What is the rate of change of a function. Derivative of a function

An alternative physical meaning of the concept of a derivative of a function.

Nikolay Brylev

An article for those who think for themselves. For those who cannot understand how one can cognize with the help of the uncognizable and for this reason cannot agree with the introduction of uncognizable concepts into the tools of cognition: “infinity”, “desperation to zero”, “infinitesimal”, “neighborhood of a point”, etc. .P.

The purpose of this article is not to criticize the idea of ​​​​introducing very useful fundamental knowledge into mathematics and physics. concepts of derivative of a function(differential), but to deeply understand it in a physical sense based on the general global dependencies of natural science. The goal is to endow the concept derivative of a function(differential) cause-and-effect structure and deep meaning interaction physics. This meaning is impossible to guess today, because the generally accepted concept is adjusted to the conditionally formal, non-rigorous, mathematical approach of differential calculus.

1.1 Classical concept of derivative of a function.

To begin with, let us turn to the universally used, generally accepted, existing for almost three centuries, which has become classic, mathematical concept (definition) of the derivative of a function (differential).

This concept is explained in all numerous textbooks in the same way and approximately this way.

Let the value u depends on the argument x as u = f(x). If f(x ) was fixed at two points in the argument values: x 2 , x 1, , then we get the quantities u 1 = f (x 1 ), and u 2 = f (x 2 ). Difference of two argument values x 2 , x 1 let's call it an increment of argument and denote it as Δ x = x 2 - x 1 (hence x 2 = x 1 + Δ x) . If the argument changed to Δ x = x 2 - x 1, , then the function has changed (increased) as the difference between two function values u 1 = f (x 1 ), u 2 = f (x 2 ) by the value of the function incrementΔf. It is usually written like this:

Δf= u 1 - u 2 = f (x 2 ) - f (x 1 ) . Or considering that x 2 = x 1 + Δ x , we can write that the change in function is equal toΔf= f (x 1 + Δx)- f (x 1 ). And this change occurred, naturally, in the range of possible values ​​of the function x 2 and x 1, .

It is believed that if the values x 2 and x 1, infinitely close in magnitude to each other, then Δ x = x 2 - x 1, - infinitesimal.

Definition of derivative: Derivative of a function f (x) at point x 0 is called the limit of the function increment ratio Δ f at this point to the increment of the argument Δх, when the latter tends to zero (infinitesimal). It is written like this.

Lim Δx →0 (Δf(x 0)/ Δx)=lim Δx→0 ((f (x + Δx)-f (x 0))/ Δx)=f ` (x0)

Finding the derivative is called differentiation . Introduced definition of a differentiable function : Function f , having a derivative at each point of a certain interval, is called differentiable on this interval.

1.2 Generally accepted physical meaning of the derivative of a function

And now about the generally accepted physical meaning of the derivative .

About her so-called physical, or rather pseudophysical and geometric meanings can also be read in any textbook on mathematics (calculus, differential calculus). I will briefly summarize their content by topic. about her physical essence:

Physical meaning of the derivative x`(t ) from a continuous function x (t) at point t 0 – is the instantaneous rate of change in the value of the function, provided that the change in argument Δ t tends to zero.

And to explain this to the students physical meaning teachers may, for example, do this.

Imagine that you are flying on a plane and you have a watch on your hand. When you fly, you have a speed equal to the speed of an airplane, right?” the teacher addresses the audience.

Yes!, the students answer.

What is the speed of you and the plane at each moment of time on your watch?

The speed is equal to the speed of an airplane! - the good and excellent students answer in unison.

Not quite like that,” explains the teacher. – Speed, as a physical concept, is the path an airplane travels in a unit of time (for example, in an hour (km/h)), and when you looked at your watch, only a moment passed. Thus, instantaneous speed (the distance traveled in an instant) is a derivative of the function that describes the aircraft’s path in time. Instantaneous speed is the physical meaning of the derivative.

1.3 Problems of the rigor of the methodology for the formation of the mathematical concept of a derivative function.

A audiencestudents, taught by the education system resignedly,immediately and completelylearn dubious truths, as a rule, does not ask the teacher more questions about concept and physical meaning of derivative. But an inquisitive, deeply and independently thinking person cannot grasp this as a strict scientific truth. He will certainly ask a number of questions to which he clearly will not receive a reasoned answer from a teacher of any rank. The questions are as follows.

1. Are such concepts (expressions) of “exact” science - mathematics as: moment - an infinitely small value, aspiration to zero, aspiration to infinity, smallness, infinity, aspiration? How can cognize some essence in the magnitude of change, while operating with unknowable concepts, having no magnitude? More The great Aristotle (384–322 BC) in the 4th chapter of the treatise “PHYSICS”, from time immemorial, said: “If the infinite, since it is infinite, is unknowable, then the infinite in quantity or magnitude is unknowable, how great it is, and the infinite in appearance is unknowable, what it is in quality. Since the principles are infinite both in quantity and in appearance, then to know those formed from them [ things] is impossible: after all, we only believe that we have known a complex thing when we find out what and how many [principles] it consists of..." Aristotle, "Physics", 4 ch..

2. How can derivative have physical meaning identical to some instantaneous speed, if instantaneous speed is not a physical concept, but a very conditional, “inexact” concept of mathematics, because it is the limit of a function, and the limit is a conditional mathematical concept?

3. Why is the mathematical concept of a point, which has only one property - a coordinate (not having other properties: size, area, interval) replaced in the mathematical definition of a derivative by the concept of a neighborhood of a point, which actually has an interval, only of indefinite magnitude. For in the concept of derivative the concepts and quantities Δ are actually identified and equated x = x 2 - x 1, and x 0.

4. Correctly at all physical meaning explain with mathematical concepts that have no physical meaning?

5. Why is the cause-and-effect relationship (function), depending on the reason (argument, property, parameter) must itself have finite concrete defined in magnitude limit changes (consequences) with an indefinitely small change in the magnitude of the cause?

6. There are functions in mathematics that do not have derivatives (non-differentiable functions in non-smooth analysis). This means that in these functions, when changing its argument (its parameter, property), the function (mathematical object) does not change. But there are no objects in nature that would not change when their own properties change. Why can mathematics take such liberties as using a mathematical model that does not take into account the fundamental cause-and-effect relationships of the universe?

I will answer. In the proposed, classical, existing in mathematics concept - instantaneous speed, derivative, physical and generally scientific, there is no correct meaning and cannot be due to the unscientific incorrectness and unknowability of the concepts used for this! It is not in the concept of “infinity”, and in the concept of “instant”, and in the concept of “striving towards zero or infinity”.

But true, cleared of the lax concepts of modern physics and mathematics (aspiration to zero, infinitesimal value, infinity, etc.)

THE PHYSICAL MEANING OF THE CONCEPT OF A DERIVATIVE FUNCTION EXISTS!

This is what we will talk about now.

1.4 True physical meaning and causal structure of the derivative.

In order to understand the physical essence, “shake off the thick layer of centuries-old noodles from your ears”, hung by Gottfried Leibniz (1646-1716) and his followers, you will have, as usual, to turn to the methodology of cognition and strict basic principles. True, it should be noted that, thanks to the prevailing relativism, at present, these principles are no longer adhered to in science.

Let me make a brief digression.

According to deeply and sincerely believers Isaac Newton (1643-1727) and Gottfried Leibniz, a change in objects, a change in their properties, did not happen without the participation of the Almighty. The study of the Almighty source of variability by any natural scientist was at that time fraught with persecution by the powerful church and was not carried out for the purpose of self-preservation. But already in the 19th century, naturalists figured out that CAUSAL ESSENCE OF CHANGING PROPERTIES OF ANY OBJECT - INTERACTION. “Interaction is a causal relation posited in its full development”, noted Hegel (1770-1831) “In the closest way, interaction is represented by the mutual causation of presupposed substances that condition each other; each is relative to the other at the same time an active and passive substance.” . F. Engels (1820-1895) specified: “Interaction is the first thing that appears to us when we consider moving (changing) matter as a whole, from the point of view of modern natural science... Thus, natural science confirms that... that interaction is the true causa finalis (ultimate cause) of things. We cannot go further than the knowledge of this interaction precisely because behind it there is nothing more to know.” Nevertheless, having formally dealt with the root cause of variability, none of the bright minds of the 19th century began to rebuild the edifice of natural science.As a result, the building remained so - with fundamental rottenness. As a result, the causal structure (interaction) is still missing in the vast majority of basic concepts of natural science (energy, force, mass, charge, temperature, speed, momentum, inertia, etc.), including mathematical concept of derivative of a function- as a mathematical model describing " magnitude of instantaneous change" of an object from an "infinitesimal" change in its causal parameter. A theory of interactions that unites even the well-known four fundamental interactions (electromagnetic, gravitational, strong, weak) has not yet been created. Nowadays, much more has already been “screwed up” and “jambs” are coming out everywhere. Practice, the criterion of truth, completely destroys all the theoretical models built on such a building that claim to be universal and global. Therefore, it will still be necessary to rebuild the building of natural science, because there is nowhere else to “swim”, science has long been developing at random - stupidly, costly and ineffective. The physics of the future, the physics of the 21st century and subsequent centuries, must become the physics of interactions. And it is simply necessary to introduce a new fundamental concept into physics - “event-interaction”. At the same time, a basic basis is provided for the basic concepts and relationships of modern physics and mathematics, and only in this case is the fundamental formula found"causa finalis" (ultimate first cause) formula to justify all the basic formulas that work in practice. The meaning of world constants and much more is clarified. And I will show you this, dear reader, now.

So, formulation of the problem.

Let us outline the model. Let an abstract object of cognition cognizable in magnitude and nature (let us denote it -u) is a relative whole, having a certain nature (dimension) and magnitude. An object and its properties are a cause-and-effect system. An object depends in magnitude on the magnitude of its properties and parameters, and in dimension on their dimension. Thus, we denote the causal parameter as – x, and the effectual parameter as – u. In mathematics, such a cause-and-effect relationship is formally described by a function (dependence) on its properties - parameters u = f (x). A changing parameter (property of an object) entails a change in the value of the function - the relative whole. Moreover, an objectively determined known quantity of a whole (number) is a relative value obtained as a ratio to its unit part (a certain objective generally accepted single standard of the whole - uet. A unit standard is a formal quantity, but generally accepted as an objective comparative measure.

Then u =k*u fl. The objective value of a parameter (property) is the ratio to the unit part (standard) of the parameter (property) -x = i* x this. The dimensions of the whole and the dimensions of the parameter and their individual standards are not identical. Odds k, iare numerically equal to u, x, respectively, since the reference values ​​of u fl andx thisare isolated. As a result of interactions, the parameter changes and this causal change consequently entails a change in the function (relative to the whole, object, system).

Need to determine formal general dependence of the amount of change in an object on interactions - the reasons for this change. This statement of the problem reflects a true, cause-and-effect, causal (according to F. Bacon) sequential approach interaction physics.

Decision and consequences.

Interaction is a general evolutionary mechanism - the cause of variability. What actually is interaction (short-range, long-range)? Since the general theory of interaction and the theoretical model of the interaction of objects, carriers of commensurate properties in natural science are still missing, we will have to create(more about this at).But since the thinking reader wants to know about the true physical essence of the derivative immediately and now, then we will make do with only brief, but strict and necessary for understanding the essence of the derivative conclusions from this work.

“Any, even the most complex interaction of objects, can be represented on such a scale of time and space (expanded in time and displayed in a coordinate system in such a way) that at each moment of time, at a given point in space, only two objects, two bearers of commensurate properties, will interact. And at this moment they will interact only with two of their commensurate properties."

« Any (linear, nonlinear) change in any property (parameter) of a certain nature of any object can be decomposed (represented) as a result (consequence) of events-interactions of the same nature, following in formal space and time, respectively, linearly or nonlinearly (uniformly or unevenly). Moreover, in each elementary, single event-interaction (short-range interaction), the property changes linearly because it is determined by the only reason for the change - an elementary commensurate interaction (which means it is a function of one variable). ... Accordingly, any change (linear or nonlinear), as a consequence of interactions, can be represented as the sum of elementary linear changes following linearly or nonlinearly in formal space and time.”

« For the same reason, any interaction can be decomposed into quanta of change (indivisible linear pieces). An elementary quantum of any nature (dimension) is the result of an elementary event-interaction along a given nature (dimension). The magnitude and dimension of the quantum is determined by the magnitude of the interacting property and the nature of this property. For example, with an ideal, absolutely elastic collision of balls (without taking into account thermal and other energy losses), the balls exchange their impulses (commensurate properties). The change in the momentum of one ball is a portion of linear energy (given to it or taken away from it) - there is a quantum that has the dimension of angular momentum. If balls with fixed momentum values ​​interact, then the state of the angular momentum of each ball at any observed interaction interval is a “allowed” value (by analogy with the views of quantum mechanics).”

In physical and mathematical formalism, it has become generally accepted that any property at any time and at any point in space (for simplicity, let’s take linear, one-coordinate) has a value that can be expressed by writing

(1)

where is the dimension.

This entry, among other things, constitutes the essence and deep physical meaning of a complex number, different from the generally accepted geometric representation (according to Gauss), in the form of a point on the plane..( Note author)

In turn, the modulus of the magnitude of change , denoted in (1) as , can be expressed, taking into account event-interactions, as follows

(2)

Physical meaning This basis for a huge number of well-known relationships of natural science, the root formula, is that in the interval of time and in the interval of homogeneous linear (single-coordinate) space, there were commensurate events-close interactions of the same nature, following in time and space in accordance with their functions -distributions of events in space and time. Each of the events changed to a certain . We can say that in the presence of homogeneity of objects of interaction on a certain interval of space and time, we are talking about some constant, linear, averaged value of elementary change - derivative value on the magnitude of the change , characteristic of the interaction environment, a formally described function characterizing the environment and the interaction process of a certain nature (dimension). Taking into account the fact that there may be different types of distribution functions of events in space and time, then there are variable space-time dimensions in as an integral of the distribution functionsevents in time and space , namely [time - t ] and[ coordinate - x ] can be to the power of k(k is not equal to zero).

If we denote, in a fairly homogeneous environment, the value of the average time interval between events - , and the value of the average interval of distance between events - , then we can write that the total number of events in the interval of time and space is equal to

(3)

This fundamental record(3) is consistent with the basic space-time identities of natural science (Maxwell’s electrodynamics, hydrodynamics, wave theory, Hooke’s law, Planck’s formula for energy, etc.) and is the true root cause of the logical correctness of physical and mathematical constructions. This entry (3) is consistent with the “mean value theorem” known in mathematics. Let's rewrite (2) taking into account (3)

(4) - for time relationships;

(5) - for spatial relationships.

From these equations (3-5) it follows general law of interaction:

the magnitude of any change in an object (property) is proportional to the number of events-interactions (close interactions) commensurate with it that cause it. At the same time, the nature of the change (the type of dependence in time and space) corresponds to the nature of the succession of these events in time and space.

We got general basic relationships of natural science for the case of linear space and time, cleared of the concept of infinity, aspirations to zero, instantaneous speed, etc. For the same reason, the designations of infinitesimal dt and dx are justifiably not used. Instead of them, the final Δti and Δxi are introduced . From these generalizations (2-6) it follows:

- the general physical meaning of the derivative (differential) (4) and gradient (5), as well as “world” constants, as values ​​of the averaged (mean) linear change of a function (object) during a single event-interaction of an argument (property) having a certain dimension ( nature) with commensurate (of the same nature) properties of other objects. The ratio of the magnitude of the change to the number of events-interactions initiating it is actually the value of the derivative of the function, reflecting the cause-and-effect dependence of the object on its property.

; (7) - derivative of a function

; (8) - function gradient

- physical meaning of the integral, as the sum of the magnitudes of changes in the function during events on the argument

; (9)

- justification (proof and clear physical meaning) of Lagrange's theorem for finite increments(formulas of finite increments), in many ways fundamental to differential calculus. Because for linear functions and the values ​​of their integrals in expressions (4)(5) and . Then

(10)

(10.1)

Formula (10.1) is in fact Lagrange's formula for finite increments [ 5].

When specifying an object with a set of its properties (parameters), we obtain similar dependencies for the variability of the object as a function of the variability of its properties (parameters) and clarify physical meaning of the partial derivative of a function several variable parameters.

(11)

Taylor formula for a function of one variable, which has also become classical,

looks like

(12)

It represents the expansion of a function (formal cause-and-effect system) into a series in which its change is equal to

is decomposed into components, according to the principle of decomposing the general flow of events of the same nature into subflows having different following characteristics. Each subflow characterizes the linearity (nonlinearity) of the sequence of events in space or time. This is physical meaning of Taylor's formula . So, for example, the first term of the Taylor formula identifies the change during events linearly occurring in time (space).

At . Second at nonlinearly following events of the type, etc.

- physical meaning of a constant rate of change (motion)[m/s], which has the meaning of a single linear movement (change, increment) of a quantity (coordinates, path), with linearly following events.

(13)

For this reason, speed is not causally dependent on a formally chosen coordinate system or time interval. Speed ​​is an informal dependence on the function of succession (distribution) in time and space of events leading to a change in coordinate.

(14)

And any complex movement can be decomposed into components, where each component is a dependence on the following linear or nonlinear events. For this reason, the kinematics of a point (the equation of a point) is expanded in accordance with the Lagrange or Taylor formula.

It is when the linear sequence of events changes to nonlinear that speed becomes acceleration.

- physical meaning of acceleration- as a quantity numerically equal to a unit displacement, with a nonlinear sequence of events-interactions causing this displacement . Wherein, or . At the same time, the total movement during a nonlinear sequence of events (with a linear change in the speed of events) for equals (15) - formula known from school

- physical meaning of the acceleration of a free fall object- as a constant value, numerically equal to the ratio of the linear force acting on an object (in fact, the so-called “instantaneous” linear displacement), correlated with the nonlinear number of subsequent events-interactions with the environment in formal time, causing this force.

Accordingly, the value is equal to the quantity nonlinearly following events, or attitude - received the name body weight , and the value is body weight , as a force acting on a body (on support) in a state of rest.Let us clarify the above, because widely used, fundamental physical concept of mass in modern physics is not structured causally from any interactions at all. And physics knows the facts of changes in the mass of bodies when certain reactions (physical interactions) occur within them. For example, during radioactive decay, the total mass of a substance decreases.When a body is at rest relative to the surface of the Earth, the total number of events-interactions of particles of this body with an inhomogeneous medium that has a gradient (otherwise called a gravitational field) does not change. This means that the force acting on the body does not change, and the inertial mass is proportional to the number of events occurring in the objects of the body and objects in the environment, equal to the ratio of the force to its constant acceleration .

When a body moves in a gravitational field (falls), then the ratio of the changing force acting on it to the changing number of events also remains constant and the ratio - corresponds to gravitational mass. this implies analytical identity of inertial and gravitational mass. When a body moves nonlinearly, but horizontally towards the Earth's surface (along the spherical equipotential surface of the Earth's gravitational field), then there is no gradient in the gravitational field in this trajectory. But any force acting on a body is proportional to the number of events both accelerating and decelerating the body. That is, in the case of horizontal movement, the reason for the movement of the body simply changes. And the nonlinearly changing number of events imparts acceleration to the body (Newton’s 2nd law). With a linear sequence of events (both accelerating and decelerating), the speed of the body is constant and the physical quantity, with such a sequence of events, in physics called impulse.

- The physical meaning of angular momentum, as the movement of a body under the influence of linear events in time.

(16)

- Physical meaning of electric charge object brought into the field, as the ratio of the force acting on the “charged” object (Lorentz force) at the field point to the magnitude of the charge of the field point. For force is the result of the interaction of the commensurate properties of an object introduced into the field and the field object. The interaction is expressed in a change in these commensurate properties of both. As a result of each single interaction, objects exchange the modules of their changes, changing each other, which is the magnitude of the “instantaneous” force acting on them, as a derivative of the acting force on a space interval. But in modern physics, the field, a special type of matter, unfortunately, does not have a charge (does not have charge carrier objects), but has a different characteristic - tension in the interval (the difference in potentials (charges) in a certain void). Thus, charge in its magnitude it shows how many times the force acting on a charged object differs from the field strength at a given point (from the “instantaneous” force). (17)

Then positive charge of an object– is seen as a charge that exceeds in absolute value (larger) the charge of a field point, and a negative charge is less than the charge of a field point. This implies a difference in the signs of the repulsive and attractive forces. Which determines the direction of the acting “repulsion-attraction” force. It turns out that the charge is quantitatively equal to the number of interaction events that change it in each event by the value of the field strength. The magnitude of the charge, in accordance with the concept of number (magnitude), is the relationship with the standard, unit, test charge - . From here . When a charge moves, when events follow linearly (the field is homogeneous), the integrals are , and when a homogeneous field moves relative to the charge. Hence the well-known physics relations ;

- Physical meaning of electric field strength, as the ratio of the force acting on a charged object to the number of events occurring - interactions of a charged object with a charged environment. There is a constant characteristic of the electric field. It is also the coordinate derivative of the Lorentz force.Electric field strength is a physical quantity numerically equal to the force acting on a unit charge during a single event-interaction () of a charged body and a field (charged medium).

(18)

-Physical meaning of potential, current, voltage and resistance (electrical conductivity).

In relation to changes in charge magnitude

(19)

(20)

(21)

Where is called the potential of a field point and it is taken as the energy characteristic of a given field point, but in fact it is the charge of a field point, which differs by a factor of times from the test (reference) charge. Or: . When the charge introduced into the field and the charge of a field point interact, an exchange of commensurate properties—charges—occurs. Exchange is a phenomenon described as “the Lorentz force acts on a charge introduced into the field,” equal in magnitude to the magnitude of the change in charge, as well as to the magnitude of the relative change in the potential of a field point. When introducing a charge into the Earth's field, the last change can be neglected due to the relative smallness of this change compared to the enormous value of the total charge of a point in the Earth's field.

From (20) it is noticeable that the current (I) is the time derivative of the magnitude of the charge change over a time interval, changing the charge in magnitude in one event-interaction (short-range interaction) with the charge of the medium (field point).

*It is still believed in physics that if: a conductor has a cross section with area S, the charge of each particle is equal to q 0, and the volume of the conductor, limited by cross sections 1 and 2 and length (), contains particles, where n is the concentration of particles. That's the total charge. If particles move in one direction with an average speed v, then during the time all particles contained in the volume under consideration will pass through cross section 2. Therefore, the current strength is equal to

.

The same, we can say in the case of our methodological generalization (3-6), only instead of the number of particles, we should say the number of events, which in meaning is more correct, because there are much more charged particles (events) in a conductor than, for example, electrons in a metal . The dependence will be rewritten as Therefore, the validity of (3-6) and other generalizations of this work is once again confirmed.

Two points of a homogeneous field, spaced apart in space, having different potentials (charges) have relative to each other potential energy, which is numerically equal to the work of changing the potential from a value to . It is equal to their difference.

. (22)

Otherwise, we can write Ohm’s law, rightly equating

. (23)

Where in this case is the resistance, showing the number of events required to change the amount of charge, provided that in each event the charge will change by a constant value of the so-called “instantaneous” current, depending on the properties of the conductor. It follows from this that current is a time-derived quantity and concept of voltage. It should be remembered that in SI units, electrical conductivity is expressed in Siemens with the dimension: cm = 1 / Ohm = Ampere / Volt = kg -1 m -2 s ³A². Resistance in physics is the reciprocal quantity equal to the product of electrical conductivity (resistance of a unit cross section of a material) and the length of the conductor. What can be written (in the sense of generalization (3-6)) as

(24)

- Physical meaning of magnetic field induction. It was experimentally established that the ratio of the maximum value of the modulus of force acting on a conductor with current ( Ampere's force) to the current strength - I to the length of the conductor - l, does not depend on either the current strength in the conductor or the length of the conductor. It was taken as a characteristic of the magnetic field in the place where the conductor is located - magnetic field induction, a value depending on the structure of the field - which corresponds

(25)

and since , then .

When we rotate the frame in a magnetic field, we first of all increase the number of events-interactions of charged objects of the frame and charged objects of the field. This implies the dependence of the emf and current in the frame on the speed of rotation of the frame and the field strength near the frame. We stop the frame - no interactions - no current. Z swirl (change) field - current flowed in the frame.

- Physical meaning of temperature. Today in physics the concept of a measure of temperature is not very trivial. One kelvin is equal to 1/273.16 of the thermodynamic temperature of the triple point of water. The beginning of the scale (0 K) coincides with absolute zero. Conversion to degrees Celsius: °C = K -273.15 (temperature of the triple point of water - 0.01 °C).
In 2005, the definition of Kelvin was refined. In the mandatory Technical Appendix to the text of ITS-90, the Advisory Committee on Thermometry established requirements for the isotopic composition of water when realizing the triple point temperature of water.

Nevertheless, physical meaning and essence of the concept of temperature much simpler and clearer. Temperature is inherently a consequence of events-interactions occurring inside a substance that have both “internal” and “external” causes. More events - more temperature, fewer events - less temperature. Hence the phenomenon of temperature changes in many chemical reactions. P. L. Kapitsa also used to say "... the measure of temperature is not the movement itself, but the randomness of this movement. The randomness of the state of a body determines its temperature state, and this idea (which was first developed by Boltzmann) that a certain temperature state of a body is not determined at all by the energy of movement, but by the randomness of this movement , and is the new concept in the description of temperature phenomena that we must use..." (Report by 1978 Nobel Prize laureate Pyotr Leonidovich Kapitsa “Properties of Liquid Helium”, read at the conference “Problems of Modern Science” at Moscow University on December 21, 1944)
The measure of chaos should be understood as a quantitative characteristic of a number interaction events per unit time in a unit volume of a substance - its temperature. It is no coincidence that the International Committee of Weights and Measures is going to change the definition of Kelvin (a measure of temperature) in 2011 in order to get rid of the difficult-to-reproduce conditions of the “triple point of water”. In the new definition, kelvin will be expressed in terms of a second and the value of Boltzmann's constant. Which exactly corresponds to the basic generalization (3-6) of this work. In this case, Boltzmann’s constant expresses the change in state of a certain amount of matter during a single event (see the physical meaning of the derivative), and the value and dimension of time characterizes the number of events in a time interval. This once again proves that causal structure of temperature - events-interactions. As a result of the events-interactions that occur, objects in each event exchange kinetic energy (angular momentum as in the collision of balls), and the medium eventually acquires thermodynamic equilibrium (the first law of thermodynamics).

- The physical meaning of energy and strength.

In modern physics, energy E has different dimensions (nature). There are as many natures as there are energies. For example:

Force multiplied by length (E ≈ F ·l≈N*m);

Pressure multiplied by volume (E ≈ P ·V≈N*m 3 /m 2 ≈N*m);

Impulse multiplied by speed (E ≈ p v≈kg*m /s*m /s≈(N* s 2 )/m*(m/s*m /s) ≈N*m);

Mass multiplied by the square of speed (E ≈ m ·v 2 ≈N*m);

Current multiplied by voltage (E ≈ I U ≈

From these relationships follows a refined concept of energy and a connection with a single standard (unit of measurement) of energy, events and change.

Energy, – is a quantitative characteristic of a change in any physical parameter of matter under the influence of events-interactions of the same dimension that cause this change. Otherwise, we can say that energy is a quantitative characteristic applied for some time (at some distance) to the property of an external acting force. The magnitude of energy (number) is the ratio of the magnitude of change of a certain nature to the formal, generally accepted standard of energy of this nature. The dimension of energy is the dimension of the formal generally accepted standard of energy. Causally, the magnitude and dimension of energy, its change in time and space, depend formally on the overall magnitude of the change in relation to the standard and the dimension of the standard, and informally depend on the nature of the sequence of events.

The total magnitude of the change - depends on the number of interaction events that change the magnitude of the total change in one event by - the average unit force - the derivative value.

The standard of energy of a certain nature (dimension) must correspond to the general concept standard (singularity, generally accepted, immutability), have the dimension of the function of the sequence of events in space-time and the changed value.

These relationships, in fact, are common to the energy of any change in matter.

About strength. and the magnitude or in essence, there is the same “instant” force that changes energy.

. (26)

Thus, the general concept of inertia should be understood as the magnitude of an elementary relative change in energy under the action of a single event-interaction (unlike force, not correlated with the value of the interval, but the assumed presence of an interval of invariability of the action), having an actual time interval (space interval) of its invariance until next event.

An interval is the difference between two moments in time of the beginning of a given and the next commensurate event-interaction, or two coordinate points of events in space.

Inertia has the dimension of energy, because energy is the integral sum of inertia values ​​in time under the influence of events-interactions. The magnitude of the energy change is equal to the sum of inertias

(27)

Otherwise, we can say that the inertia imparted to an abstract property by the interaction event is the energy of change in the property, which had some time of invariance until the next interaction event;

- physical meaning of time, as a formal way of knowing the magnitude of the duration of change (invariance), as a way of measuring the magnitude of duration in comparison with the formal standard of duration, as a measure of the duration of change (duration, duration

And it’s time to stop numerous speculations regarding the interpretation of this basic concept of natural science.

- physical meaning of coordinate space , as magnitudes (measures) of change (path, distance),

(32)

having the dimension of a formal, unit standard of space (coordinate) and the magnitude of the coordinate as an integral of the function of the sequence of events in space , equal to the total number of coordinate standards on the interval. When measuring coordinates, for convenience, a linearly varying subintegral a function whose integral is equal to the number of formally chosen standard intervals of unit coordinates;

- the physical meaning of all basic physical properties (parameters) that characterize the properties of any medium during elementary commensurate interaction with it (dielectric and magnetic permeability, Planck’s constant, coefficients of friction and surface tension, specific heat, world constants, etc.).

Thus, new dependencies are obtained that have a single initial form of recording and a single methodologically uniform causal meaning. And this causal meaning is acquired with the introduction of a global physical principle into natural science - “event-interaction”.

Here, dear reader, is what it should be like in the most general terms new mathematics endowed with physical meaning and certainty And new physics of interactions of the 21st century , cleared of a swarm of irrelevant concepts that lack definition, magnitude and dimension, and therefore common sense. Such, for example, How classical derivative and instantaneous velocity - has little in common with physical concept of speed. How concept of inertia – a certain ability of bodies to maintain speed... How inertial reference system (IRS) , which has nothing in common with the concept of a frame of reference(SO). Because ISO, unlike the usual reference frame of reference (FR) is not an objective system of cognition of the magnitude of movement (change). Relative to ISO, by its definition, bodies are only at rest or moving rectilinearly or uniformly. And also many other things that have been stupidly replicated for many centuries as unshakable truths. These pseudo-truths, which have become basic, are no longer capable of fundamentally, consistently and cause and effect describe general dependencies numerous phenomena of the universe that exist and change according to the same laws of nature.

1. Literature.

1. Hegel G.V.F. Encyclopedia of Philosophical Sciences: In 3 volumes. T. 1: The Science of Logic. M., 197 3

2. Hegel G.V.F. , Soch., t. 5, M., 1937, p. 691.

3. F. Engels. PSS. vol. 20, p. 546.

The idea is this: let's take some value (read "delta x") , which we'll call argument increment, and let’s start “trying it on” to various points on our path:

1) Let's look at the leftmost point: passing the distance, we climb the slope to a height (green line). The quantity is called function increment, and in this case this increment is positive (the difference in values ​​along the axis is greater than zero). Let's create a ratio that will be a measure of the steepness of our road. Obviously, this is a very specific number, and since both increments are positive, then .

Attention! Designation areONEsymbol, that is, you cannot “tear off” the “delta” from the “X” and consider these letters separately. Of course, the comment also concerns the function increment symbol.

Let's explore the nature of the resulting fraction more meaningfully. Let us initially be at a height of 20 meters (at the left black point). Having covered the distance of meters (left red line), we will find ourselves at an altitude of 60 meters. Then the increment of the function will be meters (green line) and: . Thus, on every meter this section of the road height increasesaverage by 4 meters...forgot your climbing equipment? =) In other words, the constructed relationship characterizes the AVERAGE RATE OF CHANGE (in this case, growth) of the function.

Note : The numerical values ​​of the example in question correspond only approximately to the proportions of the drawing.

2) Now let's go the same distance from the rightmost black point. Here the rise is more gradual, so the increment (crimson line) is relatively small, and the ratio compared to the previous case will be very modest. Relatively speaking, meters and function growth rate is . That is, here for every meter of the path there are average half a meter of rise.

3) A little adventure on the mountainside. Let's look at the top black dot located on the ordinate axis. Let's assume that this is the 50 meter mark. We overcome the distance again, as a result of which we find ourselves lower - at the level of 30 meters. Since the movement is carried out top down(in the “counter” direction of the axis), then the final the increment of the function (height) will be negative: meters (brown segment in the drawing). And in this case we are already talking about rate of decrease Features: , that is, for every meter of path of this section, the height decreases average by 2 meters. Take care of your clothes at the fifth point.

Now let's ask ourselves the question: what value of the “measuring standard” is best to use? It’s completely understandable, 10 meters is very rough. A good dozen hummocks can easily fit on them. No matter the bumps, there may be a deep gorge below, and after a few meters there is its other side with a further steep rise. Thus, with a ten-meter we will not get an intelligible description of such sections of the path through the ratio .

From the above discussion the following conclusion follows: the lower the value, the more accurately we describe the road topography. Moreover, the following facts are true:

– For anyone lifting points you can select a value (even if very small) that fits within the boundaries of a particular rise. This means that the corresponding height increment will be guaranteed to be positive, and the inequality will correctly indicate the growth of the function at each point of these intervals.

- Likewise, for any slope point there is a value that will fit completely on this slope. Consequently, the corresponding increase in height is clearly negative, and the inequality will correctly show the decrease in the function at each point of the given interval.

– A particularly interesting case is when the rate of change of the function is zero: . Firstly, zero height increment () is a sign of a smooth path. And secondly, there are other interesting situations, examples of which you see in the figure. Imagine that fate has brought us to the very top of a hill with soaring eagles or the bottom of a ravine with croaking frogs. If you take a small step in any direction, the change in height will be negligible, and we can say that the rate of change of the function is actually zero. This is exactly the picture observed at the points.

Thus, we have come to an amazing opportunity to perfectly accurately characterize the rate of change of a function. After all, mathematical analysis makes it possible to direct the increment of the argument to zero: , that is, to make it infinitesimal.

As a result, another logical question arises: is it possible to find for the road and its schedule another function, which would let us know about all the flat sections, ascents, descents, peaks, valleys, as well as the rate of growth/decrease at each point along the way?

What is a derivative? Definition of derivative.
Geometric meaning of derivative and differential

Please read carefully and not too quickly - the material is simple and accessible to everyone! It’s okay if in some places something doesn’t seem very clear, you can always return to the article later. I will say more, it is useful to study the theory several times in order to thoroughly understand all the points (the advice is especially relevant for “technical” students, for whom higher mathematics plays a significant role in the educational process).

Modeled after the tales of continuity of function, “promotion” of a topic begins with studying the phenomenon at a single point, and only then does it spread to numerical intervals.

What is a derivative?
Definition and meaning of a derivative function

Many will be surprised by the unexpected placement of this article in my author’s course on the derivative of a function of one variable and its applications. After all, as it has been since school: the standard textbook first of all gives the definition of a derivative, its geometric, mechanical meaning. Next, students find derivatives of functions by definition, and, in fact, only then they perfect the technique of differentiation using derivative tables.

But from my point of view, the following approach is more pragmatic: first of all, it is advisable to UNDERSTAND WELL limit of a function, and, in particular, infinitesimal quantities. The fact is that the definition of derivative is based on the concept of limit, which is poorly considered in the school course. That is why a significant part of young consumers of the granite of knowledge do not understand the very essence of the derivative. Thus, if you have little understanding of differential calculus or a wise brain has successfully gotten rid of this baggage over many years, please start with function limits. At the same time, master/remember their solution.

The same practical sense dictates that it is advantageous first learn to find derivatives, including derivatives of complex functions. Theory is theory, but, as they say, you always want to differentiate. In this regard, it is better to work through the listed basic lessons, and maybe master of differentiation without even realizing the essence of their actions.

I recommend starting with the materials on this page after reading the article. The simplest problems with derivatives, where, in particular, the problem of the tangent to the graph of a function is considered. But you can wait. The fact is that many applications of the derivative do not require understanding it, and it is not surprising that the theoretical lesson appeared quite late - when I needed to explain finding increasing/decreasing intervals and extrema functions. Moreover, he was on the topic for quite a long time. Functions and graphs”, until I finally decided to put it earlier.

Therefore, dear teapots, do not rush to absorb the essence of the derivative like hungry animals, because the saturation will be tasteless and incomplete.

The concept of increasing, decreasing, maximum, minimum of a function

Many textbooks introduce the concept of derivatives with the help of some practical problems, and I also came up with an interesting example. Imagine that we are about to travel to a city that can be reached in different ways. Let’s immediately discard the curved winding paths and consider only straight highways. However, straight-line directions are also different: you can get to the city along a flat highway. Or along a hilly highway - up and down, up and down. Another road goes only uphill, and another one goes downhill all the time. Extreme enthusiasts will choose a route through a gorge with a steep cliff and a steep climb.

But whatever your preferences, it is advisable to know the area or at least have a topographic map of it. What if such information is missing? After all, you can choose, for example, a smooth path, but as a result stumble upon a ski slope with cheerful Finns. It is not a fact that a navigator or even a satellite image will provide reliable data. Therefore, it would be nice to formalize the relief of the path using mathematics.

Let's look at some road (side view):

Just in case, I remind you of an elementary fact: travel happens from left to right. For simplicity, we assume that the function continuous in the area under consideration.

What are the features of this graph?

At intervals function increases, that is, each next value of it more previous one. Roughly speaking, the schedule is on down up(we climb the hill). And on the interval the function decreases– each next value less previous, and our schedule is on top down(we go down the slope).

Let's also pay attention to special points. At the point we reach maximum, that is exists such a section of the path where the value will be the largest (highest). At the same point it is achieved minimum, And exists its neighborhood in which the value is the smallest (lowest).

We will look at more strict terminology and definitions in class. about the extrema of the function, but for now let’s study another important feature: on intervals the function increases, but it increases at different speeds. And the first thing that catches your eye is that the graph soars up during the interval much more cool, than on the interval . Is it possible to measure the steepness of a road using mathematical tools?

Rate of change of function

The idea is this: let's take some value (read "delta x"), which we'll call argument increment, and let’s start “trying it on” to various points on our path:

1) Let's look at the leftmost point: passing the distance, we climb the slope to a height (green line). The quantity is called function increment, and in this case this increment is positive (the difference in values ​​along the axis is greater than zero). Let's create a ratio that will be a measure of the steepness of our road. Obviously, this is a very specific number, and since both increments are positive, then .

Attention! Designations are ONE symbol, that is, you cannot “tear off” the “delta” from the “X” and consider these letters separately. Of course, the comment also concerns the function increment symbol.

Let's explore the nature of the resulting fraction more meaningfully. Let us initially be at a height of 20 meters (at the left black point). Having covered the distance of meters (left red line), we will find ourselves at an altitude of 60 meters. Then the increment of the function will be meters (green line) and: . Thus, on every meter this section of the road height increases average by 4 meters...forgot your climbing equipment? =) In other words, the constructed relationship characterizes the AVERAGE RATE OF CHANGE (in this case, growth) of the function.

Note : The numerical values ​​of the example in question correspond only approximately to the proportions of the drawing.

2) Now let's go the same distance from the rightmost black point. Here the rise is more gradual, so the increment (crimson line) is relatively small, and the ratio compared to the previous case will be very modest. Relatively speaking, meters and function growth rate is . That is, here for every meter of the path there are average half a meter of rise.

3) A little adventure on the mountainside. Let's look at the top black dot located on the ordinate axis. Let's assume that this is the 50 meter mark. We overcome the distance again, as a result of which we find ourselves lower - at the level of 30 meters. Since the movement is carried out top down(in the “counter” direction of the axis), then the final the increment of the function (height) will be negative: meters (brown segment in the drawing). And in this case we are already talking about rate of decrease Features: , that is, for every meter of path of this section, the height decreases average by 2 meters. Take care of your clothes at the fifth point.

Now let's ask ourselves the question: what value of the “measuring standard” is best to use? It’s completely understandable, 10 meters is very rough. A good dozen hummocks can easily fit on them. No matter the bumps, there may be a deep gorge below, and after a few meters there is its other side with a further steep rise. Thus, with a ten-meter we will not get an intelligible description of such sections of the path through the ratio .

From the above discussion the following conclusion follows: the lower the value, the more accurately we describe the road topography. Moreover, the following facts are true:

– For anyone lifting points you can select a value (even if very small) that fits within the boundaries of a particular rise. This means that the corresponding height increment will be guaranteed to be positive, and the inequality will correctly indicate the growth of the function at each point of these intervals.

- Likewise, for any slope point there is a value that will fit completely on this slope. Consequently, the corresponding increase in height is clearly negative, and the inequality will correctly show the decrease in the function at each point of the given interval.

– A particularly interesting case is when the rate of change of the function is zero: . Firstly, zero height increment () is a sign of a smooth path. And secondly, there are other interesting situations, examples of which you see in the figure. Imagine that fate has brought us to the very top of a hill with soaring eagles or the bottom of a ravine with croaking frogs. If you take a small step in any direction, the change in height will be negligible, and we can say that the rate of change of the function is actually zero. This is exactly the picture observed at the points.

Thus, we have come to an amazing opportunity to perfectly accurately characterize the rate of change of a function. After all, mathematical analysis makes it possible to direct the increment of the argument to zero: , that is, to make it infinitesimal.

As a result, another logical question arises: is it possible to find for the road and its schedule another function, which would let us know about all the flat sections, ascents, descents, peaks, valleys, as well as the rate of growth/decrease at each point along the way?

What is a derivative? Definition of derivative.
Geometric meaning of derivative and differential

Please read carefully and not too quickly - the material is simple and accessible to everyone! It’s okay if in some places something doesn’t seem very clear, you can always return to the article later. I will say more, it is useful to study the theory several times in order to thoroughly understand all the points (the advice is especially relevant for “technical” students, for whom higher mathematics plays a significant role in the educational process).

Naturally, in the very definition of the derivative at a point we replace it with:

What have we come to? And we came to the conclusion that for the function according to the law is put in accordance other function, which is called derivative function(or simply derivative).

The derivative characterizes rate of change functions How? The idea runs like a red thread from the very beginning of the article. Let's consider some point domain of definition functions Let the function be differentiable at a given point. Then:

1) If , then the function increases at the point . And obviously there is interval(even a very small one), containing a point at which the function grows, and its graph goes “from bottom to top”.

2) If , then the function decreases at the point . And there is an interval containing a point at which the function decreases (the graph goes “top to bottom”).

3) If , then infinitely close near a point the function maintains its speed constant. This happens, as noted, with a constant function and at critical points of the function, in particular at minimum and maximum points.

A bit of semantics. What does the verb “differentiate” mean in a broad sense? To differentiate means to highlight a feature. By differentiating a function, we “isolate” the rate of its change in the form of a derivative of the function. What, by the way, is meant by the word “derivative”? Function happened from function.

The terms are very successfully interpreted by the mechanical meaning of the derivative :
Let us consider the law of change in the coordinates of a body, depending on time, and the function of the speed of movement of a given body. The function characterizes the rate of change of body coordinates, therefore it is the first derivative of the function with respect to time: . If the concept of “body movement” did not exist in nature, then there would be no derivative concept of "body speed".

The acceleration of a body is the rate of change of speed, therefore: . If the initial concepts of “body motion” and “body speed” did not exist in nature, then there would not exist derivative concept of “body acceleration”.

1.1 Some problems of physics 3

2. Derivative

2.1 Rate of change function 6

2.2 Derivative function 7

2.3 Derivative of a power function 8

2.4 Geometric meaning of derivative 10

2.5 Differentiation of functions

2.5.1 Differentiation of results of arithmetic operations 12

2.5.2 Differentiation of complex and inverse functions 13

2.6 Derivatives of parametrically defined functions 15

3. Differential

3.1 Differential and its geometric meaning 18

3.2 Differential properties 21

4. Conclusion

4.1 Appendix 1. 26

4.2 Appendix 2. 29

5. List of references 32

1. Introduction

1.1Some problems of physics. Let's consider simple physical phenomena: rectilinear motion and linear mass distribution. To study them, the speed of movement and density are introduced respectively.

Let us examine the phenomenon of movement speed and related concepts.

Let the body perform rectilinear motion and we know the distance , traveled by the body for any given time , i.e. we know the distance as a function of time:

The equation
called equation of motion, and the line it defines in the axle system
- traffic schedule.

Consider the movement of a body during a time interval
from some point until the moment
. During time, the body has traveled a path, and in time, a path
. This means that in units of time it traveled the distance

If the motion is uniform, then there is a linear function:

In this case, and the relation
shows how many units of path there are per unit of time; at the same time it remains constant, independent of any point in time taken, nor from what time increment is taken . It's a constant attitude called speed of uniform motion.

But if the movement is uneven, then the ratio depends

from , and from . It is called the average speed of movement in the time interval from before and denoted by :

During this interval of time, with the same distance traveled, movement can occur in a variety of ways; graphically this is illustrated by the fact that between two points on the plane (dots
in Fig. 1) you can draw a variety of lines
- graphs of movements in a given time interval, and all these various movements correspond to the same average speed.

In particular, between points passes through a straight line
, which is a graph of a uniform in the interval
movements. So the average speed shows at what speed you need to move uniformly in order to cover the same time interval same distance
.

Leaving the same , let's reduce . Average speed calculated for the modified interval
, lying within a given interval, may, of course, be different than in; throughout the entire interval . It follows from this that the average speed cannot be considered as a satisfactory characteristic of movement: it (the average speed) depends on the interval for which the calculation is made. Based on the fact that the average speed in the interval should be considered the better characterizing the movement, the smaller , Let's make it tend to zero. If there is an average speed limit, then it is taken as the current speed .

Definition. Speed rectilinear motion at a given time is called the limit of the average speed corresponding to the interval as it tends to zero:

Example. Let's write down the law of free fall:

.

For the average rate of fall in the time interval we have

and for the speed at the moment

.

This shows that the speed of free fall is proportional to the time of movement (fall).

2. Derivative

Rate of change of function. Derivative function. Derivative of a power function.

2.1 Rate of change of function. Each of the four special concepts: speed of movement, density, heat capacity,

the rate of a chemical reaction, despite the significant difference in their physical meaning, is from a mathematical point of view, as is easy to see, the same characteristic of the corresponding function. All of them are particular types of the so-called rate of change of a function, defined, just like the listed special concepts, using the concept of limit.

Let us therefore examine in general terms the question of the rate of change of the function
, abstracting from the physical meaning of the variables
.

Let first
- linear function:

.

If the independent variable gets increment
, then the function gets incremented here
. Attitude
remains constant, independent of the way in which the function is considered, and of what is taken .

This relationship is called rate of change linear function. But if the function not linear, then the relation

depends on , and from . This relationship only “on average” characterizes the function when the independent variable changes from given to
; it is equal to the speed of such a linear function which, given has the same increment
.

Definition.Attitude calledaverage speed function changes in interval
.

It is clear that the smaller the interval under consideration, the better the average speed characterizes the change in the function, so we force tend to zero. If there is a limit to the average speed, then it is taken as a measure of the rate of change of the function for a given , And is called the rate of change of a function.

Definition. Rate of change of function Vat this point is called the limit of the average rate of change of a function in the interval as it approaches zero:

2.2 Derivative function. Rate of change of function

determined through the following sequence of actions:

1) incrementally , given meaning , find the corresponding increment of the function

;

2) a relation is drawn up;

3) the limit of this ratio is found (if it exists)

as it arbitrarily tends to zero.

As already noted, if this function not linear,

then the attitude depends on , and from . The limit of this ratio depends only on the selected value and is therefore a function of . If the function linear, then the limit under consideration does not depend on , that is, it will be a constant value.

The specified limit is called derivative function of the function or simply derivative of a function and is denoted as follows:
.Reads: “ef touch from » or “ef prim from”.

Definition. Derivative of a given function is called the limit of the ratio of the increment of the function to the increment of the independent variable with an arbitrary tendency, this increment to zero:

.

The value of the derivative of a function at any given point usually designated
.

Using the introduced definition of a derivative, we can say that:

1) The speed of rectilinear motion is the derivative of

functions By (time derivative of the path).

2.3 Derivative of a power function.

Let's find derivatives of some simple functions.

Let
. We have

,

i.e. derivative
there is a constant value equal to 1. This is obvious, because it is a linear function and the rate of its change is constant.

If
, That

Let
, Then

It is easy to notice a pattern in the expressions for the derivatives of the power function
at
. Let us prove that in general the derivative of for any positive integer exponent equal to
.

.

We transform the expression in the numerator using Newton's binomial formula :

On the right side of the last equality there is a sum of terms, the first of which does not depend on , and the rest tend to zero along with . That's why

.

So, a power function with a positive integer has a derivative equal to:

.

At
from the general formula found, the formulas derived above follow.

This result is true for any indicator, for example:

.

Let us now consider separately the derivative of a constant quantity

.

Since this function does not change with changes in the independent variable, then
. Hence,

,

T. e. the derivative of the constant is zero.

2.4 Geometric meaning of the derivative.

Derivative of a function has a very simple and visual geometric meaning, which is closely related to the concept of a tangent to a line.

Definition. Tangent
to the line
at her point
(Fig. 2). is the limiting position of a line passing through a point, and another point
line when this point tends to merge with a given point.

.Tutorial

There is an average speedchangesfunctions in the direction of the straight line. 1 is called derivative functions in the direction and is indicated. So, - (1) - speedchangesfunctions at the point...

Now we know that the instantaneous rate of change of the function N (Z) at Z = +2 is equal to -0.1079968336. This means rise/fall over the period, so when Z = +2, the N(Z) curve rises by -0.1079968336. This situation is shown in Figure 3-13.

The measure of “absolute” sensitivity can be called the rate of change of the function. The measure of the sensitivity of a function at a given point (“instantaneous velocity”) is called its derivative.

We can measure the degree of absolute sensitivity of the variable y to changes in the variable x if we determine the ratio Ay/Ax. The disadvantage of this definition of sensitivity is that it depends not only on the “initial” point XQ, relative to which the change in argument is considered, but also on the very value of the interval Dx at which the speed is determined. To eliminate this drawback, the concept of derivative (the rate of change of a function at a point) is introduced. When determining the rate of change of a function at a point, the points XQ and xj are brought closer together, directing the interval Dx to zero. The rate of change of the function f(x) at point XQ is called the derivative of the function f(x) at point x. The geometric meaning of the rate of change of the function at point XQ is that it is determined by the angle of inclination of the tangent to the graph of the function at point XQ. The derivative is the tangent of the angle of inclination of the tangent to the graph of the function.

If the derivative y is considered as the rate of change of the function /, then the value y /y is its relative rate of change. Therefore, the logarithmic derivative (In y)

Directional derivative - characterizes the rate of change of the function z - f(x,y) at the point MO(ZO,UO) in the direction

Rate of change of function relative 124.188

So far, we have considered the first derivative of a function, which allows us to find the rate of change of the function. To determine whether the rate of change is constant, take the second derivative of the function. This is denoted as

Here and below, the prime means differentiation, so h is the rate of change of the function h relative to the increase in excess supply).

A measure of “absolute” sensitivity - the rate of change of a function (average (ratio of changes) or limiting (derivative))

Increment of a value, argument, function. Rate of change of function

The rate of change of a function over an interval (average rate).

The disadvantage of this definition of speed is that this speed depends not only on the point x0, relative to which the change in argument is considered, but also on the magnitude of the change in argument itself, i.e. on the value of the interval Dx at which the speed is determined. To eliminate this drawback, the concept of the rate of change of a function at a point (instantaneous rate) is introduced.

The rate of change of a function at a point (instantaneous rate).

To determine the rate of change of the function at point J Q, points x and x0 are brought closer together, directing the interval Ax to zero. The change in the continuous function will also tend to zero. In this case, the ratio of the change in the function tending to zero to the change in the argument tending to zero gives the rate of change of the function at the point x0 (instantaneous speed), more precisely on an infinitesimal interval, relative to the point xd.

It is this rate of change of the function Dx) at the point x0 that is called the derivative of the function Dx) at the point xa.

Of course, to characterize the rate of change in y, one could use a simpler indicator, say, the derivative of y with respect to L. The elasticity of substitution o is preferred due to the fact that it has the great advantage that it is constant for most production functions used in practice, i.e. i.e. not only does not change when moving along some isoquant, but also does not depend on the choice of isoquant.

Timely control means that effective control must be timely. Its timeliness lies in the commensurability of the time interval of measurements and assessments of controlled indicators, the process of specific activities of the organization as a whole. The physical value of such an interval (frequency of measurements) is determined by the time frame of the measured process (plan), taking into account the rate of change of controlled indicators and the costs of implementing control operations. The most important task of the control function remains to eliminate deviations before they lead the organization to a critical situation.

For a homogeneous system at TV = 0, M = 0, 5 also vanishes, so the right-hand side of expression (6.20) is equal to the rate of change of the total welfare function associated with heterogeneity.

Mechanical meaning of derivative. For a function y = f(x) varying with time x, the derivative y = f(xo] is the rate of change of y at time XQ.

The relative rate (rate) of change of the function y = = f(x) is determined by the logarithmic derivative

Variables x mean the magnitude of the difference between supply and demand for the corresponding type of means of production x = s - p. The function x(f) is continuously differentiated in time. Variables "x" mean the rate of change of the difference between demand and supply. Trajectory x (t) means the dependence of the rate of change of demand and supply on the magnitude of the difference between demand and supply, which in turn depends on time. State space (phase space) in our case is two-dimensional , i.e. it has the form of a phase plane.

Such properties of the quantity a explain the fact that the rate of change in the marginal rate of substitution y is characterized on its basis, and not using any other indicator, for example, the derivative of y with respect to x. Moreover, for a significant number of functions the elasticity of substitution is constant not only along isoclines, but also along isoquants. Thus, for the production function (2.20), using the fact that according to the isoclimate equation

There are many tricks that can be pulled off with short-term rates of change. This model uses a one-period