the function of two arguments, three arguments, etc. We will mostly be concerned about two arguments because three and more, etc., they are absolutely analogous. So we will have all the examples or almost all examples related to functions of two arguments. So the theme of this lecture is partial derivative of the function of two or more arguments. This lecture is part of the advanced course of mathematics for teenagers and high school students. It's presented on Unizor.com. I recommend you to watch this lecture from this website rather than from any other source because every lecture has detailed notes which you can basically consider as a textbook if you wish. And it has certain problems and exams for those students who are interested in self-evaluation of themselves. And some other functionality which is basically described on the website. And the website itself is completely free and there are no advertising. So it's completely outside of the financial world. Alright, so partial derivative. Number one, as I was saying, we will consider only functions of two arguments, for example. Or these are two functions which we will be mostly using in this particular lecture. So these are examples of function of two arguments. Alright, now, obviously the partial derivative, it's called derivative and it's definitely based on the concept of a regular derivative. I assume that everything about regular derivatives you do know and if you don't, just go to a corresponding lecture of the course. So I assume that there are no problems with regular derivatives. Okay, now, how can function of two arguments can be represented graphically? Well, for function of one argument we need coordinate plane and it's basically some kind of curve, right? For function of two arguments we are talking about three-dimensional representation and there is some kind of a surface, something like this. So for each pair of x and y we go up and get the z. So that's how it goes. Now, do you remember what is the purpose of regular derivative? You remember we're talking about, for instance, steepness of the curve. It's measured by the regular derivative at point, right? So steepness can be something like this. At this point we have this tangential line and the tangent of this is basically a derivative. So if it's steeper, the derivative will be higher. If it goes up, it's positive. If it goes down, the tangent will be negative so the function will be decreasing. If there is a maximum, local maximum at this point, our tangential line will be horizontal and obviously the steepness will be zero. So steepness is one of the things. Another thing is the rate of change. But let's talk about steepness in this particular case. If I have a curve, let's say it's some kind of an earth surface with mountains and stuff like this, what is the steepness of Mount Everest, for instance? Well, that's not such an easy thing to answer because if you go one route, you will have maybe one kind of a steepness at every point and if you go through another route it will be another. How can we measure the steepness of a particular surface at a particular point? Well, in the case of the surfaces, we can't do it in one particular number but what we can do is we can use these two major dimensions, x and y, and have two different sections, if you wish. So we will cut with a surface which is parallel to the x, so it's something like this. And as a result of this surface, we will have some kind of a curve. And within this surface, this intersection between the plane and the surface of our function, which represents the graph of our function, so I have certain curve. And on this plane, at that particular point, I can determine my steepness. But this is a steepness related to x direction. Now I can do the similar thing and cut it with a plane which is parallel to the y and you will have some other curve. So I have one plane and one curve and another plane perpendicular, obviously, to the first plane and I have another curve. Now these two curves, I would say major, well, I can say probably major, curves which basically describe the behavior of the whole surface in this point. Is it a complete description of the surface? Of course not because I can actually slightly change the direction of the cut but I will have a different curve as a result. The intersection would be different and it might be greater or smaller than any one of those. However, what's important is that these are two major components and in some way they do describe the steepness of the surface. Not completely, but to a certain extent because if along this surface the curve is very steep and along that curve the curve is very, very steep then we can assume that the whole surface is relatively steep in this particular case. So these two, the derivative of this curve within this plane and derivative of this curve in that plane they do describe to a certain extent the behavior of the surface itself. Now if it's something like a local mountain of the surface let's consider you have a surface which has this local maximum so it's like a top of the hill. Then if we will cut it with this surface or with that surface you will still have a curve which will have a local maximum at that point. So imagine you are talking about the surface of the globe and the point is a north pole it actually is the top of the surface in the proper orientation, north, up, etc. And then any section of this globe surface and a plane which is parallel to the axis this way or that way or this way all of these intersections will actually produce meridians which will have a local maximum exactly at north pole. So this is my earth, this is my north pole and any plane which goes through the axis something like this it will produce some kind of a meridian which will have a maximum at this point. So the maximum of the surface results in the local maximum of each of these two major intersection curves. So it does make sense to learn something about the surface by analyzing these two major curves. Okay, so how can I actually get it? Well, it's very easy actually. What does it mean that I am in this particular plane for instance? So this is the plane parallel to the y where x is equal to a, let's say. Now if x is equal to a then within this plane my function is actually f of a, y. a is fixed, right? So all these points are described by this where a is where this plane intersects the x-axis. And obviously as y is changing I am moving along this curve so I can actually make I can take a derivative by y of this function and I get some kind of a function as a result of that it's a derivative of this curve at any point, at any particular y. And if I will take this plane which is parallel to x and it intersects something as y is equal to b it intersects the y-axis then my function is f of xb it's a function of only one argument and I can actually have a derivative of this function. So my point is that I can actually take derivative along these curves by substituting instead of x some kind of a constant and then differentiating by y or substituting instead of y some kind of a constant and differentiating by x because in this case the function actually becomes the function of one argument y in this case function becomes some kind of a function of one argument x. Now let me just give the in this particular case let me just calculate what it is so my first function z is equal to x squared plus y squared if x is a constant a then my function becomes well it's not z actually anymore function becomes a squared plus y squared and if I will differentiate it by y a is a constant so the derivative of a sum is sum of derivative a is a constant so it's zero so it's differentiating basically only y squared which is 2y. Now if I differentiate x squared plus b squared by x I'll have similar I will have 2x right if I differentiate my another example x squared by cube if I differentiate if x is equal to a and differentiate by y I will have what? a squared is a constant so it's factored out right so differentiating y to the power of 3 will be 3y squared so it's 3a squared y squared and finally if I have x squared b cubed differentiate by x I will have 2b cubed x to the first degree right so this is basically the my derivatives of this or that function which describe the behavior of this or that curve its derivative actually so for instance if I wanted to know when the function reached the maximum I can just equate it to zero and I could find that if my x is fixed to a then by resolving this is equal to zero I can find the local maximum for instance now when it's positive I can say that the curve actually is growing if it's negative I can say it's decreasing etc so all these normal properties which we used to have with regular functions of one argument are applicable in this particular case but what's important is we have to fix one of one of the argument to get the behavior as it depends on the other argument now the first step which I would like actually to make in this particular case is that we don't really have to substitute x is equal to a we assume that x is a constant now if x is a constant because it's true for any a right so which means it's actually a function of a as well but we are talking about behavior as it depends on why if my x is fixed to this particular value a but it can be fixed to any other value so what does make sense is forget about a or b and just say that this and this and this and this can basically contain x and y and x and y but this is important you see this index this means we are differentiating by y having x only as a fixed variable same thing here and same thing here so now let me give you a definition of the partial derivative partial derivative of function of let's say two arguments is by one of them is a derivative by this particular argument assuming that the other argument is a constant is fixed so if you have this function and you have to have a partial derivative by y you assume x is a constant and then differentiating by y gives you the constant give 0 and this gives you 2i now this function if you would like to differentiate it by x then you assume that y is constant so y cube actually retain is retained and only x is differentiating so this is like a factor it goes out so differentiating x square would give you 2x so this is the definition of the partial derivative so you need a function and you need one particular argument you are differentiating by because in case of a function of one argument there is only one argument so you know what you are differentiating by in case of more than one argument you have to specify this is a partial derivative of this function by y or differential or partial derivative by x and then you can actually do the calculation you assume that all other arguments are constant you just pay attention to one particular function argument and differentiate by this function using the regular type of our calculations whatever is necessary to take a derivative so that's where we are now now let me introduce the symbolics now this is one of the way how you can actually symbolize that you would like to differentiate by this particular argument traditionally however there are other symbols used for this purpose and let me just explain what is this symbolics now you remember that in case of regular derivative sometimes we are using d by dx of let's say x square y cube we don't need second x square so we are differentiating function x square by its argument so you remember this d by dx and in this case it's equal to 2x right this is the derivative similar symbolics is used for partial derivatives the only thing instead of regular letter d like this we are using a different incarnation of this letter d it's also d in some font I don't even know which one in some languages this is preferable symbol for the same letter d but in any case that's what we are using and now we are doing this d by dx means we are differentiating partially by argument x you see in case of function of one argument you don't really need this detail kind of a symbol for differentiating because you have only one argument and you don't have to specify so you can just have some kind of prime for first derivative right so x square prime would be 2x so that's easier and shorter than d by dx right but in case of a partial derivative you still have to specify the argument and this is a preferable way of specifying the argument so now you can say this the derivative by x is equal to 2x and y cube so 2x y cube and derivative by y of the same thing is equal to 3x square y square 3y square is the derivative of this and x square is a constant we assume right so we are always using a specific argument by which we are differentiating right so that's very important and this is a preferable symbolics for partial derivatives now now I have two very important notes let's consider this function x square y cube and let's do its partial derivative by dx and it's kind of all about symbolics as well now the result of this is by dx it's 2x y cube now this is also a function of two arguments one which was considered to be the argument we are differentiating and the second argument is the one which we considered to be fixed as a constant and that's why we can and that's why we can just write this particular expression but as a result we still have a function of two arguments which can be differentiating again differentiated again either by x or by y so what will be if I would like to do d by dx of d by dx of x square y cube what happens in this case so it's a partial derivative by x of partial derivative by x now y is constant so basically factor out x to the first degree so the result would be this but I can also differentiate twice by y let's say if I have this by y so x square is constant so y to the power of 3 it's 3y squared so it's 3x squared y squared so let's differentiate twice by y so I'm differentiating by y again which is what 2y times this so it's 6x square y now I can actually do it in a mixed way first by x and then by y nobody prohibits it right so let's differentiate by dx x square y cube so first we're differentiating by y of this expression and then we differentiate by x so what happens by y would be this and now by x so x square derivative is 2x so it's 6xy squared right or I can do vice versa I can first differentiate by x getting this and then differentiate by y which is 3y square so it's 6x y square now first about symbolics this is kind of a cumbersome symbol and it's really done like this d2 by dx square you see dx and dx it's pure symbolic right so that's why we are doing it this way now this is obviously d2 by dy square and what is this how can I shorten this particular notation well it's d2 x dy and this is obviously d2 dy dx okay so these are shorter notations which are actually the ones which we are using in mathematics and the only thing which I left here for this particular case is look at this is that a coincidence well no under relatively broad conditions these two mixed partial derivatives of the function of two arguments by one and then by two or by second and then by first are supposed to be the same but this is a separate matter we'll talk about this later so it's not a coincidence that they are the same these are obviously different okay so this is my first important note the second important note is very short it's basically something which I was talking to the very beginning in as much as we can talk about partial derivatives about of the functions of two arguments we can talk about partial derivative of function of many argument by one specific argument which means that specific argument we are actually considering a variable while other arguments we are considering to be constant just as an example if you have something like function f of x, y, z equals x square plus y square plus z square and you would like to do d by let's say of this function so you consider x and z are constant so in addition they are disappearing and all it has as a result is 2y now if your function is a little bit more complex which is involved which involves other type of arguments now I'm exemplifying this as the function of three arguments I can put 23 but it doesn't really matter let's say it's x square, y cube z to the power of 4 something like this and you are looking for d by dz of this function so x and y are constant z is a function so we will have 4, 4z cube right? x square y cube and z cube so it's the definition of the partial derivative is very easily expanded to the function of any number of arguments however we will probably mostly be concentrating on the functions of two arguments in our partial definitions thing that's it for today I do recommend you to go to this website and read the notes for this lecture and other than that, that's it, thank you very much good luck