Development of a lesson on the topic: "Derivative of a complex function." Derivative of a complex function Topic derivative of a complex function

Lesson topic: Derivative complex function.

Lesson type: combined

Lesson objectives:

educational:

– formation of the concept of a complex function;

Learning the rules of findingderivative of a complex function.

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

developing:

Develop logic, ability to analyze, plan your educational activities express your thoughts logically

Develop cognitive interest.

educational:

Education and development of diverse interests of the individual;

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

Lesson plan:

1. Organizational moment: group readiness for the lesson, checking those absent from the lesson.

2.Checking homework.

3. Updating knowledge: repeating the material covered.

4.Learning new material.

5. Fixing the material

6. Homework

Lesson progress:

1.Org.moment: Greeting, checking the readiness of the group for the lesson, communicating the topic and purpose of the lesson, motivating learning activities.

2. Checking homework: Students demonstrate their homework on the topic covered.

3. Updating students’ knowledge:

1. Guys, let's remember what the derivative of a function is?

Answer:derivative of a function at a pointis called the limit of the function increment ratioto the argument increment that caused itat this point at.

2. The geometric meaning of the derivative in which equation is expressed?

Answer: Expressed as a tangent equation.

3. In a mechanical sense, what is the first derivative of a path with respect to time?

Answer: Speed

4. What is another name for the points of extremum and minimum?

Answer: Critical points derivative.

5.What is the derivative of a constant?

Answer: 0

6. Cards with examples:

a) y=5x+3 x 2 ; b) y = ;c) y= ; d) y= ; e)2x 7 +; e) y=

7. Staging problematic situation: find the derivative of a function

y =ln( sinx).

We have here logarithmic function, whose argument is not an independent variableX , and the functions in x this variable.

1.What do you think these functions are called?

Answer: functions are called complex functions or functions of functions.

2. Do we know how to find derivatives of complex functions?

Answer: No.

3. So, what should we get to know now?

Answer: With finding the derivative of complex functions.

4.What will the topic of our lesson today be?

Answer: Derivative of a complex function

4. Studying new material.

The rules and formulas of differentiation, which we examined in the last lesson, are basic when calculating derivatives. But if for not complex expressions the use of basic rules does not represent special labor, then for complex expressions, use general rule may turn out to be a very difficult matter.

The goal of our lesson today is to consider the concept of a complex function and master the technique of using 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 formy = f(g(x)) calledcomplex function , composed of functionsf ug, orsuperposition of functions f Andg.

Example: Functiony =ln( sinx) there is a complex function made up of functions

y = ln u Andu = sinx .

Therefore, a complex function is often written in the form

y = f(u), Whereu = g(x)

External function Intermediate function

In this case, the argumentX calledindependent variable , Au - 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 functionu = g(x) differentiable at some pointX 0 , and the functiony=f(u) differentiable at the pointu 0 = g(x 0 ), then a complex functiony=f(g(x)) differentiable at a given point x 0 .

Rule:

To find the derivative of a complex function, you need to read it correctly;

We read the function in the reverse order of actions;

We find the derivative as we read the function.

Now let's look at this with an example:

Example1: Functiony =ln( sinx) is obtained by sequentially performing two operations: taking the sine of the angleX and finding from this number natural logarithm:

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

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

. We will use the augmented table of derivatives for differentiation.

Next we get (u) ’ =(s in x) ’ = cosx

U ’ = ’ ==ctg x

Example2: Find the derivative of a functionh( x)=(2 x+3) 100 .

Solution: Functionhcan be represented as a complex functionh( x) = g( f( x)), Whereg( y)= y 100 , y= f( x)=2 x+3, becausef I ( x)=2, g I ( y)=100 y 99 , h I ( x)=2*100 y 9 =200(2 x+3) 99 .

5.Reinforcement of the material: (Students come to the board and solve examples)

№1. Find the domain of the function.

A) y = ; b) y =;

IN); d) y=

№2. Find the derivative of the function:

A) (2 x -7) 14

B) (3+5 x ) 10

B) (7 x -1) 3

G) (8 x +6) 55

E) (7 x -1) 5

№3. Functions are set f ( x ) = 2- x - x 2 ; g ( x ) = ; p ( x ) = .

Define functions using formulas:

A) f ( g ( x )) ; b) g ( f ( x )); V) f ( p ( x ))

6. Homework:

Find the derivative of the function: a) (5 x -7) 17 ; b) (7 x +6) 14 ; IN) y =; G) y =;

This lesson is a learning lesson new topic. The presented lesson development reveals methodological approaches to introducing the concept of a complex function and 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 during the search mathematical dependencies, continue the formation of self-esteem when implementing differentiated instruction, and 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.

Heuristic conversation to introduce the concept of a complex function.
Oral frontal work in order to consolidate the definition of a complex function.
Teacher's message about the algorithm for calculating the derivative of a complex function.
Primary fixation of the algorithm for calculating the derivative of a complex function frontally.
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 blackboard 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.

Progress of the lesson.

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 the 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)).

define the inner function g(x).
find the derivative of the internal function g"(x)
define the outer function h(g)
find the derivative of the external function h"(g)
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

g(x) = 3-5x
g"(x) = -5
h(g) = g 5
h"(g)=5g 4
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) =

Literature used:

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 written exam in mathematics per 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 #19Date:

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 visually 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 educational 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.

LESSON PROGRESS:

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, great job, 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 Task 1: Find the derivative of a function y = (x 2 + sinx) 3 (write on the board)

Is this function basic 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 this task 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 of the real world...”

Therefore, summarizing our knowledge, we will devote the solution to the next task to connections with physical phenomena(at the board 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 AC I.

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

7. Carrying out 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”

Evaluation criteria : “3 answers” - 3 points; “2 answers” - 2 points; “1 answer” - 1 point

Answer Keys(Slide 7)

№ assignments	1 option	2 option	3 option	4 option
№ assignments	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 min) (Slide 9)

Solve an individual task using task cards.

Giving grades based on work results.

10. Reflection (2 min)

"I want to ask"

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.

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;

Developing 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.

Checking homework completion – 5 minutes (frontal check, self-control).
Comprehensive knowledge test – 10 min (frontal work, mutual control).
Preparing to learn (learn) new things educational material through repetition and updating of basic knowledge – 5 minutes (problem situation).
Assimilation of new knowledge – 15 minutes (frontal work under the guidance of a teacher).
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).
Consolidating new knowledge – 15 min ( independent work– test in two versions, with differentiated tasks).
Information about homework, instructions for its implementation – 2 min.
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, assessment criteria 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.

Pay attention to the correct solution and correct formatting of the solution in 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 (mastering) 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 of 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:

VI. Primary comprehension and understanding of new material.

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

Solve examples:

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 grade 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

Topic: “Derivative

complex function."

Lesson type: – lesson on learning new material.

Lesson format: application of information technology.

Place of the lesson in the system of lessons for this section: first lesson.

teach to recognize complex functions, be able to apply the rules for calculating derivatives; improve subject, including computational, skills and abilities; computer skills;
develop readiness for information and educational activities through the use of information technologies.
cultivate adaptability to modern learning conditions.

Equipment: electronic files with printed material, individual computers.

Progress of the lesson.

I. Organizational moment (0.5 min.).

II. Setting goals. Motivating students (1 min.).

Educational goals: learn to recognize complex functions, know the rules of differentiation, be able to apply the formula for the derivative of a complex function when solving problems; improve subject, including computational, skills and abilities; computer skills.
Developmental goals: develop cognitive interests through the use of information technology.
Educational goals: to cultivate adaptability to modern conditions training.

III. Updating of reference knowledge
(5 min.).

Name the rules for calculating the derivative.

3. Oral work.

Find the derivatives of the functions.

a) y = 2x 2 + xі;

b) f(x) = 3x 2 – 7x + 5;

d) f(x) = 1/2x 2 ;

e) f(x) = (2x – 5)(x + 3).

4. Rules for calculating derivatives.

Repetition of formulas on the computer with sound accompaniment.

IV. Programmed control
(5 min.) .

Find the derivative.
Option 1.		Option 2.


y = tan x + cot x.		y = tg x – ctg x.

Y = x 2 + 7x + 5		Y = 2x 2 – 5x + 7
Answer options .



1/cos 2 x + 1/sin 2 x	1/cos 2 x – 1/sin 2 x	1/sin 2 x – 1/cos 2 x
1.6x 0.6 + 2.5x 1.5	2.6x 0.6 + 1.5x 1.5	1.5x 0.5 + 4x 3	2.5x 0.5 + 4x 3

Exchange notebooks. In the diagnostic cards, mark correctly completed tasks with a + sign, and incorrectly completed tasks with a “–”.

V. New material
(5 min.) .

Complex function.

Consider the function given by the formula f(x) =

In order to find the derivative of a given function, you must first calculate the derivative of the internal function u = v(x) = xI + 7x + 5, and then calculate the derivative of the function g(u) = .

They say the function f(x) – there is a complex function made up of functions g And v , and write:

f(x) = g(v(x)) .

The domain of definition of a complex function is the set of all those X from the domain of the function v , for which v(x) is within the scope of the function g.

Let the complex function y = f(x) = g(v(x)) be such that the function y = v(x) is defined on the interval U, and the function u = v(x) is defined on the interval X and the set of all its values is included into the interval U. Let the function u = v(x) have a derivative at each point inside the interval X, and the function y = g(u) have a derivative at each point inside the interval U. Then the function y = f(x) has a derivative at each point inside the interval X, calculated by the formula

x = y" u u" x .

The formula can be read as follows: derivative y By x equal to the derivative y By u , multiplied by the derivative u By x .

The formula can also be written like this:

f" (x) = g" (u) v" (x).

Proof.

At the point X

X let's set the increment of the argument, (x+ x) X. Then the functionu = v(x) will receive an increment , and the function y = g(u) will receive increment Dy. It should be taken into account that since the function u=v(x) at the point x has a derivative, then it is continuous at this point and at .

Provided that

Examination.

VIII. Individual tasks
(7 min.) .

On the computer desktop.

Folder: “Derivative of a complex function.” Document: “Individual assignments”.