This lesson is a learning lesson new topic. The presented lesson development reveals methodological approaches to introducing the concept complex function, an algorithm for calculating its derivative. The development is intended for conducting lessons among first-year students of vocational education institutions.

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Derivative of a complex function

Goals: 1) educational - formulate the concept of a complex function, study the algorithm for calculating the derivative of a complex function, show its application in calculating derivatives.

2) developing - to continue developing the skills to reason logically and reasonedly, using generalizations, analysis, comparison when studying the derivative of a complex function.

3) educational - to cultivate observation in the process of finding mathematical dependencies, to continue the formation of self-esteem when implementing differentiated learning, and to increase interest in mathematics.

Equipment: table of derivatives, presentation for the lesson.

Lesson outline:

I. AZ.

1. Mobilizing beginning (setting the goal of work in the lesson).

2. Oral work for the purpose of updating background knowledge.

3. Checking homework to motivate learning new material.

4. Summing up the results of the first stage and setting tasks for the next one.

II. FNZ and SD.

1. Heuristic conversation to introduce the concept of a complex function.
2. Oral frontal work in order to consolidate the definition of a complex function.
3. Teacher's message about the algorithm for calculating the derivative of a complex function.
4. Primary fixation of the algorithm for calculating the derivative of a complex function frontally.
5. Summing up the results of stage II and setting tasks for the next one.

III. FUN.

1. Solving a problem based on an algorithm for calculating the derivative of a complex function frontally at the board by a student.

2. Differentiated work on solving problems, followed by checking frontally at the board.

3. Summing up the lesson

4. Handing out homework.

During the classes.

I AZ

1. The outstanding Russian mathematician and shipbuilder Academician Alexei Nikolaevich Krylov (1863-1945) once noted that a person turns to mathematics “not to admire innumerable treasures. First of all, he needs to become familiar with centuries-old proven instruments and learn to use them correctly and skillfully.” We have become acquainted with one of these tools – this is a derivative. Today in class we continue to study the topic “Derivative” and our task is to consider the new question “Derivative of a complex function”, i.e. We will find out what a complex function is and how its derivative is calculated.

2. Now let's remember how the derivative of various functions is calculated. To do this you must complete 7 tasks. For each task, answer options are offered, encrypted in letters. The correct solution to each task allows you to open the desired letter of the surname of the scientist who introduced the designation y" , f " (x).

Find the derivative of the function.

1) y = 5 y " = 0 L

Y" = 5x N

Y" = 1 B

2) y = -x y " = 1 V

Y" = -1 A

Y" = x 2 And

3) y = 2x+3 y " = 3 Y

Y " = x And

Y" = 2 G

4) y = - 12 y " = P

Y" = 1 T

Y" = -12 G

5) y=x 4 y "= P

Y" = 4x 3 A

y "= x 3 C

6) y=-5x 3 y "= -15x 2 N

Y" = -5x 2 O

y " = 5x 2 Р

7) y=x-x 3 y "= 1-x 2 D

Y" = 1-3x 2 F

Y" = x-3x 2 A

(Tasks on slides 2 – 3).

So, the scientist’s name is Lagrange, and we thereby repeated the calculation of derivatives of various functions.

3. One of the students fills out the table: (slide 4).

f(x)	f(1)	f" (x)	f" (1)
1) 4-x
2) 2x5		10x4
5) (4-x) 5

What questions do you have? As a result of the conversation, we come to the conclusion that we do not know how to calculate ()"; ((4-x) 3 )"

4. What is the name of the function 1), 2), 3), 4).

1) – linear, 2) power, 3) power, 4) -?, 5) -?

Now we will find out what such functions are called and how their derivatives are calculated.

II. FNZ and SD.

1. In order to do this, consider the function Z = f(x) =

What is the sequence for calculating the function values?

A) g = 4-x

B) h =

What is the relationship between g and h called?

Function

This means g and h can be represented as:

G = g(x) = 4-x

H = h(g) =

As a result of sequential execution of functions g and h for a given value x, the value of which function will be calculated?

F(x)

Z = f(x) = h(g) = h(g(x))

Thus f(x) = h(g(x)).

They say that f is a complex function made up of g and h. Function

g – internal, h – external.

In our example, 4-x is an internal function, and √ is an external one.

G(x) = 4-x

H(g) =

2. Which of them following functions are complex? In the case of a complex function, name the internal and external functions (the following functions are written on slide 8:

a) f(x) = 5x+1; b) f(x) = (3-5x) 5 ; c) f(x) = cos3x.

3. So, we found out what a complex function is. How to calculate its derivative?

Algorithm for calculating the derivative of a complex function f(x) = h(g(x)).

1. define the inner function g(x).
2. find the derivative of the internal function g"(x)
3. define the outer function h(g)
4. find the derivative of the external function h"(g)
5. find the product of the derivative of the internal function and the derivative of the external function g"(x) ∙ h"(g)

Everyone is given a monument with an algorithm.

4. Teacher at the blackboard: f(x) = (3-5x) 5

1. g(x) = 3-5x
2. g"(x) = -5
3. h(g) = g 5
4. h"(g)=5g 4
5. f "(x) = g"(x) ∙ h"(g) = -5 ∙ 5g 4 = -5 ∙ 5(3-5x) 4 = -25(3-5x) 4

5. So, we have found out what a complex function is and how its derivative is calculated.

III. FUN.

1. Now let's learn how to find derivatives of various complex functions. Performed by advanced students.

Find the derivative of the function f(x) =

1) g(x) = 4-x

2) g"(x) = -1

3) h(g) =

4) h"(g) =

5) f "(x) = g"(x) ∙ h"(g) = -1 ∙ = -

2. Find the derivative of the function:

“3” f(x) = (1 – 2x) 4

“4” f(x) = (x 2 – 6x + 5) 7

“5” f(x) = - (1 – x) 3

3. Summing up.

4. D/Z: learn the algorithm. Find the derivative.

"3" - f(x) = (2+4x) 9

"4" - f(x) =

"5" - f(x) =

Used Books:

1. Kolmogorov A.N. Algebra and the beginnings of analysis. Textbook for 10 – 11 grades. – M.: Education, 2010.

2. Ivlev B.M., Sahakyan S.M. Didactic materials on algebra and the beginnings of analysis for 10th grade. M.: Education - 2006.

3. Dorofeev G.V. “Collection of tasks for conducting a written exam in mathematics for the course high school" - M.: Bustard, 2007.

4. Bashmakov M.I. Algebra and the beginnings of analysis. Textbook for 10 – 11 grades. 2nd ed. – M.: 1992.- 351 p.

Lesson type: combined

educational:

– formation of the concept of a complex function;

Formation of the ability to find the derivative of a complex function according to the rule;

Development of an algorithm for applying the rule for finding the derivative of a complex function when solving examples.

developing:

Develop the ability to generalize, systematize based on comparison, and draw conclusions;

Develop visually effective creative imagination;

Develop cognitive interest.

educational:

Fostering a responsible attitude towards academic work, will and perseverance to achieve final results when finding derivatives of complex functions;

Formation of the ability to rationally and accurately write out a task on the board and in a notebook.

Cultivating friendly relations between students during lessons.

The student must know:

the concept of a complex function, the rule for finding its derivative.

The student must be able to:

find the derivative of a complex function according to the rule, use this rule when solving examples.

Interdisciplinary connections: physics, geometry, economics.

Lesson equipment: multimedia projector, magnetic board, blackboard, chalk, handouts for the lesson.

Lesson plan:

Communicating the purpose, objectives of the lesson and motivation for learning activities – 3 min.

1. Checking homework completion – 5 minutes (frontal check, self-control).
2. Comprehensive knowledge test – 10 min (frontal work, mutual control).
3. Preparing to learn (learn) new things educational material through repetition and updating of basic knowledge – 5 minutes (problem situation).
4. Assimilation of new knowledge – 15 minutes (frontal work under the guidance of a teacher).
5. Initial comprehension and understanding of new material - 20 minutes (front work: one student shows the solution to the example on the board, the rest solve in notebooks).
6. Consolidation of new knowledge - 15 minutes (independent work - test in two versions, with differentiated tasks).
7. Information about homework, instructions for completing it – 2 min.
8. Summing up the lesson, reflection – 5 min.

I. Lesson progress: Communicating the goals, objectives and lesson plan, motivation for learning activities:

Check the preparedness of the audience and the readiness of students for the lesson, mark those who are absent.

Please note that this lesson continues work on the topic “Derivative of a function.”

II. Checking homework.

Examples for finding the derivative of a function are given at home:

5) at point x=0.

The answers are projected onto a multimedia projector.

Students individually check their answers and give themselves a (self-control) grade on the control sheet. Each student has a control sheet, an assessment criterion for homework and a sample control sheet in the handout for the lesson

Control sheet

Call a student to the board to show the design of the solution to example No. 5 with a commentary on the actions performed.

Emphasize on correct solution and correct execution of the solution to home example No. 5.

III. Comprehensive knowledge test.

The game “Mathematical Lotto” is a test of knowledge of the rules of differentiation, tables of derivatives.

In a special envelope, each pair of students is offered a set of cards (10 cards in total). These are formula cards. There is another set of cards. These are answer cards, of which there are more, since among the answers there are false answers. The student finds the answer to the task, and with this card (answer) covers the corresponding number in a special card. Students work in pairs, so they evaluate each other, put marks on the control sheet according to the criterion: “5” - knows 9-10 formulas; “4” - knows 7-8 formulas; “3” - knows 5-6 formulas; “2” - knows less than 5 formulas.

Knowledge of formulas is being tested and assessed on a magnetic board. If the answers on the magnetic board are correct, the backs of the answer cards form a larger picture for the entire group to see. The numbers on the special card match the numbers on the formula cards. If you open the answers on the magnetic board from the reverse side, then all the cards as a whole form a picture.

IV. Preparation for (learning) the study of new educational material through repetition and updating of basic knowledge.

Statement of the problem situation: find the derivative of the function ;

In previous lessons we learned how to find derivatives elementary functions. Functions complex. Do we know how to find derivatives of complex functions?

So, what should we get to know today?

[With finding the derivative of complex functions.]

Students themselves formulate the topic and objectives of the lesson, the teacher writes the topic on the board, and the students write it in their notebooks.

Historical background, connection with future professional activities.

V. Assimilation of new knowledge.

Show on the board how to find derivatives of functions: ;

Solve examples:

3)

VI. Primary comprehension and understanding of new material.

Repeat the algorithm for finding the derivative of a complex function;

Solve examples:

2)

3)

4) ;

VII. Consolidate new knowledge using a test based on options.

Test tasks are differentiated: examples from No. 1-3 are rated at “3”, up to No. 4 – at “4”, all five examples – at “5”.

Students solve in notebooks and check each other's answers using multimedia and evaluate each other (mutual control) on the control sheet.

Option 1.

Find derivatives of functions. (A., B., S. – answers)

1
2
3
4
5
4
5

ALGEBRA

Grade 10

"Derivative of a complex function"

Subject: Derivative of a complex function.

The purpose of the lesson:familiarization with the formula for the derivative of a complex function; applying the formula to solve problems.

Tasks:contribute to the formation of knowledge on finding the derivative of various functions;

Develop the ability to find derivatives of functions; promote the development of students’ cognitive interests and quick calculations;

Cultivate accuracy in decisions, determination, and attentiveness.

Lesson type:learning new material.

Forms: collective, individual

Methods: conversation, research, independent work.

During the classes.

Organizing time.

Hello. Today in the lesson we will get acquainted with the formula for finding the derivative of a complex function.

Slide No. 2

The lesson will go through the stages of the Olympiad program.

Slide No. 3

1. Qualifying round.

2. Application.

3.Admission to competitions.

4. Training camps.

5. Competitions.

6. Rewarding.

Oral work

Each Olympiad begins with a qualifying round, where you need to answer questions and complete tasks

Slide No. 4

Qualifying round.

1. What is a function?

2. What is the scope of a function?

3. Which function is called continuous on an interval?

4. Determine whether the function is continuous at point x0

5. Is the function continuous at points x1, x2, x3

Slide number 5

6. What is the derivative of a function?

7. What is function increment?

8. What is argument increment?

9. Formulate the definition of a tangent to the graph of a function.

10. Calculate the derivative:

The qualifying round has been completed.

You know all the topics, but for further work you need to fill out an application form.

Individual work.

You need to fill out the sheet by answering the questions using your PIN code

1. What is the physical meaning of the derivative?

2. What is it? geometric meaning derivative?

3. Write down the tangent equation for the function y = ax 2 + in + s

at point x 0 =d

Next stage: Admission to competitions.

Solve the tasks:

Compose a complex function and calculate the derivative:

a) f=x 2 +3 g=7x-2 y=f(g)

b) f= sin x g=2x y=f(g)

c)f=3x 5 -2x 4 +3x g=x+6 y=f(g)

The first two tasks do not cause any difficulties, but the third requires additional knowledge.

We will use the rule for finding the derivative of a complex function.

Y = f(g(x)) Y / =f / (g).g / (x)

Using the formula, we will check the examples under the letters a) and b) and compare them with the answers received earlier.

a) f(g)= (7x-2) 2 +3

b) f(g)=sin2x

The results were the same. Therefore, the formula can be applied to the third example: f=3x 5 -2x 4 +3x g=x+6 y=f(g)

f ( g ) =3(x+6) 5 -2(x+6) 4 +3(x+6)

Systematization of knowledge.

Next step: competition.

Each of you will try your hand at solving complex derivatives using the formula.

We complete tasks from the Unified State Exam collection (Part 2), increasing the level of difficulty.

№ 336,355,359,377,379

Reflection

Every achievement must be evaluated.

You are invited to rateyour knowledge and skills on the topic “Derivative of a complex function”, how much you understood the topic, determining your place on the podium.

Summarizing.

What new did you learn?

How clear is the presentation?

How did you work in class?

Can you cope at home?

Write down the homework assignment: 380 - 410.

THANK YOU FOR THE LESSON!

Lesson #19Date of:

TOPIC: Derivative of a complex function

Lesson objectives:

educational:

formation of the concept of a complex function;

developing the ability to find the derivative of a complex function according to the rule;

development of an algorithm for applying the rule for finding the derivative of a complex function when solving problems.

developing:

develop the ability to generalize, systematize based on comparison, and draw conclusions;

develop visual and effective creative imagination;

develop cognitive interest.

contribute to the formation of the ability to rationally and accurately write out a task on the board and in a notebook.

educational:

to cultivate a responsible attitude towards academic work, will and perseverance to achieve final results when finding derivatives of complex functions;

contribute to the development of friendly relations between students during the lesson.

The student must know:

rules and formulas of differentiation;

concept of complex function;

rule for finding the derivative of a complex function.

The student must be able to:

calculate derivatives of complex functions using derivative tables and differentiation rules;

apply acquired knowledge to solve problems.

Lesson type : reflection lesson.

Lesson provision:

presentation; table of derivatives; table Rules of differentiation;

cards – tasks for individual work; cards - tasks for test work.

Equipment :

computer, TV.

DURING THE CLASSES:

1. Organizational moment (1 min).

Introduction

Readiness of the class for work.

General mood.

2. Motivational stage (2-3 min).

(Let's show ourselves that we are ready to confidently comprehend knowledge that may be useful to us!)

Tell me, what homework did you do for this lesson? (in the last lesson, we were asked to study the material on the topic “Derivative of a complex function” and, as a result, make notes).

What sources did you use to study this topic? (video, textbook, additional literature).

What additional literature did you use? (literature from the library).

So the topic of the lesson is...? ("Derivative of a complex function")

Open your notebooks and write down: number, Classwork, and the topic of the lesson. (Slide 1)

Based on the topic, let's outline the goals and objectives of the lesson (formation of the concept of a complex function; development of the ability to find the derivative of a complex function according to the rule; work out an algorithm for applying the rule for finding the derivative of a complex function when solving problems).

3. Updating knowledge and implementing primary action (7-8 min)

Let's move on to achieving the lesson's goals.

Let us formulate the concept of a complex function (function of the form y = f ( g (x)) called complex function, composed of functions f And g, Where f– external function and g- internal) (Slide 2 )

Let's consider Exercise 1: Find the derivative of a function y = (x 2 + sinx) 3 (write on the board)

This function is it elementary or complex? (difficult)

Why? (since the argument is not the independent variable x, but the function x 2 + sinx of this variable).

To find the derivative of a given function, you need to know the basic formulas for the derivative of elementary functions and know the rules of differentiation. Let's remember them by spending dictation: (Slide 3)

1) C ’ =0; 2) (x n) ' = nx n-1 ; ; 4) a x = a x ln a; 5)

The dictation result is checked (Slide 4)

Let us select from the table of derivatives and differentiation rules those that are needed to solve of this assignment and write them down in the form of a diagram on the board.

4. Identifying individual difficulties in implementing new knowledge and skills (4 min)

Let's solve example 1 and find the derivative of the function y ’ = ( ( x 2 + sin x) 3) '

What formulas are needed to solve the problem? ((x n) ’ = nx n -1 ;

Work at the board:

( x 2 + sin x) 3 = U;

y ’ = (U 3) ’ = 3 U 2 U`=3 ( x 2 + sin x) 2 ( 2x +cos x)

It can be noted that without knowledge of formulas and rules it is impossible to take the derivative of a complex function, but for correct calculation you need to see the main function in differentiation.

5. Building a plan to resolve the difficulties that have arisen and its implementation (8 - 9 min)

Having identified the difficulties, let's build an algorithm for finding the derivative of a complex function: (Slide 5)

Algorithm:

1. Define external and internal functions;

2. We find the derivative as we read the function.

Now let's look at this with an example

Task 2: Find the derivative of the function:

When simplifying, we get: (5-4x) = U,

y ’ = ’ =

Task 3: Find the derivative of the function:

1. Define external and internal functions:

y = 4 U – exponential function

2. Find the derivative as we read the function:

6. Generalization of identified difficulties (4 min)

N.I. Lobachevsky “... there is not a single area in mathematics that will ever not be applicable to the phenomena real world…”

Therefore, summarizing our knowledge, we will devote the solution to the next task to connections with physical phenomena (at the blackboard if desired)

Task 4:

During electromagnetic oscillations arising in an oscillatory circuit, the charge on the capacitor plates changes according to the law q = q 0 cos ωt, where q 0 is the amplitude of charge oscillations on the capacitor. Find the instantaneous value of the force alternating current I.

‘ = - . If you add initial phase, then using the reduction formulas we get - .

7. Implementation independent work(6 min)

Students perform testing using individual cards in a notebook. One answer is not enough, there must be a solution. (Slide 6)

Cards “Independent work for lesson No. 19”

Criteria for evaluation : “3 answers” ​​- 3 points; “2 answers” ​​- 2 points; “1 answer” - 1 point

Answer Keys(Slide 7)

№ tasks	1 option	2 option	3 option	4 option
	answer	answer	answer	answer

After checking (Slide 8)

8. Implementation of a plan to resolve difficulties (6 - 7 min)

Answers to students’ questions regarding difficulties encountered during independent work, discussion typical mistakes.

Examples - tasks to answer questions that arise***:

9. Homework(2 minutes) (Slide 9)

Solve an individual task using task cards.

Giving grades based on work results.

10. Reflection (2 min)

"I want to ask you"

The student asks a question, starting with the words “I want to ask...”. In response to the response received, he expresses his emotional attitude: “I am satisfied...” or “I am not satisfied because...”.

Summarize the students’ answers, finding out whether the lesson objectives were achieved.

OPEN CLASS ON THE DISCIPLINE ELEMENTS OF HIGHER MATHEMATICS FOR THE SPECIALTY COMPUTING EQUIPMENT AND AUTOMATED SYSTEMS SOFTWARE

LESSON PLAN

1 ORGANIZING TIME

1.1 Introduction

1.2 Group readiness to work

1.3 Setting the goal of the lesson

2 REPEATING THE MATERIAL COVERED

2.1 Frontal survey

2.2 Individual work using cards

2.3 Domino game

2.4 Oral work

3 EXPLANATION OF NEW MATERIAL

3.1 Derivative of a complex function

4 APPLYING KNOWLEDGE IN SOLVING TYPICAL PROBLEMS

5.1 Verification work with selective response system

6 CONCLUSION

6.1 Summing up

6.2 Homework

TOPIC: DERIVATIVE OF COMPLEX FUNCTION

Type of lesson: combined

Objectives of studying the topic:

educational:

1. formation of the concept of a complex function;
2. developing the ability to find the derivative of a complex function according to the rule;
3. development of an algorithm for applying the rule for finding the derivative of a complex function when solving examples.

developing:

1. develop the ability to generalize, systematize based on comparison, and draw conclusions;
2. develop visual and effective creative imagination;
3. develop cognitive interest.

educational:

1. nurturing a responsible attitude towards academic work, will and perseverance to achieve final results when finding derivatives of complex functions;
2. developing the ability to rationally and accurately write out a task on the board and in a notebook.
3. nurturing friendly relations between students during lessons.

Providing classes:

1. table of derivatives;
2. table Rules of differentiation;
3. cards for playing dominoes;
4. cards – tasks for individual work;
5. cards - tasks for test work.

The student must know:

1. definition of derivative;
2. rules and formulas of differentiation;
3. concept of complex function;
4. rule for finding the derivative of a complex function.

The student must be able to:

1. calculate derivatives of complex functions using derivative tables and differentiation rules;
2. apply acquired knowledge to solve problems.

PROGRESS OF THE CLASS

I ORGANIZATIONAL MOMENT

1. Introduction
2. Group readiness to work
3. Setting a lesson goal

II HOMEWORK CHECK

a) Questions for frontal survey:

1. What is the derivative of a function at a point?
2. . What is differentiation?
3. Which function is called differentiable at a point?
4. What does it mean to calculate the derivative using an algorithm?
5. What rules of differentiation do you know?
6. How are the continuity of a function at a point and its differentiability at this point related?

b) Individual work using cards

c) Game "Dominoes"

X /		() /
WITH /		() /
() /		f/(x)
() /		() /
() /
() /		() /
() /		() /
() /		() /
() /		() /
() /	2 x	() /

The Domino set contains 20 cards. Pairs shuffle their cards, divide in half and begin to lay out dominoes from a card in which only the right or left side is filled. Next, you must find an expression on another card that is identically equal to the expression on the first card, etc. The result is a chain.

A domino is considered to be laid out only when all the cards are used and the outer halves of the last and first cards are empty.

If not all the cards are laid out, it means you made a mistake somewhere and you need to find it.

Students working in pairs must evaluate each other and put marks on the control sheet. The evaluation criteria are written on the envelopes.

Criteria for evaluation:

1. “5” – no errors;
2. “4” – 1-2 errors;
3. “3” – 3-4 errors.

d) Oral work

Example 1 Find the derivative of a function.

Solution: .

Example 2 Find the derivative of the function.

Solution: .

Example 3 Find the derivative of the function.

Solution: .

Example 4 Statement of the problem situation: find the derivative of the function

y =ln(cos x).

We have here a logarithmic function whose argument is not an independent variable x, and the function cos x this variable.

What are these kinds of functions called?

[This kind of function is called complex

Functions or functions from functions.]

Do we know how to find derivatives of complex functions?

[No.]

So, what should we get to know now?

[With finding the derivative of complex functions.]

What will the topic of our lesson today be?

[Derivative of a complex function]

Students themselves formulate the topic and goals of the lesson, the teacher writes the topic on the board, and the students write it in their notebooks.

III STUDYING NEW MATERIAL

The rules and formulas of differentiation, which we discussed in the last lesson, are basic when calculating derivatives.

However, if for not complex expressions using the basic rules is not particularly difficult, but for complex expressions, using general rule It can be a very painstaking task.

The goal of our lesson today is to consider the concept of a complex function and master the technique of differentiating a complex function, i.e. technique of applying basic formulas in differentiating complex functions.

Derivative of a complex function

The example shows that a complex function is a function of a function. Therefore, we can give the following definition of a complex function:

Definition: Function of the form

y = f(g(x))

called complex function, composed of functions f u g, or superposition of functions f and g.

Example: Function y =ln(cos x) there is a complex function made up of functions

y = ln u and u = cos x.

Therefore, a complex function is often written in the form

y = f(u), where u = g(x).

External function Intermediate

Function

In this case, the argument x is called independent variable, and u - intermediate argument.

Let's go back to the example. We can calculate the derivative of each of these functions using a derivative table.

How to calculate the derivative of a complex function?

The answer to this question is given by the following theorem.

Theorem: If the function u = g(x) differentiable at some point x 0, and the function y=f(u) differentiable at the point u 0 = g(x 0 ), then a complex function y=f(g(x)) differentiable at a given point x 0 .

Wherein

or

those. derivative of y by variable x equal to the derivative of y by variable and , multiplied by the derivative of and by variable x.

Rule:

1. To find the derivative of a complex function, you need to read it correctly;
2. To read a function correctly, you need to determine the order of actions in it;
3. Read the function in reverse order action direction;
4. We find the derivative as we read the function.

Now let's look at this with an example:

Example 1: Function y =ln(cos x) is obtained by sequentially performing two operations: taking the cosine of the angle X and finding the natural logarithm of this number:

The function reads like this: logarithmic function of a trigonometric function.

Let's differentiate the function: y = ln(cos x)=ln u, u=cos x.

In practice, such differentiation is made much shorter and simpler, at least without introducing the notation And .

The art of differentiating a complex function lies in the ability to see at the moment of differentiation only one function (namely, the one being differentiated in this moment), not noticing others for now, postponing their vision until the moment of differentiation.

We will use the augmented table of derivatives for differentiation.

Example2: Find the derivative of a function y = (x 3 - 5x + 7) 9 .

Solution : Having designated in the “mind” u = x 3 – 5x +7, we get y = u 9. Let's find:

According to the formula we have

4 APPLYING KNOWLEDGE IN SOLVING TYPICAL PROBLEMS

1) ;

2) ;

3) ;

4) ;

5) ;

5 INDEPENDENT APPLICATION OF KNOWLEDGE, ABILITIES AND SKILLS

5.1 Test work in the form of a test

Test Specification:

1. The test is homogeneous;
2. Closed form test;
3. Number of tasks – 3;
4. Task completion time – 5 minutes;
5. For a correct answer, the subject receives 1 point.

For an incorrect one - 0 points.

Instructions: choose the correct answer.

Criteria for evaluation :

“5” – 3 points

“4” – 2 points

“3” - 1 point

Students solve on the slips of paper and check their answers using the key provided on the board. Put the assessment on the control sheet (self-control).

Option 1

1. The derivative of the function is equal to:

A) ; b) ; V) .

1. The derivative of the function is equal to:

A) ; b) ; V) .

A) ; b) ; V) .

Option 2

Choose the correct answer

1. The derivative of the function is equal to:

A) ; b) ; V) .

1. The derivative of the function is equal to:

A) ; b) ; V) .

1. Calculate derivative for function:

A) ; b) ; V) .

Option 3

Choose the correct answer

1. The derivative of the function is equal to:

A) ; b) ; V) .

1. The derivative of the function is equal to:

A) ; b) ; V) .

1. Calculate derivative for function:

A) ; b) ; V) .

Option 4

Choose the correct answer

1. The derivative of the function is equal to:

A) ; b) ; V) .

1. The derivative of the function is equal to:

A) ; b) ; V) .

1. Calculate derivative for function:

A) ; b) ; V) .

Answer Keys

Job No.	1 option	Option 2	Option 3	Option 4
	answer	answer	answer	answer