The geometric meaning of the derivative is associated with. What is a derivative? Definition and meaning of a derivative function. I. Organizational moment

Derivative(functions at a point) - basic concept differential calculus characterizing the rate of change of the function (at a given point). Defined as limit the ratio of the increment of a function to its increment argument when trying to increment the argument to zero if such a limit exists. A function that has a finite derivative (at some point) is called differentiable (at a given point).

The process of calculating the derivative is called differentiation. Reverse process - finding primitive - integration.

If a function is given by a graph, its derivative at each point is equal to the tangent of the slope of the tangent to the graph of the function. And if the function is given by a formula, the table of derivatives and the rules of differentiation will help you, that is, the rules for finding the derivative.

4. Derivative of a complex and inverse function.

Let now given complex function , i.e. a variable is a function of a variable, and a variable is, in turn, a function of an independent variable.

Theorem . If And  differentiable functions of its arguments, then a complex function is a differentiable function and its derivative is equal to the product of the derivative of the given function with respect to the intermediate argument and the derivative of the intermediate argument with respect to the independent variable:

.

The assertion is easily obtained from the obvious equality (valid for and ) passing to the limit at (which, due to the continuity of the differentiable function, implies ).

Let's move on to the consideration of the derivative inverse function.

Let a differentiable function on a set have a set of values ​​and on the set there exists inverse function .

Theorem . If at the point derivative , then the derivative of the inverse function at the point exists and is equal to the reciprocal of the derivative of the given function: , or

This formula is easily obtained from geometric considerations.

T as there is a tangent of the angle of inclination of the tangent line to the axis, that is, the tangent of the angle of inclination of the same tangent (the same line) at the same point to the axis.

If they are sharp, then , and if they are blunt, then .

In both cases . This equality is equivalent to the equality

5.Geometric and physical meaning of the derivative.

1) The physical meaning of the derivative.

If the function y = f(x) and its argument x are physical quantities, then the derivative is the rate of change of the variable y relative to the variable x at a point. For example, if S \u003d S (t) is the distance traveled by a point in time t, then its derivative is the speed at the time. If q = q(t) is the amount of electricity flowing through the cross section of the conductor at time t, then is the rate of change in the amount of electricity at time, i.e. current strength at a time.

2) The geometric meaning of the derivative.

Let be some curve, be a point on the curve.

Any line that intersects at least two points is called a secant.

The tangent to the curve at the point is the limiting position of the secant if the point tends to, moving along the curve.

It is obvious from the definition that if a tangent to a curve exists at a point, then it is unique.

Consider the curve y = f(x) (i.e., the graph of the function y = f(x)). Let at the point it has a non-vertical tangent. Its equation is: (the equation of a straight line passing through a point and having a slope k).

By definition of the slope coefficient , where is the angle of inclination of the straight line to the axis.

Let be the angle of inclination of the secant to the axis, where. Since is tangent, then

Hence,

Thus, we have obtained that is the slope of the tangent to the graph of the function y = f(x) at the point (geometric meaning of the derivative of a function at a point). Therefore, the equation of the tangent to the curve y = f(x) at the point can be written in the form

Definition of a derivative. Her physical meaning. Definition of a differentiable function. Formulate a theorem on the relationship between differentiability and continuity of a function.

Derivative - the basic concept of differential calculus, which characterizes the rate of change of a function.

Derivative is the limit of the ratio of the increment of the function to the increment of its argument as the increment of the argument tends to zero, if such a limit exists.

A function that has a finite derivative is called differentiable.
The process of calculating the derivative is called differentiation

If the position of a point during its movement along a number line is given by the function S= f(t), Where t is the time of motion, then the derivative of the function S- instantaneous speed of movement at a moment of time t. By analogy with this model, they generally say that the derivative of the function at= f(x) – rate of change functions at the point X.

Theorem(a necessary condition for the function to be differentiable). If a function is differentiable at a point, then it is continuous at that point.

Proof. Let the function y=f(x) differentiable at a point X 0 . Let's give an increment to the argument at this point Dx. The function will be incremented Do. Let's find .

Hence, y=f(x) continuous at point X 0 .

Consequence. If X 0 is a discontinuity point of the function, then the function is not differentiable at it.

Statement, converse theorem, not true. Continuity does not imply differentiability.

Example. y=|x| , X 0 = 0.

Dx> 0, ;

Dx< 0, .

At the point X 0 = 0 the function is continuous, but the derivative does not exist.

The geometric meaning of the derivative. Tangent and normal equations

The geometric meaning of the derivative. Consider the graph of the function y= f (x):

It can be seen from Fig. 1 that for any two points A and B of the graph of the function:

Where is the angle of inclination of the secant AB.

Thus, the difference ratio is equal to the slope of the secant. If we fix point A and move point B towards it, then it decreases indefinitely and approaches 0, and the secant AB approaches the tangent AC. Therefore, the limit of the difference ratio is equal to the slope of the tangent at point A. It follows from this: the derivative of a function at a point is the slope of the tangent to the graph of that function at that point. This is what it consists geometric meaning derivative.

Tangent equation. Let us derive the equation of the tangent to the graph of the function at the point A ( x 0 , f (x 0)). In the general case, the equation of a straight line with a slope f ’(x 0) has the form:

y = f ’(x 0) · x + b .

To find b, we use the fact that the tangent passes through the point A:

f (x 0) = f ’(x 0) · x 0 +b,

from here b = f (x 0) – f ’(x 0) · x 0 , and substituting this expression for b, we will get tangent equation:

y =f (x 0) + f ’(x 0) · ( x-x 0) .

normal to the graph of the function y = f (x) at point A ( x 0 ; y 0) is called a line passing through the point A and perpendicular to the tangent to that point. It is given by the equation

which follows from the property of slope coefficients perpendicular each other direct.

In the case of an infinite derivative, the tangent at the point x 0 becomes vertical and is given by the equation x = x 0 , and the normal is horizontal: y = y 0 .

Function derivative.

1. Definition of the derivative, its geometric meaning.

2. Derivative of a complex function.

3. Derivative of the inverse function.

4. Derivatives of higher orders.

5. Parametrically predefined functions and implicitly.

6. Differentiation of functions given parametrically and implicitly.

Introduction.

source differential calculus there were two issues raised by the demands of science and technology in the 17th century.

1) The question of calculating the speed for an arbitrarily given law of motion.

2) The question of finding (with the help of calculations) a tangent to an arbitrarily given curve.

The problem of drawing a tangent to some curves was solved by the ancient Greek scientist Archimedes (287-212 BC), using the drawing method.

But only in the 17th and 18th centuries, in connection with the progress of natural science and technology, these issues were properly developed.

One of important issues when studying any physical phenomenon usually the question is about the speed, the speed of the occurring phenomenon.

The speed at which an aircraft or car is moving is always the most important indicator of its performance. The rate of population growth of a given state is one of the main characteristics of its social development.

The original idea of ​​speed is clear to everyone. However, this general idea is not enough to solve most practical problems. It is necessary to have such a quantitative definition of this quantity, which we call speed. The need for such accurate quantification has historically been one of the main motivators for the creation of mathematical analysis Yu. A whole section of mathematical analysis is devoted to the solution of this basic problem and the conclusions from this solution. We now turn to the study of this section.

Definition of the derivative, its geometric meaning.

Let a function defined in some interval be given (a, c) and continuous in it.

1. Let's give an argument X increment , then the function will get

increment :

2. Compose a relation .

3. Passing to the limit at at and, assuming that the limit

exists, we get the value , which is called

derivative of a function with respect to the argument X.

Definition. The derivative of a function at a point is the limit of the ratio of the increment of the function to the increment of the argument when →0.

The value of the derivative obviously depends on the point X, in which it is found, so the derivative of the function is, in turn, some function of X. Designated .

By definition, we have

or (3)

Example. Find the derivative of the function .

1. ;

Lesson Objectives:

Students should know:

• what is called the slope of a straight line;
• the angle between the line and the x-axis;
• what is the geometric meaning of the derivative;
• the equation of the tangent to the graph of the function;
• a method for constructing a tangent to a parabola;
• be able to apply theoretical knowledge in practice.

Lesson objectives:

Educational: to create conditions for students to master the system of knowledge, skills and abilities with the concepts of the mechanical and geometric meaning of the derivative.

Educational: to form a scientific worldview in students.

Developing: to develop students' cognitive interest, creativity, will, memory, speech, attention, imagination, perception.

Methods of organizing educational and cognitive activities:

• visual;
• practical;
• on mental activity: inductive;
• according to the assimilation of the material: partially exploratory, reproductive;
• by degree of independence: laboratory work;
• stimulating: encouragement;
• control: oral frontal survey.

Lesson Plan

1. Oral exercises (find the derivative)
2. Student's report on the topic “The reasons for the appearance of mathematical analysis”.
3. Learning new material
4. Phys. Minute.
5. Problem solving.
6. Laboratory work.
7. Summing up the lesson.
8. Commenting on homework.

Equipment: multimedia projector (presentation), cards ( laboratory work).

“A person achieves something only where he believes in himself”

L. Feuerbach

The organization of the class throughout the lesson, the readiness of students for the lesson, order and discipline.

Setting learning goals for students, both for the entire lesson and for its individual stages.

Determine the significance of the material being studied both in this topic and in the entire course.

Verbal counting

1. Find derivatives:

" , ()" , (4sin x)", (cos2x)", (tg x)", "

2. Logic test.

a) Insert the missing expression.

The general direction of the development of science is ultimately determined by the requirements of the practice of human activity. The existence of ancient states with a complex hierarchical system of government would have been impossible without a sufficient development of arithmetic and algebra, because the collection of taxes, the organization of army supplies, the construction of palaces and pyramids, the creation of irrigation systems required complex calculations. During the Renaissance, ties between various parts of the medieval world expanded, trade and crafts developed. A rapid rise in the technical level of production begins, new sources of energy are being used industrially, not connected with the muscular efforts of humans or animals. In the XI-XII centuries, fullers and looms appeared, and in the middle of the XV - a printing press. In connection with the need for the rapid development of social production during this period, the essence of the natural sciences, which have been descriptive since antiquity, changes. The goal of natural science becomes an in-depth study of natural processes, not objects. The descriptive natural science of antiquity corresponded to mathematics, which operated with constant values. It was necessary to create a mathematical apparatus that would describe not the result of the process, but the nature of its flow and its inherent patterns. As a result, by the end of the 12th century, Newton in England and Leibniz in Germany completed the first stage in the creation of mathematical analysis. What is "mathematical analysis"? How can one characterize and predict the features of any process? Use these features? To penetrate deeper into the essence of this or that phenomenon?

Let's go along the path of Newton and Leibniz and see how we can analyze the process, considering it as a function of time.

Let us introduce some notions that will help us further.

The graph of the linear function y=kx+ b is a straight line, the number k is called the slope of the straight line. k=tg, where is the angle of a straight line, that is, the angle between this straight line and the positive direction of the Ox axis.

Picture 1

Consider the graph of the function y \u003d f (x). Draw a secant through any two points, for example, the secant AM. (Fig.2)

The slope of the secant k=tg. In a right triangle AMC<МАС = (объясните почему?). Тогда tg = = , что с точки зрения физики есть величина средней скорости протекания любого процесса на данном промежутке времени, например, скорости изменения расстояния в механике.

Figure 2

Figure 3

The term “speed” itself characterizes the dependence of a change in one quantity on a change in another, and the latter does not have to be time.

So, the tangent of the slope of the secant tg = .

We are interested in the dependence of the change in values ​​in a shorter period of time. Let us tend the increment of the argument to zero. Then the right side of the formula is the derivative of the function at point A (explain why). If x -> 0, then point M moves along the graph to point A, which means that line AM approaches some line AB, which is tangent to the graph of the function y \u003d f (x) at point A. (Fig.3)

The angle of inclination of the secant tends to the angle of inclination of the tangent.

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

The mechanical meaning of the derivative.

The tangent of the slope of the tangent is a value that shows the instantaneous rate of change of the function at a given point, that is, a new characteristic of the process under study. Leibniz called this quantity derivative, and Newton said that the instantaneous speed.

No. 91(1) page 91 - show on the board.

The slope of the tangent to the curve f (x) \u003d x 3 at the point x 0 - 1 is the value of the derivative of this function at x \u003d 1. f '(1) \u003d 3x 2; f'(1) = 3.

No. 91 (3.5) - under dictation.

No. 92 (1) - on the board at will.

No. 92 (3) - independently with oral verification.

No. 92 (5) - at the board.

Answers: 45 0, 135 0, 1.5 e 2.

Purpose: development of the concept of “mechanical meaning of the derivative”.

Applications of the derivative to mechanics.

The law is given rectilinear motion points x = x(t), t.

1. The average speed of movement in the specified period of time;
2. Velocity and acceleration at time t 04
3. stopping points; whether the point continues to move in the same direction after the moment of stopping or starts moving in the opposite direction;
4. The highest speed of movement for a specified period of time.

The work is performed according to 12 options, the tasks are differentiated by the level of complexity (the first option is the lowest level of complexity).

Before starting work, a conversation on the following questions:

1. What is the physical meaning of displacement derivative? (Speed).
2. Can you find the derivative of speed? Is this quantity used in physics? What is it called? (Acceleration).
3. Instant Speed equals zero. What can be said about the movement of the body at this moment? (This is the stopping point).
4. What is the physical meaning of the following statements: the derivative of motion is equal to zero at the point t 0; does the derivative change sign when passing through the point t 0? (The body stops; the direction of movement changes to the opposite).

Sample work for students.

x (t) \u003d t 3 -2 t 2 +1, t 0 \u003d 2.

Figure 4

In the opposite direction.

Let's draw a schematic speed graph. The highest speed is reached at the point

t=10, v (10) =3 10 2 -4 10 =300-40=260

Figure 5

1) What is the geometric meaning of the derivative?
2) What is the mechanical meaning of the derivative?
3) Make a conclusion about your work.

Page 90. No. 91 (2,4,6), No. 92 (2,4,6,), p. 92 No. 112.

Used Books

• Textbook Algebra and the beginning of analysis.
  Authors: Yu.M. Kolyagin, M.V. Tkacheva, N.E. Fedorova, M.I. Shabunin.
  Edited by A. B. Zhizhchenko.
• Algebra 11th grade. Lesson plans according to the textbook by Sh. A. Alimov, Yu. M. Kolyagin, Yu. V. Sidorov. Part 1.
• Internet resources: http://orags.narod.ru/manuals/html/gre/12.jpg

For clarification geometric value derivative, consider the graph of the function y = f(x). Take an arbitrary point M with coordinates (x, y) and a point N close to it (x + $\Delta $x, y + $\Delta $y). Let us draw the ordinates $\overline(M_(1) M)$ and $\overline(N_(1) N)$, and draw a line parallel to the OX axis from the point M.

The ratio $\frac(\Delta y)(\Delta x) $ is the tangent of the angle $\alpha $1 formed by the secant MN with the positive direction of the OX axis. As $\Delta $x tends to zero, the point N will approach M, and the tangent MT to the curve at the point M will become the limiting position of the secant MN. Thus, the derivative f`(x) is equal to the tangent of the angle $\alpha $ formed by the tangent to curve at the point M (x, y) with a positive direction to the OX axis - the slope of the tangent (Fig. 1).

Figure 1. Graph of a function

When calculating the values ​​using formulas (1), it is important not to make a mistake in the signs, because increment can be negative.

The point N lying on the curve can approach M from any side. So, if in Figure 1, the tangent is given the opposite direction, the angle $\alpha $ will change by $\pi $, which will significantly affect the tangent of the angle and, accordingly, the slope.

Conclusion

It follows that the existence of the derivative is connected with the existence of a tangent to the curve y = f(x), and the slope -- tg $\alpha $ = f`(x) is finite. Therefore, the tangent must not be parallel to the OY axis, otherwise $\alpha $ = $\pi $/2, and the tangent of the angle will be infinite.

At some points, a continuous curve may not have a tangent or have a tangent parallel to the OY axis (Fig. 2). Then the function cannot have a derivative in these values. There can be any number of such points on the function curve.

Figure 2. Exceptional points of the curve

Consider Figure 2. Let $\Delta $x tend to zero from negative or positive values:

\[\Delta x\to -0\begin(array)(cc) () & (\Delta x\to +0) \end(array)\]

If in this case relations (1) have a finite aisle, it is denoted as:

In the first case, the derivative on the left, in the second, the derivative on the right.

The existence of a limit speaks of the equivalence and equality of the left and right derivatives:

If the left and right derivatives are not equal, then at this point there are tangents that are not parallel to OY (point M1, Fig. 2). At points M2, M3, relations (1) tend to infinity.

For N points to the left of M2, $\Delta $x $

To the right of $M_2$, $\Delta $x $>$ 0, but the expression is also f(x + $\Delta $x) -- f(x) $

For point $M_3$ on the left $\Delta $x $$ 0 and f(x + $\Delta $x) -- f(x) $>$ 0, i.e. expressions (1) are both positive on the left and right and tend to +$\infty $ both when $\Delta $x approaches -0 and +0.

The case of the absence of a derivative at specific points of the line (x = c) is shown in Figure 3.

Figure 3. Absence of derivatives

Example 1

Figure 4 shows the graph of the function and the tangent to the graph at the point with the abscissa $x_0$. Find the value of the derivative of the function in the abscissa.

Solution. The derivative at a point is equal to the ratio of the increment of the function to the increment of the argument. Let's choose two points with integer coordinates on the tangent. Let, for example, these be points F (-3.2) and C (-2.4).