The physical meaning of the derivative. Instant function change rate, acceleration and gradient

The idea is this: take some meaning (read "delta x") , which we will call argument increment, and we will start "trying on" it to various points of our path:

1) Let's look at the leftmost point: bypassing the distance, we climb the slope to a height (green line). The quantity is called by function increment, and in this case this increment is positive (the difference in values ​​along the axis is greater than zero). Let's make up the relation, which will be the 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" the "delta" from the "x" and consider these letters separately. Of course, the comment also applies to the function increment symbol.

Let us investigate the nature of the resulting fraction more meaningfully. Let us initially be at a height of 20 meters (at the left black point). Having overcome 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 4 meters… Have you forgotten your climbing equipment? =) In other words, the constructed relation characterizes the AVERAGE RATE of CHANGE (in this case, growth) of the function.

Note : numerical values of the example under consideration correspond to the proportions of the drawing only approximately.

2) Now let's go the same distance from the rightmost black point. Here the rise is shallower, 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 makes up. That is, here for every meter of the path there is average half a meter rise.

3) A small adventure on the side of the mountain. Let's look at the top black dot located on the ordinate. Let's say it's 50 meters. Again we cover the distance, 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 "opposite direction" to the direction of the axis), then the final the increment of the function (height) will be negative: meters (brown line in the drawing). And in this case we are already talking about decay rate functions: , that is, for every meter of the path of this section, the height decreases average by 2 meters. Protect your clothing at the fifth point.

Now let's ask ourselves the question: what is the best value of the “measuring standard” to use? Quite understandably, 10 meters is very rough. A good dozen bumps can easily fit on them. Why are there bumps, there may be a deep gorge below, and after a few meters - its other side with a further steep rise. Thus, at ten meters, we will not get an intelligible characteristic of such sections of the path by means of a ratio.

The conclusion follows from the above reasoning - how less value , the more accurately we will describe the relief of the road. Moreover, the following facts are true:

– For any lifting points you can choose a value (albeit very small) that fits within the boundaries of one or another rise. This means that the corresponding increment in height is guaranteed to be positive, and the inequality will correctly indicate the growth of the function at each point of these intervals.

- Similarly, for any slope point there is a value that will fully fit on that slope. Consequently, the corresponding increment in height is uniquely negative, and the inequality will correctly show the decrease in the function at each point of the given interval.

- Of particular interest is the case when the rate of change of the function is equal to zero:. First, a zero height increment () is a sign of a flat path. And secondly, there are other curious situations, examples of which you see in the picture. Imagine that fate has taken 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, then the change in height will be negligible, and we can say that the rate of change of the function is virtually zero. Such a picture is 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 allows you to direct the increment of the argument to zero:, that is, to make it infinitely small.

As a result, another logical question arises: is it possible to find for the road and its schedule another function which would tell us about all flat areas, ascents, descents, peaks, lows, as well as the rate of increase / decrease at each point of the path?

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

Please read thoughtfully and not too quickly - the material is simple and accessible to everyone! It's okay if in some places something does not 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 qualitatively understand all the points (the advice is especially relevant for students-"techies", for whom higher mathematics plays a significant role in the educational process).

Modeled on the legends of continuity of function, "Promotion" of the topic begins with the study of the phenomenon at a single point, and only then it spreads to numerical intervals.

Now we know that the instantaneous rate of change of the function N (Z) at Z = +2 is -0.1079968336. This means up / down 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 the 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 the 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 a 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, tending the interval Dx to zero. The rate of change of the function f (x) at the point XQ is called the derivative of the function f (x) at the point x. The geometric meaning of the rate of change of the function at the point XQ is that it is determined by the angle of inclination of the tangent to the graph of the function at the point XQ. The derivative is the tangent of the slope 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 of 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 (ЖО, УО) in the direction

The rate of change of the function is relative 124.188

So far, we have considered the first derivative of a function, which allows us to find the rate of change of a function. To determine whether the rate of change is constant, the second derivative of the function should be taken. 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 marginal (derivative))

Increment of value, argument, function. Function change rate

The rate of change of the function on the interval (average rate).

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

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

To determine the rate of change of the function at the point J Q, the points x and x0 are brought together, tending the interval Ax to zero. In this case, 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 velocity), more precisely on an infinitely small 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 the value of y, a simpler indicator, say, the derivative of y with respect to L. that is, not only does it not change when moving along some isoquant, but also does not depend on the choice of the isoquant.

Timeliness of control means that effective control must be timely. Its timeliness lies in the commensurability of the time interval of measurements and assessments of the monitored indicators, the process of specific activities of the organization as a whole. The physical value of such an interval (measurement frequency) is determined by the time frame of the measured process (plan), taking into account the rate of change of the monitored 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 that the right-hand side of expression (6.20) is equal to the rate of change of the total welfare function associated with inhomogeneity.

The mechanical meaning of the derivative. For the function y = f (x), changing with time x, the derivative y = f (xo] is the rate of change of y at the time XQ.

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

Variables x mean the value of the difference between demand and supply 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 supply and demand. Trajectory x (t) means the dependence of the rate of change of supply and demand on the magnitude of the difference between demand and supply, which in turn depends on time. The state space (phase space) in our case is two-dimensional , that is, 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 with the help of some 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. So, for the production function (2.20), using the fact that according to the isoclite equation

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

1.1 Some problems of physics 3

2. Derivative

2.1 Function change rate 6

2.2 Derived function 7

2.3 Derivative of a power function 8

2.4 Geometric meaning of the derivative 10

2.5 Differentiation of functions

2.5.1 Differentiating the results of arithmetic operations 12

2.5.2 Differentiating 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 Properties of the differential 21

4. Conclusion

4.1 Appendix 1.26

4.2 Appendix 2.29

5. List of used literature 32

1. Introduction

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

Let us examine such a phenomenon as the speed of movement and the concepts associated with it.

Let the body perform rectilinear motion and we know the distance , traversed by the body for each given time , that is, we know the distance as a function of time:

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

Consider the movement of the body during the time interval
from some moment until the moment
. During the time the body has passed the way, and during the time - the way
. This means that in units of time it has passed the way

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

In this case, and the ratio
shows how many units of path are there per unit of time; at the same time, it remains constant, independent of any moment in time taken, nor from what time increment is taken . This is an ongoing relationship are called speed of uniform movement.

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 time interval, at the same distance traveled, movement can occur in a variety of ways; This is graphically illustrated by the fact that between two points on the plane (points
in fig. 1) a wide variety of lines can be drawn
- graphs of movements in a given time interval, and all these various movements correspond to the same average speed.

In particular, between the points passes a straight line segment
, which is a graph of a uniform in the interval
movement. So the average speed shows how fast you need to move evenly in order to pass in the same time interval the same distance
.

Leaving the same , we will decrease. Average speed calculated for the changed interval
, lying within a given interval, may, of course, be different than in; all interval . It follows from this that the average speed cannot be regarded as a satisfactory characteristic of movement: it (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 less , let's make it tend to zero. If at the same time there is a limit of the average speed, then it is taken as the speed of movement at the moment .

Definition. Speed rectilinear motion at a given time is called the limit of the average speed corresponding to the interval, when tending to zero:

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

.

For the average falling speed in the time interval, we have

and for the speed at the moment

.

From this it can be seen that the speed of free fall is proportional to the time of movement (fall).

2. Derivative

The rate of change of the function. Derived function. Derivative of a power function.

2.1 The rate of change of the 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 it 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 the function, which is determined, as well as the listed special concepts, using the concept of a limit.

Therefore, let us examine in general terms the question of the rate of change of the function
, distracting from the physical meaning of the variables
.

Let first
- linear function:

.

If the independent variable gets incremented
, then the function gets incremented here
. Attitude
remains constant, independent of either the function at which the function is considered, or the one taken .

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

also depends on , and from . This ratio only "on average" characterizes the function when the independent change changes from a given to
; it is equal to the speed of such a linear function, which for the taken has the same increment
.

Definition.Attitude calledaverage speed function changes in the interval
.

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

Definition. The rate of change of function vthis point is called the limit of the average rate of change of the function in the interval when tending to zero:

2.2 Derived function. Function change rate

is determined by the following sequence of actions:

1) incrementally , given this value , find the corresponding increment of the function

;

2) a relationship is drawn up;

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

with an arbitrary approach to zero.

As already noted, if the given function not linear,

that attitude also 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 considered limit does not depend on, i.e., it will be a constant value.

The specified limit is called derived function of function or simply derivative of the function and is denoted like this:
. Read: "eff stroke from » or "eff approx from".

Definition. Derivative This 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 denoted
.

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

1) The speed of rectilinear motion is a derivative of

function on (derivative of the path with respect to time).

2.3 Derivative of a power function.

Let us find the derivatives of some of the simplest functions.

Let be
... We have

,

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

If
, then

Let be
, then

It is easy to see the 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 is equal to
.

.

The expression in the numerator is transformed by the binomial Newton formula :

The right-hand side of the last equality contains the sum of terms, the first of which does not depend on, and the rest tend to zero together with . That's why

.

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

.

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

This result is true for any metric, for example:

.

Let us now consider separately the derivative of a constant value

.

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

,

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

2.4 Geometric meaning of the derivative.

Function derivative has a very simple and intuitive 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 called the limiting position of the straight line passing through the point, and another point
lines when this point tends to merge with this point.

.Tutorial

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Limit and Continuity of a Function

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Differential calculus concept that characterizes speedchangesfunction; P. is function, defined for each x ... the continuous derivative (differential calculus characterizing speedchangesfunction at this point). Then and ...

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It exists and is finite) will be speedchangesfunction at a point in the direction of the vector. Its ... and stands for or. In addition to the value speedchangesfunction, allows you to determine the nature of changesfunction at a point in the direction of the vector ...

The derivative of a function is one of the difficult topics in the school curriculum. Not every graduate will answer the question what a derivative is.

This article explains simply and clearly what a derivative is and what it is for.... We will not now strive for mathematical rigor of presentation. The most important thing is to understand the meaning.

Let's remember the definition:

The derivative is the rate of change of the function.

The figure shows graphs of three functions. Which one do you think is growing faster?

The answer is obvious - the third. It has the highest rate of change, that is, the largest derivative.

Here's another example.

Kostya, Grisha and Matvey got a job at the same time. Let's see how their income has changed over the year:

You can see everything on the chart right away, isn't it? Kostya's income has more than doubled in six months. And Grisha's income also increased, but only slightly. And Matvey's income dropped to zero. The starting conditions are the same, but the rate of change of the function, that is derivative, - different. As for Matvey, the derivative of his income is generally negative.

Intuitively, we can easily estimate the rate of change of a function. But how do we do it?

We are actually looking at how steeply the function graph goes up (or down). In other words, how fast does y change with changing x. Obviously, the same function at different points can have a different value of the derivative - that is, it can change faster or slower.

The derivative of the function is denoted.

Let's show you how to find it using the graph.

A graph of some function is drawn. Let's take a point with an abscissa on it. Let us draw at this point the tangent to the graph of the function. We want to estimate how steeply up the function graph is. A convenient value for this is tangent of the angle of inclination of the tangent.

The derivative of the function at a point is equal to the tangent of the angle of inclination of the tangent drawn to the graph of the function at this point.

Pay attention - as the angle of inclination of the tangent, we take the angle between the tangent and the positive direction of the axis.

Sometimes students ask what a tangent function is. This is a straight line that has a single common point with the graph in this section, and as shown in our figure. It looks like a tangent to a circle.

We'll find it. We remember that the tangent of an acute angle at right triangle equal to the ratio of the opposite leg to the adjacent one. From the triangle:

We found the derivative using the graph without even knowing the function formula. Such problems are often found in the exam in mathematics under the number.

There is another important relationship. Recall that the straight line is given by the equation

The quantity in this equation is called slope of the straight line... It is equal to the tangent of the angle of inclination of the straight line to the axis.

.

We get that

Let's remember this formula. It expresses the geometric meaning of the derivative.

The derivative of a function at a point is equal to the slope of the tangent drawn to the graph of the function at that point.

In other words, the derivative is equal to the tangent of the angle of inclination of the tangent.

We have already said that the same function can have different derivatives at different points. Let's see how the derivative is related to the behavior of the function.

Let's draw a graph of some function. Let this function increase in some areas, and decrease in others, and at different rates. And let this function have maximum and minimum points.

At a point, the function increases. A tangent to the graph drawn at a point forms an acute angle; with a positive direction of the axis. This means that the derivative is positive at the point.

At the point, our function decreases. The tangent at this point forms an obtuse angle; with a positive direction of the axis. Since the tangent of an obtuse angle is negative, the derivative at the point is negative.

Here's what happens:

If the function is increasing, its derivative is positive.

If it decreases, its derivative is negative.

And what will happen at the maximum and minimum points? We see that at the points (maximum point) and (minimum point) the tangent is horizontal. Therefore, the tangent of the angle of inclination of the tangent at these points is zero, and the derivative is also zero.

Point is the maximum point. At this point, the increase in the function is replaced by a decrease. Consequently, the sign of the derivative changes at the point from "plus" to "minus".

At the point - the minimum point - the derivative is also zero, but its sign changes from "minus" to "plus".

Conclusion: using a derivative, you can learn everything that interests us about the behavior of a function.

If the derivative is positive, then the function is increasing.

If the derivative is negative, then the function is decreasing.

At the maximum point, the derivative is zero and changes sign from "plus" to "minus".

At the minimum point, the derivative is also zero and changes sign from "minus" to "plus".

Let's write these conclusions in the form of a table:

	is increasing	maximum point	decreases	minimum point	is increasing
+	0	-	0	+

Let's make two small clarifications. You will need one of them when solving the problem. Another - in the first year, with a more serious study of functions and derivatives.

The case is possible when the derivative of a function at any point is equal to zero, but the function has no maximum or minimum at this point. This is the so-called :

At a point, the tangent to the graph is horizontal and the derivative is zero. However, up to the point the function increased - and after the point it continues to increase. The sign of the derivative does not change - as it was positive, it remains.

It also happens that the derivative does not exist at the maximum or minimum point. On the graph, this corresponds to a sharp bend, when a tangent at a given point cannot be drawn.

And how to find the derivative if the function is given not by a graph, but by a formula? In this case, the

Alternative physical meaning of the concept of the derivative of a function.

Nikolay Brylev

An article for those who think independently. For those who cannot understand how it is possible to cognize with the help of the unknowable and for this reason cannot agree with the introduction of unknowable concepts into the toolkit of cognition: "infinity", "ascent to zero", "infinitesimal", "neighborhood of a point", etc. .NS.

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

1.1 The classical concept of the derivative of a function.

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

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

Let the quantity 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 values u 1 = f (x 1), and u 2 = f (x 2 ). Difference of two argument values x 2, x 1 will be called the increment of the argument and denoted as Δ x = x 2 - x 1 (hence, x 2 = x 1 + Δ x) ... If the argument has changed to Δ x = x 2 - x 1, , then the function has changed (increased) as the difference between two values ​​of the function u 1 = f (x 1), u 2 = f (x 2 ) by the increment of the functionΔf... 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 down that the change in the 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 quantities x 2 and x 1, infinitely close in magnitude to each other, then Δ x = x 2 - x 1, - infinitely few.

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

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

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

1.2 The generally accepted physical meaning of the derivative of a function

And now on 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, calculus). I will briefly summarize their content by topic about her physical entity :

The physical meaning of the derivative x `(t ) of 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 the argument Δ t tends to zero.

And to explain to the students given physical meaning teachers can, for example, so.

Imagine that you are flying in an airplane and you have a watch on your wrist. When you fly, you have a speed equal to the speed of an airplane ?, - the teacher addresses the audience.

Yes !, - the students answer.

And what speed do you and the plane have at each moment of time on your watch?

The speed is equal to the speed of the plane !, - the good and the excellent students answer in chorus.

Not quite so, - the teacher explains. - Speed, as a physical concept, is the path of an airplane traversed in a unit of time (for example, in an hour (km / h)), while for you, when you looked at your watch, only a moment has passed. Thus, the instantaneous speed (the value of the distance traveled in an instant) is the derived value from the function that describes the plane's path in time. Instantaneous velocity is the physical meaning of the derivative.

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

A uditoriastudents, accustomed by the education system meekly,at once and entirelyassimilate questionable truths, as a rule, does not ask the teacher more questions about concept and physical sense of the derivative. But an inquisitive, deeply and independently thinking person cannot assimilate this as a strict scientific truth. He will certainly ask a number of questions, to which a reasoned answer from a teacher of any rank is clearly not expected. The questions are as follows.

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

2. How can derivative have physical meaning identical to some kind of instantaneous velocity, if instantaneous velocity is not a physical concept, but a very conditional, "imprecise" concept of mathematics, because this 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 (which has no other properties: size, area, interval), is substituted in the mathematical definition of the derivative by the concept of the neighborhood of a point, which actually has an interval, only indefinite in magnitude? For in the concept of the derivative, the concepts and quantities Δ x = x 2 - x 1, and x 0.

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

5. Why causation (function), depending on the reason (argument, property, parameter) should itself have final concrete defined in magnitude limit changes (consequences) with an indefinitely small, non-significant change in the magnitude of the cause?

6. There are functions in mathematics that do not have a derivative (nondifferentiable functions in nonsmooth analysis). This means that in these functions, when its argument (its parameter, property) changes, the function (mathematical object) does not change. But after all, there are no objects in nature that would not change when their own properties change. Why can mathematicians take such liberty as the use of 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 concept that exists in mathematics - instantaneous speed, derivative, there is no physical and scientific, correct meaning in general and cannot be due to the unscientific incorrectness and unknowability of the concepts used for this! It is absent in the concept of "infinity", and in the concept of "instant", and in the concept of "striving to zero or to infinity."

But true, cleared of the loose concepts of modern physics and mathematics (striving to zero, infinitely small value, infinity, etc.)

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

This is what will be discussed now.

1.4 True physical meaning and causal structure of the derivative.

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

I will allow myself a brief digression.

According to the deeply and sincere believers Isaac Newton (1643-1727) and Gottfried Leibniz, the change of 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 self-preservation. But already in the 19th century, naturalists figured out that THE CAUSAL ESSENCE OF CHANGE IN THE PROPERTIES OF ANY OBJECT - INTERACTION. "Interaction is a causal relationship in its full development.", noted Hegel (1770-1831) “In the closest way, the interaction appears to be the mutual causality of the substances that are supposed to condition each other; each is relative to the other at the same time both active and passive substance " ... F. Engels (1820-1895) specified: “Interaction is the first thing that comes before us when we consider moving (changing) matter as a whole, from the point of view of today's natural science ... This is how natural science confirms ... that interaction is the true causa finalis (ultimate root cause) of things. We cannot go beyond the knowledge of this interaction precisely because there is nothing more to cognize behind it " Nevertheless, having formally dealt with the root cause of variability, none of the bright minds of the 19th century began to rebuild the building of natural history.As a result, the building remained so - with a fundamental "rottenness". As a result, the causal structure (interactions) is still absent in the overwhelming majority of basic concepts of natural science (energy, force, mass, charge, temperature, speed, momentum, inertia, etc.), including in the mathematical concept of the derivative of a function- as a mathematical model describing " instantaneous change"object" from "infinitesimal" changes in its causal parameter. The theory of interactions, uniting even the well-known four fundamental interactions (electromagnetic, gravitational, strong, weak) has not yet been created. Now it is much more "nakosyacheno" and "jambs" everywhere crawl out. Practice - the criterion of truth, completely breaks all theoretical models built on such a building, claiming universality and globality. Therefore, all the same, the building of natural science will have to be rebuilt, because there is nowhere else to "swim", science has long been developing by the "poke" method - it is stupid, costly and ineffective. Physics of the future, physics of the 21st century and subsequent centuries should become the physics of interactions. And in physics it is simply necessary to introduce a new fundamental concept - "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 radical formulacausa finalis formula to justify all the basic formulas that work in practice. The meaning of world constants and much more is being clarified. And I will show you this, dear reader, now.

So, formulation of the problem.

Let's outline the model. Let the abstract object of cognition cognizable in size and nature (we denote it - u) is a relative whole, which has 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, parameters, and in dimension on their dimension. The causal parameter, therefore, will be denoted by - x, and the investigative parameter will be denoted as - u. In mathematics, such a causal relationship is formally described by a function (dependence) on its properties - the parameters u = f (x). A changing parameter (property of an object) entails a change in the value of the function - a relative whole. Moreover, the objectively determined known value of the whole (number) is a relative value obtained as a relation to its unitary part (some objective generally accepted unitary standard of the whole - u et, the unitary standard is a formal value, but generally accepted as an objective comparative measure.

Then u = k * u floor The objective value of the 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 dimension of the parameter and their unit standards are not identical. Odds k, iare numerically equal to u and x, respectively, since the reference values ​​u et andx thisisolated. As a result of interactions, the parameter changes and this causal change consequently entails a change in the function (relative whole, object, system).

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

Decision and Consequences.

Interaction is a common evolutionary mechanism - the cause of variability. What is interaction really (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 absent, we will have to create(more on this at).But since the thinking reader wants to know about the true physical essence of the derivative right away and now, then we will manage only with 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 (deployed in time and displayed in a coordinate system so that at any moment in time, at a given point in space, only two objects will interact, two carriers of commensurate properties.) And at this moment they will interact only with two of their commensurate properties. "

« Any (linear, nonlinear) change of any property (parameter) of a certain nature of any object can be decomposed (represented) as a result (consequence) of interaction events of the same nature, following in formal space and time, respectively, linearly or nonlinearly (uniformly or unevenly). At the same time, in each elementary, single event-interaction (short-range), the property changes linearly because it is caused by the only reason for the change - an elementary proportional interaction (which means there 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 in formal space and time, linearly or nonlinearly. "

« For the same reason, any interaction can be decomposed into quanta of changes (indivisible linear pieces). An elementary quantum of any nature (dimension) is the result of an elementary event-interaction in a given nature (dimension). The size and dimension of the quantum is determined by the value of the interacting property and the nature of this property. For example, in the case of 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 (imparted to it or or taken from it) - there is a quantum that has the dimension of the angular momentum. If the balls interact with fixed values ​​of momentum, then the state of the value of the angular momentum of each ball at any observed interaction interval is the "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, we will take a linear, one-dimensional) has a value that can be expressed by writing

(1)

where is the dimension.

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

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

(2)

Physical sense this basic for a huge number of well-known relationships in natural science, the root formula, is that on the interval of time and on the interval of a homogeneous linear (one-coordinate) space, there have been - commensurate short-range events of the same nature that followed in time and space in accordance with their functions -distributions of events in space - and time. Each of the events changed to a certain one. We can say that in the presence of homogeneity of objects of interaction in a certain interval of space and time, we are talking about some constant, linear, average value of elementary change - derived quantity from the magnitude of the change , characteristic of the interaction environment with a formally described function that characterizes the environment and the interaction process of a certain nature (dimension). Taking into account that may take place different kinds distribution functions of events in space and time, then there are variable space-time dimensions y as an integral of the distribution functionsevents in time and space , namely [time - t] and[coordinate - x] can be in power k(k is not equal to zero).

If we denote, in a sufficiently homogeneous environment, the value of the average time interval between events - and the value of the average interval of the distance between events -, then we can write down 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 fidelity of physical and mathematical constructions. This notation (3) is consistent with the "mean theorem" known in mathematics. 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 (short-range) 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 sequence in time and space of these events.

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

- the general physical meaning of the derivative (differential) (4) and gradient (5), as well as the "world" constants, as values ​​of the averaged (mean) linear change of a function (object) in 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 magnitude of the derivative of the function reflecting the cause-and-effect dependence of the object on its property.

; (7) - derivative of the function

; (8) is the gradient of the function

- the physical meaning of the integral, as the sum of the values ​​of the function change during events by the argument

; (9)

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

(10)

(10.1)

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

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

(11)

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

has the form

(12)

It is an expansion of a function (formal causal system) in a series, in which its change is equal to

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

At . Second at nonlinearly following view events, etc.

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

(13)

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

(14)

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

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

- physical meaning of acceleration- as a quantity numerically equal to a unit displacement, in the case of nonlinear succession of interaction events that cause this displacement ... Wherein, or ... In this case, the total displacement in the case of nonlinear sequencing of events (with a linear change in the rate of sequencing of events) for equals (15) - the formula known from school

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

Accordingly, a value equal to the amount nonlinearly following events, or relation - received the name body weight , and the quantity - body weight , as a force acting on the body (on the support) at rest.Let us explain the above, because widely used, fundamental physical concept of mass in modern physics is not causally structured from any interactions at all. And physics knows the facts of the change in the mass of bodies when certain reactions (physical interactions) occur inside them. For example, with radioactive decay, the total mass of the substance decreases.When a body is at rest relative to the Earth's surface, the total number of events-interactions of particles of this body with an inhomogeneous medium with a gradient (otherwise it is called the gravitational field) does not change. This means that the force acting on the body does not change, and the inert mass is proportional to the number of events occurring in the objects of the body and objects in the environment, and is 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 the gravitational mass... this implies analytic identity of inertial and gravitational mass... When the body moves nonlinearly, but horizontally to 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 the 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 gives acceleration to the body and (Newton's second law). With a linear sequence of events (both accelerating and decelerating), the speed of the body is constant and physical quantity, with such a sequence of events, in physics is called impulse.

- The physical meaning of the angular momentum, as body movements under the influence of events linearly following in time.

(16)

- Physical sense electric charge an object introduced into the field as the ratio of the force (Lorentz force) acting on the "charged" object at the point of the field to the value of the charge of the point of the field. For force is the result of the interaction of the commensurate properties of the object introduced into the field and the object of the field. Interaction is expressed in the change of these commensurate properties of both. As a result of each single interaction, the objects exchange the modules of their changes, changing each other, which is the value of the "instantaneous" force acting on them, as a derivative of the acting force on the interval of space. But in modern physics, a field, a special kind of matter, unfortunately, does not have a charge (does not have objects of charge carriers), but has a different characteristic - the intensity 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 the object- is seen as a charge exceeding in absolute value (greater) the charge of the field point, and negative - less than the charge of the field point. Hence follows the difference in the signs of the forces of repulsion and attraction... This is what determines the presence of a direction in the acting force of "repulsion - attraction". It turns out that the charge is quantitatively equal to the number of events-interactions that change it in each event by the value of the field strength. The magnitude of the charge, in accordance with the concept of a number (magnitude), is a ratio with a standard, unit, test charge -. From here ... When the charge moves, when events follow linearly (the field is uniform), the integrals, and when moving uniform field regarding charge. Hence the well-known relations of physics ;

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

(18)

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

Applied to a change in the amount of charge

(19)

(20)

(21)

Where the potential of the field point is called and it is taken as the energy characteristic of a given point of the field, but in fact it is the charge of the point of the field, which differs in times from the test (reference) charge. Or: . With the interaction of the charge introduced into the field and the charge of the point of the field, an exchange of commensurate properties - charges takes place. Exchange is a phenomenon described as “the Lorentz force acts on the charge introduced into the field”, which is equal in magnitude to the magnitude of the change in the charge, as well as to the magnitude of the relative change in the potential of the point of the field. When a charge is introduced into the Earth's field, the latter change can be neglected due to the relative smallness of this change compared to the huge 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 change in charge over the time interval, which changes the charge in magnitude in one event-interaction (short-range) with the charge of the medium (field point).

* Until now, it is believed in physics that if: the 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 is the total charge. If the particles move in one direction with an average velocity v, then during the time all the particles contained in the considered volume will pass through the cross section 2. Therefore, the current strength is

.

Same, it can be said 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 terms of meaning, is more true, because there are much more charged particles (events) in the conductor than, for example, electrons in the metal ... The dependency will be rewritten as , therefore, once again the validity of (3-6) and other generalizations of this work is confirmed.

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

. (22)

Otherwise, you 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 that depends on the properties of the conductor. It also follows that current is a time-derivative quantity and a 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 equal product specific electrical conductivity (resistance of a single section of the material) per conductor length. What can be written (in the sense of generalization (3-6)) as

(24)

- The physical meaning of induction magnetic field. It was experimentally found that the ratio of the maximum value of the modulus of the force acting on the conductor with current (Ampere's force) to the current strength - I to the length of the conductor - l, does not depend on the current strength in the conductor, nor on the length of the conductor. It was taken for the characteristic of the magnetic field in the place where the conductor is located - the induction of the magnetic field, a value depending on the structure of the field - which corresponds to

(25)

and since, then.

When we rotate the frame in a magnetic field, then we first of all increase the number of events-interactions between charged objects of the frame and charged objects of the field. From here follows the dependence of the EMF and current in the frame on the rotation speed of the frame and the field strength around the frame. We stop the frame - there are no interactions - there is no current either. Z swirl (change) field - the current went and in the frame.

- The physical meaning of temperature. Today in physics the concept - a measure of temperature is not quite 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: ° С = 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 Supplement to the text of ITS-90, the Advisory Committee on Thermometry established the requirements for the isotopic composition of water when realizing the temperature of the triple point of water.

Nevertheless, the physical meaning and essence of the concept of temperature much simpler and clearer. Temperature, in essence, is a consequence of events-interactions occurring inside the substance, which have both "internal" and "external" causes. More events - more temperature, fewer events - less temperature. Hence the phenomenon of temperature change at many chemical reactions... P. L. Kapitsa used to say "... the measure of temperature is not the movement itself, but the chaos of this movement. The chaotic 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 at all determined by the energy of motion, but the chaotic nature of this movement , and is that new concept in the description of temperature phenomena, which we should use ... " (Laureate's report Nobel Prize 1978 Petr 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 the number events-interactions per unit of time in a unit volume of matter - its temperature... It is no coincidence that the International Committee for Weights and Measures is going to change the definition of kelvin (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, the kelvin will be expressed in a second and the value of the Boltzmann constant. Which exactly corresponds to the basic generalization (3-6) of this work. In this case, the Boltzmann constant expresses the change in the state of a certain amount of matter in a single event (see, the physical meaning of the derivative), and the magnitude and dimension of time characterizes the number of events in a time interval. This proves once again that the causal structure of temperature - events-interactions. As a result of the events-interactions, the 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 a different dimension (nature). How many natures, so many energies. For example:

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

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

Impulse multiplied by the 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 the speed (E ≈ m · v 2 ≈N * m);

Current multiplied by voltage (E ≈ I U ≈

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

Energy, - there is a quantitative characteristic of the change in any physical parameter of matter under the influence of events-interactions of the same dimension, causing this change. Otherwise, we can say that energy is a quantitative characteristic of the applied for some time (at some distance) to the property of an external acting force. The amount of energy (number) is the ratio of the amount 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, formally depend on the total amount of 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 amount of change - depends on the number of events-interactions that change the amount of total change in one event by - the average unit force - the derived value.

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

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

About strength. and the value or in fact, there is the same "instantaneous" force that changes energy.

. (26)

Thus, under general concept inertia should be understood as the value of an elementary relative change in energy under the action of a single event-interaction (as opposed to a force that is not correlated with the size of the interval, but the assumed presence of an interval of invariability of the action), which has an actual interval of time (interval of space) of its invariability until the next event.

The interval is the difference between two points in time of the beginning of this and the next commensurate events-interactions, or two points-coordinates of events in space.

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

(27)

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

- the physical meaning of time, as a formal way of knowing the magnitude of the duration of change (immutability), 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 about the interpretation of this basic concept of natural science.

- physical meaning of coordinate space , as a magnitude (measure) of change (path, distance),

(32)

having the dimension of a formal, unit standard of space (coordinates) and the value of the coordinate, as an integral of the function of the succession of events in space , equal to the total number of reference standards on the interval. When measuring coordinates, for convenience, a linearly changing integrand a function, the integral of which is equal to the number of formally selected reference intervals of unit coordinates;

- the physical meaning of all basic physical properties(parameters) characterizing the properties of any medium with an elementary commensurate interaction with it (dielectric and magnetic permeability, Planck's constant, coefficients of friction and surface tension, specific heat capacity, 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 into natural science of the global physical principle - "events-interactions".

Here, dear reader, what should be in the most general terms new, endowed physical sense and certainty mathematician and new physics of interactions of the 21st century , cleared of a swarm of non-relative, without definiteness, magnitude and dimension, and hence common sense concepts. Such, for example, how classical derivative and instantaneous velocity - having little to do with physical concept of speed... How concept of inertia - some ability of bodies to maintain speed ... inertial reference system (ISO) , which has nothing to do with frame of reference(CO). For ISO, in contrast to the usual reference frame of reference (CO) is not an objective system of cognition of the magnitude of movement (change). Relative to IFR, by its definition, bodies only rest or move in a straight line or uniformly. And also a lot of other things that have been stupidly replicated for many centuries as unshakable truths. These, which have become basic, pseudo-truths are no longer able to fundamentally, consistently and causally describe by common dependencies numerous phenomena of the universe, existing and changing according to the uniform laws of nature.

1. Literature.

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

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

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