The physical meaning of the derivative. Instantaneous Rate of Change of Function, Acceleration and Gradient

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

This article simply and clearly explains what a derivative is and why it is needed.. 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 grows the fastest?

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

Here is another example.

Kostya, Grisha and Matvey got jobs at the same time. Let's see how their income changed during the year:

You can see everything on the chart right away, right? Kostya's income has more than doubled in six months. And Grisha's income also increased, but just a little bit. And Matthew's income decreased to zero. The starting conditions are the same, but the rate of change of the function, i.e. 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?

What we are really looking at is how steeply the graph of the function goes up (or down). In other words, how fast y changes with x. It is obvious that the same function in different points may have a different value of the derivative - that is, it may change faster or slower.

The derivative of a function is denoted by .

Let's show how to find using the graph.

A graph of some function is drawn. Take a point on it with an abscissa. Draw a tangent to the graph of the function at this point. We want to evaluate how steeply the graph of the function goes up. A handy value for this is tangent of the slope of the tangent.

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

Please note - 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 is the tangent to the graph of a function. This is a straight line, which has the only common point with a graph, and as shown in our figure. It looks like a tangent to a circle.

Let's find . We remember that the tangent of an acute angle in right triangle equal to the ratio of the opposite leg to the adjacent one. From triangle:

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

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

The quantity in this equation is called slope of a 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 angular coefficient tangent drawn to the graph of the function at that point.

In other words, the derivative is equal to the tangent of the slope 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, decrease in others, and with different speed. And let this function have maximum and minimum points.

At a point, the function is increasing. The tangent to the graph, drawn at the point, forms an acute angle; with positive axis direction. So the derivative is positive at the point.

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

Here's what happens:

If a 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 (maximum point) and (minimum point) the tangent is horizontal. Therefore, the tangent of the slope of the tangent at these points is zero, and the derivative is also zero.

The point is the maximum point. At this point, the increase of 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 equal to zero, but its sign changes from "minus" to "plus".

Conclusion: with the help of the derivative, you can find out everything that interests us about the behavior of the 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.

We write these findings in the form of a table:

	increases	maximum point	decreases	minimum point	increases
+	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.

A case is possible when the derivative of a function at some point is equal to zero, but the function has neither a maximum nor a minimum at this point. This so-called :

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

It also happens that at the point of maximum or minimum, the derivative does not exist. On the graph, this corresponds to a sharp break, when it is impossible to draw a tangent at a given point.

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

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

But from my point of view, the following approach is more pragmatic: first of all, it is advisable to UNDERSTAND the limit of the function WELL, and, in particular, infinitesimals. The fact is that

the definition of the derivative is based on the concept of a limit , which is poorly considered in the school course. That is why a significant part of young consumers of granite knowledge poorly penetrate into the very essence of the derivative. Thus, if you are poorly oriented in differential calculus, or a wise brain for long years successfully disposed of this baggage, please start from function limits . At the same time master / remember their decision.

The same practical sense suggests that it is profitable first

learn to find derivatives, including derivatives of complex functions . Theory is a theory, but, as they say, you always want to differentiate. In this regard, it is better to work out the listed basic lessons, and maybe become differentiation master without even realizing the essence of their actions.

I recommend starting the materials on this page after reading the article. The simplest problems with a derivative, where, in particular, the problem of the tangent to the graph of a function is considered. But it can be delayed. 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 intervals of increase/decrease and extremums functions. Moreover, he was in the subject for quite a long time " Functions and Graphs”, until I decided to put it in 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 study guides lead to the concept of a derivative with the help of some practical problems, and I also came up with interesting example. Imagine that we have to travel to a city that can be reached in different ways. We immediately discard the curved winding paths, and we will consider only straight lines. However, straight-line directions are also different: you can get to the city along a flat autobahn. Or on a hilly highway - up and down, up and down. Another road goes only uphill, and another one goes downhill all the time. Thrill-seekers will choose a route through the gorge with a steep cliff and a steep ascent.

But whatever your preferences, it is desirable to know the area, or at least have a topographical map of it. What if there is no such information? After all, you can choose, for example, a flat path, but as a result, stumble upon a ski slope with funny Finns. Not the fact that the navigator and even

satellite image will give reliable data. Therefore, it would be nice to formalize the relief of the path by means of mathematics.

Consider some road (side view):

Just in case, I remind you of an elementary fact: the journey occurs from left to right. For simplicity, we assume that the function is continuous on the section under consideration.

What are the features of this chart?

At intervals the function is increasing, that is, each subsequent value of it is greater than the previous one. Roughly speaking, the graph goes from the bottom up (we climb the hill). And on the interval, the function decreases - each next value is less than the previous one, and our graph goes from top to bottom (we go down the slope).

Let's also pay attention to special points. At the point we

we reach the maximum , that is, there is such a section of the path on which the value will be the largest (highest). At the same point, a minimum is reached, and there is such a neighborhood in which the value is the smallest (lowest).

More rigorous terminology and definitions will be considered in the lesson. about the extrema of the function while we study one more important feature: in between the function is increasing, but it is increasing at different speeds. And the first thing that catches your eye is that the interval graph soars up much more cool than on the interval. Is it possible to measure the steepness of the road using mathematical tools?

Function change rate

The idea is this: take some value

(read "delta x") , which we will callargument increment, and let's start "trying it on" 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 function increment, and in this case this increment is positive (the difference of values ​​along the axis is greater than

zero). Let's make the ratio , 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! The designation is a SINGLE symbol, that is, you cannot “tear off” the “delta” from the “x” and consider these letters separately. Of course, the comment also applies to the function's increment symbol.

Let's explore the nature of the resulting fraction more meaningful. Let

initially we are at a height of 20 meters (in the left black dot). Having overcome the distance of meters (left red line), we will be at a height of 60 meters. Then the increment of the function will be

meters (green line) and:. So

Thus, on every meter of this section of the road height increases an average of 4 meters ... did you forget your climbing equipment? =) In other words, the constructed ratio characterizes the AVERAGE RATE OF CHANGE (in this case, growth) of the function.

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

2) Now let's go the same distance from the rightmost black dot. Here the rise is more gentle, so the increment

(magenta line) is relatively small, and the ratio

compared with 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 is an average of half a meter of ascent.

3) A little adventure on the mountainside. Let's look at the top black dot located on the y-axis. Let's assume that this is a mark of 50 meters. Again we overcome the distance, as a result of which we find ourselves lower - at the level of 30 meters. Since the movement was carried out from top to bottom (in the "opposite" direction of the axis), the final the increment of the function (height) will be negative:meters (brown line in the drawing). And in this case we are talking about speed

descending function: , that is, for each meter of the path

In this area, the height decreases by an average of 2 meters. Take care of clothes on the fifth point.

Now let's ask the question: what is the best value of "measuring standard" to use? It is clear that 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 ascent. Thus, with a ten-meter we will not get an intelligible characterization of such sections of the path through

relationship .

From the above discussion, the following conclusion follows: the smaller the value, the more accurately we will describe the relief of the road. Moreover, fair

We now know that the instantaneous rate of change of the N(Z) function at Z = +2 is -0.1079968336. This means up/down over the period, so when Z = +2, the N(Z) curve goes up by -0.1079968336. This situation is shown in Figure 3-13.

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

We can measure the degree of absolute sensitivity of the variable y to changes in the variable x if we define the ratio Ay/Ax. The disadvantage of such a 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, on which the speed is determined. To eliminate this shortcoming, 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 together, tending the interval Dx to zero. The rate of change of the function f (x) at the point XQ and 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 function graph.

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)

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

Rate of function change relative 124.188

So far, we have considered the first derivative of the function , which allows you to find the rate of change of the function. To determine if 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 that 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 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 magnitude of the change in the argument itself, i.e. on the value of the interval Dx, on which the speed is determined. To eliminate this shortcoming, the concept of the rate of change of a function at a point (instantaneous velocity) is introduced.

The rate of change of a 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. 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, 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 a great advantage - it is constant for most production functions used in practice, t i.e. not only does 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 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 that the right side of expression (6.20) is equal to the rate of change of the total welfare function associated with heterogeneity.

The mechanical meaning of the derivative. For a function y = f(x) changing 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

The 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 differentiable in time. The variables x" mean the rate of change in the difference between supply and demand. The trajectory x (t) means the dependence of the rate of change in supply and demand on the magnitude of the difference between supply and demand, which in turn depends on time. The state space (phase space) in our case is two-dimensional , i.e., has the form of a phase plane.

Such properties of the quantity a explain the fact that the rate of change of the marginal rate of substitution y is characterized on its basis, and not with the help of 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. So, for the production function (2.20), using the fact that, according to the isocli-

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

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

Nikolay Brylev

An article for those who think on their own. For those who cannot understand how it is possible to know with the help of the unknowable and for this reason cannot agree with the introduction of unknowable concepts into the tools of cognition: "infinity", "going to zero", "infinitely small", "neighborhood of a point", etc. .P.

The purpose of this article is not to denigrate the idea of ​​introducing a very useful fundamental concept into mathematics and physics. concepts derivative of a function(differential), and deeply understand it 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 interaction physics. This meaning today is impossible to guess, because the generally accepted concept is 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, existing for almost three centuries, which has become a 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 so.

Let the value u depends on the x argument as u = f(x). If f(x ) was fixed at two points in the argument values: x2, x1, , 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 will be called the increment of the argument and denoted as Δ x = x 2 - x 1 (hence x 2=x1+ Δ x) . If the argument has changed to Δ x \u003d x 2 - x 1, , then the function has changed (increased) as the difference between the two values ​​of the function u 1 \u003d f (x 1), u 2 \u003d f (x 2 ) by the increment of the function∆f. It is usually written like this:

∆f= u 1 - u 2 \u003d f (x 2) - f (x 1 ) . Or considering that x 2 = x 1 + Δ x , we can write that the change in the function is equal to∆f= f (x 1 + Δx)- f (x 1 ). And this change occurred, of course, on the range of possible values ​​of the function x2 and x1, .

It is believed that if the values x 2 and x 1, infinitely close in magnitude to each other, then Δ x \u003d x 2 - x 1, - infinitesimal.

Derivative definition: Derivative function f (x) at the point x 0 is called the limit of the increment ratio of the function Δ f at this point to the increment of the argument Δx when the latter tends to zero (infinitely small). Recorded like this.

Lim Δx →0 (∆f(x0)/ 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 , which has a derivative at each point of some interval, is called differentiable on this interval.

1.2 The 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 (material analysis, differential calculus). I summarize briefly their content on the subject about her physical nature:

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

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

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

Yes, the students answer.

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

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

Not really, says the teacher. - Speed, as a physical concept, is the path of an aircraft traveled per unit of time (for example, per hour (km / h)), and when you looked at your watch, only a moment passed. Thus, instantaneous speed (the distance traveled in an instant) is the derivative of the function that describes the path of the aircraft in time. Instantaneous speed - this is the physical meaning of the derivative.

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

A audiencestudents, accustomed by the education system meekly,immediately and completelylearn doubtful truths, as a rule, does not ask the teacher more questions about concept and physical meaning 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 he will obviously not wait for a reasoned answer from a teacher of any rank. The questions are as follows.

1. Are exact (correct, scientific, having an objective value, causal essence) such concepts (expressions) of "exact" science - mathematics as: moment - an infinitesimal value, aspiration to zero, aspiration to infinity, smallness, infinity, aspiration? How can to know some entity in the magnitude of the change, 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, broadcast: “If the infinite, because it is infinite, is unknowable, then the infinite in quantity or magnitude is unknowable, how great it is, and the infinite in kind is unknowable, what is its quality. Since the beginnings are infinite both in quantity and in kind, then to know those formed from them [ things] is impossible: after all, only then do we believe that we have known complicated thing when we find out from what and how many [beginnings] it consists of ... " Aristotle, "Physics", 4 ch..

2. How can derivative have a physical meaning identical to some instantaneous speed, if instantaneous speed is not a physical concept, but a very conditional, "inaccurate" 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 - the coordinate (having no other properties: size, area, interval) replaced 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 a 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 cause (argument, property, parameter) must itself have final concrete defined in magnitude limit changes (consequences) with an indefinitely small, not having a magnitude change in the magnitude of the cause?

6. There are functions in mathematics that do not have a derivative (non-differentiable functions in non-smooth analysis). This means that in these functions, when its argument (its parameter, property) changes, the function (mathematical object) does not change. But there are no objects in nature that would not change when their own properties change. Why, then, can mathematics afford such liberties 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, physical and scientific in general, there is no correct meaning and cannot be due to the unscientific incorrectness and unknowability of the concepts used for this! It does not exist in the concept of "infinity", and in the concept of "instant", and in the concept of "striving towards zero or infinity".

But the true one, cleansed of the lax concepts of modern physics and mathematics (tendency to zero, infinitesimal 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 a thick layer of centuries-old noodles”, hung still by Gottfried Leibniz (1646-1716) and his followers, one will have, as usual, to turn to the methodology of knowledge and strict basic principles. True, it should be noted that due to the prevailing relativism, at present, these principles are no longer adhered to in science.

Let me digress briefly.

According to deeply and sincerely believers Isaac Newton (1643-1727) and Gottfried Leibniz, changing objects, changing 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 a powerful church and was not carried out for self-preservation purposes. But already in the 19th century, natural scientists figured out that CAUSAL ESSENCE OF CHANGING THE PROPERTIES OF ANY OBJECT - INTERACTIONS. "Interaction is a causal relationship posited in its full development", noted Hegel (1770-1831) “In the closest way, interaction appears as the mutual causality of presupposed, mutually conditioning substances; each is, relative to the other, both an active and a 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 modern natural science ... Thus, natural science confirms that ... 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 know behind it. Nevertheless, having formally dealt with the root cause of variability, none of the bright heads of the 19th century began to rebuild the building of natural science.As a result, the building remained the same - with a 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 the derivative of a function- as a mathematical model describing " amount of instantaneous change" of an object from an "infinitely small" change in its causal parameter. A theory of interactions that combines even the known four fundamental interactions (electromagnetic, gravitational, strong, weak) has not yet been created. Now it’s already “mowed up” much more and “jambs” are crawling out everywhere. Practice - the criterion of truth, completely breaks all the theoretical models built on such a building that claim to be universal and global. Therefore, all the same, it will be necessary to rebuild the building of natural science, because there is nowhere else to “swim”, science has long been developing by the “poke” method - stupidly, costly and inefficient. The physics of the future, the physics of the 21st century and subsequent centuries, must 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 foundation is provided for the basic concepts and relationships of modern physics and mathematics, and only in this case is the root formula"causa finalis" (final first cause) formula to substantiate all the basic formulas that work in practice. The meaning of world constants and much more is clarified. And it's me to you dear reader, I will show you now.

So, formulation of the problem.

Let's outline in in general terms model. Let an abstract object of cognition, cognizable in size and nature (we denote it -u) is a relative whole having a definite nature (dimension) and magnitude. The object and its properties are a causal system. An object depends in value on the value 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 by - u. In mathematics, such a causal 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 - a relative integer. Moreover, the objectively determined cognized value of the whole (number) is a relative value obtained as a relation to its individual part (to some objective generally accepted single standard of the whole - u at, A single 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 relation to the unit part (standard) of the parameter (property) -x= i* x this. The dimensions of the integer and the dimension of the parameter and their unit standards are not identical. Odds k , iare numerically equal to u, x, respectively, since the reference values ​​of u andx thisare single. As a result of interactions, the parameter changes and this causal change consequently entails a change in the function (relative whole, object, system).

Required to define formal the general dependence of the magnitude of the change of the object on the interactions - the reasons for this change. This statement of the problem reflects the true, causal, causal (according to F. Bacon) consistent, approach interaction physics.

Decision and consequences.

Interaction is a common evolutionary mechanism - the cause of variability. What is actually an interaction (short-range, long-range)? Because the general theory interaction and a theoretical model of the interaction of objects, carriers of commensurate properties in natural science is still missing, 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 immediately and now, then we will manage with only brief, but strict and necessary conclusions from this work for understanding the essence of the derivative.

"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 carriers of commensurate properties, will interact. And at this moment they will interact only with their two proportionate properties.

« Any (linear, non-linear) change of 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 non-linearly (uniformly or unevenly). At the same time, in each elementary, single event-interaction (close interaction), the property changes linearly because it is due to the only reason for the change - an elementary commensurate interaction (and therefore there is a function of one variable). ... Accordingly, any change (linear or non-linear), as a result of interactions, can be represented as the sum of elementary linear changes following in formal space and time linearly or non-linearly.”

« For the same reason, any interaction can be decomposed into change quanta (indivisible linear pieces). An elementary quantum of any nature (dimension) is the result of an elementary event-interaction according to a given nature (dimension). The magnitude and dimension of a 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 momenta (commensurate properties). A 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 the angular momentum. If balls with fixed momentum values ​​interact, then the state of the angular momentum value of each ball on any observed interval of interaction 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, let's take a linear, one-coordinate) has a value that can be expressed by writing

(1)

where is the dimension.

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

In turn, the modulus of change , denoted in (1) as , can be expressed, taking into account interaction events, as

(2)

physical meaning This basic for a huge number of the most famous relations of natural science, the root formula, is that on the interval of time and on the interval of a homogeneous linear (single-coordinate) space, there were - commensurate events-short-range 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 some . 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 about some constant, linear, average value of elementary change - derivative value on the magnitude of the change , a formally described function that is characteristic of the interaction medium and characterizes the environment and the interaction process of a certain nature (dimension). Considering that there may be different kinds distribution functions of events in space and time , then there are variable space-time dimensions y as an integral of distribution functionsevents in time and space , namely [time - t] and[ coordinate - x ] can be to the power of k(k - not equal to zero).

If we designate, in a sufficiently homogeneous environment, the value of the average time interval between events - , and the value of the average distance interval 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 well-known in mathematics "theorem of the mean". Let's rewrite (2) taking into account (3)

(4) - for time ratios;

(5) - for spatial relationships.

From these equations (3-5) it follows common law interactions:

the value 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 sequence in time and space of these events.

We got general basic ratios 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 infinitely small dt and dx are not used for the same reason. Instead of them, finite Δti and Δxi . From these generalizations (2-6) follow:

- the general physical meaning of the derivative (differential) (4) and gradient (5), as well as "world" constants, as the values ​​of the averaged (average) linear change of the function (object) with a single event-interaction of the argument (property) having a certain dimension ( nature) with proportionate (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 causal dependence of the object on its property.

; (7) - derivative of the function

; (8) - function gradient

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

; (9)

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

(10)

(10.1)

Formula (10.1) is actually 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 the 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,

has the form

(12)

Represents the decomposition of a function (formal causal system) into a series in which its change is equal to

is decomposed into components, according to the principle of decomposition of 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 the Taylor formula . So, for example, the first term of Taylor's formula identifies the change in linearly following events in time (space).

At . Second at non-linear 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 a value (coordinates, paths), with linearly following events.

(13)

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

(14)

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

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

- physical meaning of acceleration- , as a value numerically equal to a single displacement , with a non-linear succession of events-interactions that cause this displacement . Wherein, or . At the same time, the total displacement in the case of non-linear succession of events (with a linear change in the rate of succession of events) for equals (15) - formula known from school bench

- the physical meaning of the free fall acceleration 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 non-linear number of subsequent events-interactions with the environment in formal time, causing this force.

Accordingly, a value equal to the number non-linear following events, or relation - received the name body weight , and the value - body weight , as the forces 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 structured causally from any interactions at all. And physics knows the facts of changes in the mass of bodies during the course of certain reactions (physical interactions) inside them. For example, during radioactive decay, the total mass of matter 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 called a gravitational field) does not change. And this means that the force acting on the body does not change, and the inertial mass is proportional to the number of events occurring objects of the body and objects of the environment, equal to the ratio of the force to its constant acceleration .

When a body moves in a gravitational field (falls), 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 analytical identity of inertial and gravitational mass. When a body moves non-linearly, but horizontally to the Earth's surface (along the spherical equipotential surface of the Earth's gravitational field), then the gravitational field has no gradient 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 horizontal movement, the reason for the movement of the body simply changes. And a non-linearly changing number of events gives acceleration to the body and (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 is called momentum.

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

(16)

- The physical meaning of electric charge object introduced into the field, as the ratio of the force acting on the "charged" object (Lorentz force) 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 proportionate properties of the object introduced into the field and the object of the field. The interaction is expressed in the change of these proportionate properties of both. As a result of each single interaction, objects exchange 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 an interval of space. But in modern physics, the field, a special kind of matter, unfortunately, does not have a charge (it does not have charge carrier objects), but has a different characteristic - the tension on the interval (the difference in potentials (charges) in a certain void). Thus, charge in its magnitude 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 the positive charge of the object– is seen as a charge exceeding in absolute value (larger) the charge of the field point, and negative - less than the charge of the field point. This implies the difference in the signs of the forces of repulsion and attraction. Which determines the presence of a direction for 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 magnitude of the field strength. The magnitude of the charge, in accordance with the concept of number (value), is a relationship with a reference, unit, trial charge -. From here . When the charge moves, when the events follow linearly (the field is homogeneous), the integrals , and when the homogeneous field moves relative to the charge . Hence the known relations of physics ;

- The physical meaning of the electric field strength, as the ratio of the force acting on the charged object to the number of ongoing events-interactions of the charged object with the charged medium. There is a constant characteristic of the electric field. It is also the derivative with respect to the coordinate of the Lorentz force.Electric field strength- this 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).

With regard to the change in the magnitude of the charge

(19)

(20)

(21)

Where is called the potential of the field point and it is taken as the energy characteristic of a given field point, but in fact it is the charge of the field point, which differs by a factor of the test (reference) charge. Or: . During the interaction of the charge introduced into the field and the charge of the point of the field, an exchange of commensurate properties - charges occurs. Exchange is a phenomenon described as “the Lorentz force acts on the charge introduced into the field”, equal in absolute value to the magnitude of the change in charge, as well as to the magnitude of the relative change in the potential of the field point. When a charge is introduced into the Earth's field, last 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 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 points).

* Until now, in physics, it is believed that if: a conductor has a cross section of 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 the same direction with an average speed v, then in time all the particles enclosed in the volume under consideration will pass through the cross section 2. Therefore, the current strength is

.

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 true, because there are much more charged particles (events) in a conductor than, for example, electrons in a metal . Dependency will be rewritten in the form , therefore, in Once again the validity of (3-6) and other generalizations of this work is confirmed.

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

. (22)

Otherwise, one can write Ohm's law by rightly equating

. (23)

Where in this case is the resistance, showing the number of events required to change the magnitude of the 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. From this it follows that the current is a time derivative of the quantity and the concept of voltage. It should be remembered that in SI units, electrical conductivity is expressed in Siemens with the dimension: Cm \u003d 1 / Ohm \u003d Ampere / Volt \u003d kg -1 m -2 s ³A². Resistance in physics is the reciprocal of the product of electrical conductivity (resistance of a unit section of the 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. Empirically, it was found that the ratio of the maximum value of the modulus of force acting on a current-carrying conductor (Ampère 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 as a 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, we first of all increase the number of events-interactions of charged objects of the frame and charged objects of the field. From this follows 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 - there are no interactions - there is no current. W swirl (change) field - the current went 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 \u003d K -273.15 (the temperature of the triple point of water is 0.01 ° C).
In 2005, the definition of kelvin was refined. In the mandatory Technical Annex to the ITS-90 text, the Advisory Committee on Thermometry established the requirements for the isotopic composition of water at the implementation of the temperature of the triple point of water.

Nevertheless, physical meaning and essence of the concept of temperature much easier and clearer. Temperature, in its essence, is a consequence of events-interactions occurring inside the substance that have both "internal" and "external" causes. More events - more temperature, fewer events- lower temperature. Hence the phenomenon of temperature change 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 the body determines its temperature state, and this idea (which was first developed by Boltzmann) that a certain temperature state of the body is not at all determined by the energy of movement, but by the randomness of this movement , and is that new concept in the description of temperature phenomena, which we must use ... " (Report of the 1978 Nobel Prize winner Petr Leonidovich Kapitsa "Properties of liquid helium", read at the conference "Problems modern science"at Moscow University on December 21, 1944)
Under the measure of chaos one should understand the quantitative characteristic of the number event-interactions per unit 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 (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, the kelvin will be expressed in terms of the 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 during 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 causal structure of temperature - events-interactions. As a result of events-interactions, objects in each event exchange kinetic energy (moments of impulses 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, the energy E has a different dimension (nature). How many natures, so many energies. For example:

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

Pressure times volume (E ≈ P V≈N*m 3 /m 2 ≈N*m);

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

Mass times the square of the 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, causing 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 a 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 magnitude of the change in relation to the standard and the dimension of the standard, and informally depend on the nature of the succession of events.

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

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

These ratios, in fact, are common for 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 (unlike force, not correlated with the magnitude of the interval, but the supposed presence of an interval of invariance of the action), which has an actual time interval (interval of space) of its invariance until the next event.

An interval is the difference between two points in time of the beginning of this and the next comparable 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 property change, which had some time of invariance until the next event-interaction;

- the 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 about the interpretation of this basic concept of natural science.

- physical meaning of coordinate space , as values ​​(measures) of change (paths, distances),

(32)

which has the dimension of a formal, unitary 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 total coordinate standards on the interval . When measuring the coordinate, for convenience, a linearly changing integrand a function , the integral of which is equal to the number of formally chosen reference intervals of unit coordinates ;

- physical meaning of all basic physical properties(parameters) characterizing the properties of a 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 original form of notation and a single methodologically uniform causal meaning. And this causal meaning is acquired with the introduction of a global physical principle - "event-interaction" into natural science.

Here, dear reader, what should be in the most general terms a new mathematics endowed with physical meaning and certainty And new interaction physics of the 21st century , cleared of a swarm of irrelevant, not having certainty, size and dimension, and hence common sense concepts. Such, for example, How classical derivative and instantaneous velocity - having little in common with physical concept speed. How concept of inertia - a certain ability of bodies to maintain speed ... How inertial reference system (ISO) , which has nothing to do with the concept of a frame of reference(CO). For ISO, unlike the usual reference frame of reference (CO) is not an objective system of knowledge of the magnitude of movement (change). Relative to ISO, by its definition, bodies only rest or move in a straight line 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 causally describe with general dependencies numerous phenomena of the universe, existing and changing according to the uniform laws of nature.

1. Literature.

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

2. Hegel G.W.F. , Soch., vol. 5, M., 1937, p. 691.

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

1.1 Some problems of physics 3

2. Derivative

2.1 Function change rate 6

2.2 Derivative 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 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 used literature 32

1. Introduction

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

Let us analyze such a phenomenon as the speed of movement and related concepts.

Let the body move in a straight line and we know the distance , passed by the body for each given time , i.e. 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
- movement schedule.

Consider the motion of the body during the time interval
from some moment until the moment
. In time, the body has traveled a path, and in time, a path
. So, in units of time it has traveled a distance

.

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

In this case
, and the relation
shows how many units of the path are per unit of time; at the same time, it remains constant, regardless of what moment in time is taken, not on what increment of time is taken . It's a permanent attitude called uniform speed.

But if the motion is uneven, then the ratio depends

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

During this interval of time, with the same distance traveled, movement can occur in the most diverse ways; graphically, this is illustrated by the fact that between two points on the plane (points
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 goes through a straight line
, which is the graph of the uniform in the interval
movement. So the average speed shows how fast you need to move uniformly in order to pass in the same time interval the same distance
.

Leaving the same , let's decrease . Average speed calculated for the changed interval
, lying inside the given interval, can, of course, be different than in; throughout the interval . It follows from this that the average speed cannot be considered as a satisfactory characteristic of the 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 go to zero. If at the same time there is a limit of average speed, then it is taken as the speed of movement in this moment .

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 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 motion (fall).

2. Derivative

The rate of change of the function. Derivative 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,

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

Let us therefore analyze general view question about the rate of change of a function
, abstracting from the physical meaning of the variables
.

Let first
- linear function:

.

If the independent variable gets an increment
, then the function gets increment here
. Attitude
remains constant, independent of which function is considered, nor of which one is taken .

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

also depends on , and from . This ratio 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 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 on 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. Function change rate Vgiven point is called the limit of the average rate of change of the function in the interval when going to zero:

2.2 Derivative function. Function change rate

determined by the following sequence of actions:

1) by increment , assigned to this value , find the corresponding increment of the function

;

2) a relation is drawn up;

3) find the limit of this ratio (if it exists)

with an arbitrary tendency to zero.

As already noted, if this function not linear

then the relation 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., will be a constant value.

This limit is called derivative of a function or simply function derivative and is marked like this:
.Read: "ef stroke from » or "ef prim from".

Definition. derivative of 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 aspiration, 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 the derivative of

functions By (derivative of the path with respect to time).

2.3 Derivative of a power function.

Let us find derivatives of some simple functions.

Let
. We have

,

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

If
, That

Let
, Then

It is easy to notice a pattern in the expressions for derivatives of a 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 Newton binomial formula :

On the right side of the last equality is the sum of the 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
the formulas derived above follow from the found general formula.

This result is true for any indicator, for example:

.

Consider now separately the derivative of the constant

.

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 clear 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 limit position of the line passing through the point, and another point
lines when this point tends to merge with the 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...

Limit and continuity of a function

Study

The physical meaning of the derivative. The derivative characterizes speedchanges one physical quantity relative to ... . At what value of the argument are equal speedchangesfunctions and Decision. , and, and. Using the physical meaning of the derivative...

The concept of a function of one variable and methods for specifying functions

Document

The concept of differential calculus characterizing speedchangesfunctions; P. is function, defined for each x ... continuous derivative (differential calculus characterizing speedchangesfunctions at this point). Then and...

§ 5 Partial derivatives of complex functions differentials of complex functions 1 Partial derivatives of a complex function

Document

It exists and is finite) will be speedchangesfunctions at a point in the direction of the vector. His ... and denote or. In addition to the magnitude speedchangesfunctions, allows you to determine the nature changesfunctions at a point in the direction of the vector...