Physical meaning of derivative. Instantaneous rate of change of function, acceleration and gradient

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

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

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

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

Note : numeric values The example under consideration corresponds 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 more gradual, so the increment (crimson line) is relatively small, and the ratio compared to the previous case will be very modest. Relatively speaking, meters and function growth rate is . That is, here for every meter of the path there are average half a meter of rise.

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

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

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

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

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

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

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

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

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

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

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

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

This article explains in a simple and clear way what a derivative is and why it is needed.. We will not now strive for mathematical rigor in the presentation. The most important thing is to understand the meaning.

Let's remember the definition:

The derivative is the rate of change of a 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 jobs at the same time. Let's see how their income changed during the year:

The graph shows everything at once, isn’t it? Kostya’s income more than doubled in six months. And Grisha’s income also increased, but just a little. And Matvey’s income decreased to zero. The starting conditions are the same, but the rate of change of the function, that is derivative, - different. As for Matvey, his income derivative is generally negative.

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

What we're really looking at is how steeply the graph of a function goes up (or down). In other words, how quickly does y change as x changes? Obviously, the same function at different points can have different meaning derivative - that is, it can change faster or slower.

The derivative of a function is denoted .

We'll show you how to find it using a graph.

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

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

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

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

We found the derivative using a graph without even knowing the formula of the function. Such problems are often found in the Unified State Examination 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 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 the function at a point is equal to slope tangent drawn to the graph of the function at this point.

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

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 with at different speeds. And let this function have maximum and minimum points.

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

At the point our function decreases. 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.

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 tangent at these points is zero, and the derivative is also zero.

Point - 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 the derivative we can find out everything that interests us about the behavior of a function.

If the derivative is positive, then the function increases.

If the derivative is negative, then the function decreases.

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:

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.

It is possible that 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 is the 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 remains 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.

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

1.1 Some problems of physics 3

2. Derivative

2.1 Rate of change function 6

2.2 Derivative function 7

2.3 Derivative of a power function 8

2.4 Geometric meaning of derivative 10

2.5 Differentiation of functions

2.5.1 Differentiation of results arithmetic operations 12

2.5.2 Differentiation of complex and inverse functions 13

2.6 Derivatives of parametrically defined functions 15

3. Differential

3.1 Differential and its geometric meaning 18

3.2 Differential properties 21

4. Conclusion

4.1 Appendix 1. 26

4.2 Appendix 2. 29

5. List of references 32

1. Introduction

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

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

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

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

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

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

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

But if the movement is uneven, then the ratio depends

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

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

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

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

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

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

.

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

and for the speed at the moment

.

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

2. Derivative

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

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

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

Let us therefore examine in general 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 increment
, then the function gets incremented here
. Attitude
remains constant, independent of the way in which the function is considered, and of what is taken .

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

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

Definition.Attitude calledaverage speed function changes in interval
.

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

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

2.2 Derivative function. Rate of change of function

determined through the following sequence of actions:

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

;

2) a relation is drawn up;

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

as it arbitrarily tends to zero.

As already noted, if this function not linear,

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

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

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

.

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

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

1) The speed of rectilinear motion is the derivative of

functions By (time derivative of the path).

2.3 Derivative of a power function.

Let's find derivatives of some simple functions.

Let
. We have

,

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

If
, That

Let
, Then

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

.

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

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

.

So, power function when the integer is positive, it has a derivative equal to:

.

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

This result is true for any indicator, for example:

.

Let us now consider separately the derivative of a constant quantity

.

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

,

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

2.4 Geometric meaning of the derivative.

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

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

.Tutorial

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

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

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

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

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

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

Rate of change of function relative 124.188

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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