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

This lesson is a lesson in learning a 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 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 to update basic 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 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).

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)).

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) =

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 a written exam in mathematics for a high school course” - 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 topic: Derivative of a 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, the ability to analyze, plan your educational activities, logically express your thoughts

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

y =ln( sinx).

We have here a 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 simple expressions the use of basic rules is not particularly difficult, then for complex expressions, applying a general rule can be very difficult.

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 the natural logarithm of this number:

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

D)

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 =;

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.

1. 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.
2. Developmental goals: develop cognitive interests through the use of information technology.
3. Educational goals: to cultivate adaptability to modern learning conditions.

1. 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.

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

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.

The formula is 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.

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.

1. Checking homework completion - 5 minutes (frontal check, self-control).
2. Comprehensive knowledge test – 10 min (frontal work, mutual control).
3. Preparation for mastering (studying) new 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.

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.

Pay attention to the correct solution and correct formatting of the solution in home example No. 5.

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 whole 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.

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.

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

Solve examples:

3)

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

Solve examples:

2)

3)

4) ;

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)

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