Lecture 2. Partial derivatives
We have considered the concept of a function of two variables, and today we have a rather complex and interesting transition to new concepts related to differentiation.
Interestingly, there is no concept of a derivative function of two variables. We will talk about partial derivatives. First, let's talk a little bit about continuity and limits.
We often pronounce the word neighborhood of a point in the plane. What do we mean by this? So, the coordinate m0(x0, y0) is a point in the plane, r is a number greater than 0. What is called the neighborhood of a point M0 of radius r? This is the set of all points in the plane that is at a distance from point M0 less than r, it is the interior of the circle centered at point M0 of radius r. Knowing the formulas, we can easily write such an inequality (see the video).
This inequality is satisfied by all points (x, y) lying inside the circle. It is worth remembering the definition of the limit of a function of one variable, because the definition of the limit for a function of two variables sounds almost exactly the same.
Thus, number A is called a limit of the function f(x, y) at the point (x0, y0) if, whatever the neighborhood of a point A of radius ε, is the interval of point A, there is a δ neighborhood of a point (x0, y0), is already in the plane XOY that whatever point (x, y) of the area determining function f if it is a punctured (проколотой) δ neighborhood of the point (x0, y0), that is different from point M0, punctured the word (проколотое слово?) means that is the value of the function belongs to ε neighbourhood of the point A.
Many concepts, many theorems retain their truth for a function of two variables. We will not dwell on this, but we will give the continuity the same definition, which in generally agrees with the definition of continuity of a function of one variable. Let's say, without specifying anything more strictly: let the function of two variables be defined in some neighborhood of the point M0, and at the point itself, too.
A function is called continuous at a point if the limit of the function at that point is equal to the value in it.
And now we will gradually move on to differential calculus.
Let function f be defined in some neighborhood of the point M0, since there are two variables, x and y, each of them can change by a certain amount, we will give an increment of Δx and Δy to the variables x and y. See if it can happen that one variable has changed and the other has not. So, when only point x of the variable changed, we got M1, when only variable y - we got point M2, and when both changed, we got the point M. What does it look like in the picture?
Look, point M0, when we changed the first variable by value Δx wevgot point M1 if we changed only the second variable Δy we got point M2, and when changing both variables we get point M. At each of these points, the function is defined and takes some value corresponding to z, which means that when moving from point M0 to any of these points, the values of the functions change.
There are several increments: when moving to point M1, we get an increment on variable x at point M0, and we add the following adjective: partial increment; if we changed only variable y, moved to point M2, we got a partial increment on variable y; if we changed both variables, the increment is complete.
Well, now the definition of the partial derivative. The partial derivative of function f of variable x at point (x0, y0) is called the limit of the ratio of the increment of private variable x of function f at point (x0, y0) to increment Δx, Δx tends to 0. If we reveal private increment to variable x, we obtain such formula (see the video).
For the partial derivative of variable x, we can find other notations in books on mathematics, it is worth getting acquainted with them. Note that we use a round letter d to denote the partial derivative, another one, not like one used for the function of a single variable. It is the spelling of this letter d in this way that tells us that we are finding partial derivatives. Also, it looks like a fraction, but it's not a fraction. This is a symbolic notation that denotes the partial derivative of variable x of functions f. Well, it is read in the first case: dz by dx at point (x0, y0); z stroke by x at point (x0, y0); df by dx at point (x0, y0).
Similarly, the partial derivatives are determined with respect to the variable y. Only in the numerator we consider the partial increment with respect to variable y, divide by Δy, while Δy tends to 0.
How to calculate partial derivatives? It is clear that this is the main task that we have to master. Let's go back to the differential calculus of functions of a single variable. So, we have point (x0, y0). So, we will fix y0, and leave x as a variable. We get the function φ(x) of one variable. Let's try to calculate its value at point x0.
By definition, this is the limit of the ratio of the increment of the function to the increment of the argument when the latter tends to 0. So, we reveal the increment by the formula, remember how the function φ is calculated - this is f (x, y0).and what we see: the numerator turned out to be a partial increment of function f at point x0. So this is the partial derivative of variable x of function f at point (x0, y0).
So, the partial derivative of variable x is actually the derivative of a function of a single variable x. Well, similarly, partial derivatives with respect to the variable y. Look (see the video), the partial derivatives of a function of two variables are exactly the derivatives of functions of one variable. In the first case, these are functions of variable x, in the second case, these are functions of variable y.
What is the geometric meaning? You can start from the geometric meaning of the derivative of a function of one variable. Look: at point M0 of the coordinate value (x0, y0), the point of the graph is on the surface. Draw the secant planes through point (x0, y0) parallel to x and y coordinate axes. So, the tangents to the curves in these planes intersect the axes of the coordinate Ox Oy at angles α and β. In this figure (see the video), the partial derivative with respect to variable x is tg α - this is the tangent between the tangent and the Ox axis, and f’ with respect to y is tg β, the angle between the tangent and the Oy axis.
Let's try to calculate (see the video). Let's say we need to calculate the partial derivative at point (1, 2). We can use different approaches, remember that in fact it is a derivative. If we find by the variable x, you will find yourself by the variable y, then we form a function of the variable x, y say it is y0, that is 2. See, φ(x) in this case is actually a function of one variable, y is a constant and is equal to 2, once at the point (1, 2). Then everything is simple: φ’ is found, value x=1 is substituted and we got 82.
In reality, of course, we will not solve problems in this way.
The second way: immediately we start to reason, when we find the derivative of variable x, we assume that y is a constant, without going into details, what exactly this number is, 2 or some other. y is a constant, a number, and when differentiating by x, the variables turn out to be only x. so, this is a function of the variable x.
Then, applying the differentiation rule, in the first term 7*y^2 is a constant, we take the derivative for the sign, in the second term the derivative is -2, and 3*y-1 is a constant and the derivative is 0. We got a function of variables x and y. Well, in order to calculate the value of the derivative at point 12, we substituted 1 instead of x, 2 instead of y. Obviously, the value will be the same as we got.
Higher-order partial derivatives. How many of them are there? How to find them? We have already seen that if we find a partial derivative without fixing a specific value of x and y, at each point of some area there are partial derivatives with respect to x, and this again turns out to be functions of variables x and y.
Similarly, f' with respect to y is again a function of two variables x and y, so they can again be differentiated with respect to each of the variables. Look, when calculating first-order partial derivatives, we have 2 of them, in the next step they will be twice as many, in the next step, which we do not implement here, they will be twice as many again, and so on. So, when we increase the order of the partial derivative, the number of derivatives is doubled.
In this picture (see the video), let's understand how the recording is performed. Look, if we use strokes, for example, in the first entry at the top (see the video), this is the partial derivative (z'x)’x, the brackets were removed - it turned out z"xx. If we use the round letters d, then they are written on the left: (see the video).
Removing the brackets, we get as if two letters d (see the video), we read: d2 x by dx twice. The following: z’xy is got by differentiating (z'x)'y, if we go to round letters d, then it turns out that differentiation must be read from right to left, so that the difference is in writing with strokes and with round letters d.
What are other terms? If the first and second differentiation is carried out by the same variable, then the partial derivative is called pure. So here are two pure partial derivatives of the second order (watch the video), and if the second differentiation was carried out by another variable, then such a partial derivative is called mixed. So, among the second-order partial derivatives, two are pure and two are mixed (watch the video).
Well, you have an exercise: list all pure, all mixed partial derivatives of the third order of a function of two variables.
Different symbols are also used to denote partial derivatives. Pay attention to them (watch the video). For example, the sixth-order partial derivative. The superscript indicates the order of the derivative, the subscript must contain exactly the same number of variables x and y, and the order and number of variables must match.
Differentiation was carried out three times by y, then by x, then by y, then by x. To reduce the entries, we can perform the following action: write y^3 and use, for example, the Arabic numeral 6, but then in parentheses, not to confuse with the power if we use the round letter d, remember that the order of differentiation is written from right to left: dy^3, then dx, dy, dx. Higher-order partial derivatives.
Let's continue calculations and consider an example: find all second-order derivatives of the function z=e^(x*y). What to begin with? We first find, of course, a partial derivative of the first order, differentiating by x, we consider y to be a constant. Obviously, we get the answer (watch the video). Differentiating by y, we consider x to be a constant. We get the answer (watch the video). Well, then it will not be difficult to find a partial derivative of the second order, there are four of them, as we said - two pure and two mixed.
What will we pay attention to? The clean ones, of course, turned out to be different, but the mixed ones, look, they coincided. Is it accidental or not? It turns out not. The equality of mixed derivatives theorem is valid. So, if a function of two variables has a second-order partial derivative, the function is continuous at the point (x0, y0), then the values of these mixed derivatives are equal.
It follows from this theorem that if the mixed derivative, no matter what order, is continuous at a point, then the value does not depend on the order in which you performed the differentiation on the specified variables.
Well, such an exercise: how did you understand this theorem? Name all equal third-order partial derivatives of a function of two variables, of course, if the condition of the theorem – continuity-is satisfied.