The graph of an exponential function is a curved smooth line without kinks, to which a tangent can be drawn at each point through which it passes. It is logical to assume that if it is possible to draw a tangent, then the function will be differentiable at each point of its domain of definition.
Let's display in the same coordinate axes several graphs of the function y \u003d x a, For a \u003d 2; a = 2.3; a = 3; a = 3.4.
At the point with coordinates (0;1). The slope angles of these tangents will be approximately 35, 40, 48 and 51 degrees, respectively. It is logical to assume that in the interval from 2 to 3 there is a number at which the angle of inclination of the tangent will be 45 degrees.
Let's give the exact formulation of this statement: there is such a number greater than 2 and less than 3, denoted by the letter e, that the exponential function y = e x at the point 0 has a derivative equal to 1. That is: (e ∆x -1) / ∆x tends to 1 as ∆x tends to zero.
Given number e is irrational and is written as an infinite non-periodic decimal fraction:
e = 2.7182818284…
Since the number e is positive and non-zero, there is a logarithm to the base e. This logarithm is called natural logarithm. Denoted ln(x) = log e (x).
Derivative of exponential function
Theorem: The function e x is differentiable at every point of its domain, and (e x)’ = e x .
The exponential function a x is differentiable at each point of its domain of definition, and moreover (a x)’ = (a x)*ln(a).
A consequence of this theorem is the fact that the exponential function is continuous at any point in its domain of definition.
Example: find the derivative of the function y = 2 x .
According to the formula for the derivative of the exponential function, we get:
(2x)' = (2x)*ln(2).
Answer: (2x)*ln(2).
Antiderivative of the exponential function
For an exponential function a x given on the set of real numbers, the antiderivative will be the function (a x)/(ln(a)).
ln(a) is some constant, then (a x / ln(a))’= (1 / ln(a)) * (a x) * ln(a) = a x for any x. We have proved this theorem.
Consider an example of finding an antiderivative exponential function.
Example: find the antiderivative to the function f(x) = 5 x . Let's use the above formula and the rules for finding antiderivatives. We get: F(x) = (5 x) / (ln(5)) +C.
Lesson Objectives: form an idea of a number e; prove the differentiability of a function at any point X;consider the proof of the differentiability theorem for the function ; checking the formation of skills and abilities when solving examples for their application.
Lesson objectives.
Educational: repeat the definition of a derivative, the rules of differentiation, the derivative of elementary functions, remember the graph and properties of an exponential function, form the ability to find the derivative of an exponential function, control knowledge using a test task and a test.
Developing: to promote the development of attention, the development of logical thinking, mathematical intuition, the ability to analyze, apply knowledge in non-standard situations.
Educational: to educate information culture, to develop the skills of working in a group and individually.
Teaching methods: verbal, visual, active.
Forms of training: collective, individual, group.
Equipment : the textbook “Algebra and the Beginnings of Analysis” (edited by Kolmogorov), all tasks of group B “Closed Segment” edited by A.L. Semenov, I.V. Yashchenko, multimedia projector.
Lesson steps:
- Reporting the topic, goals, objectives of the lesson (2 min.).
- Preparation for the study of new material through the repetition of previously studied (15 min.).
- Introduction to new material (10 min.)
- Primary comprehension and consolidation of new knowledge (15 min.).
- Homework (1 min.).
- Summing up (2 min.).
During the classes
1. Organizational moment.
The topic of the lesson is announced: “The derivative of the exponential function. Number e.”, goals, tasks. Slide 1. Presentation
2. Activation of basic knowledge.
To do this, at the first stage of the lesson, we will answer questions and solve tasks for repetition. Slide 2.
At the blackboard, two students work on cards, completing tasks like B8 USE.
Task for the first student:
Task for the second student:
The rest of the students complete independent work according to the options:
| Option 1 | Option 2 | ||
| 1. |  |
1. |  |
| 2. |  |
2. |  |
| 3. |  |
3. |  |
| 4. |  |
4. |  |
| 5. |  |
5. |  |
Pairs exchange solutions and check each other's work, referring to the answers on slide 3.
The solutions and answers of students working at the blackboard are considered.
Checking homework No. 1904. Show slide 4.
3. Updating the topic of the lesson, creating a problem situation.
The teacher asks to give a definition of the exponential function and list the properties of the function y \u003d 2 x. Graphs of exponential functions are shown as smooth lines, to which a tangent can be drawn at each point. But the existence of a function tangent to the graph at a point with abscissa x 0 is equivalent to its differentiability at x 0.
For the graphs of the function y \u003d 2 x and y \u003d 3 x, we draw tangents to them at the point with abscissa 0. The angles of inclination of these tangents to the abscissa axis are approximately equal to 35 ° and 48 °, respectively. Slide 5.
Conclusion: if the base of the exponential function a increases from 2 to, for example, 10, then the angle between the tangent to the graph of the function at the point x=0 and the x-axis gradually increases from 35° to 66.5°. It is logical to assume that there is a reason a, for which the corresponding angle is 45
It is proved that there is such a number greater than 2 and less than 3. It is customary to denote it by the letter e. In mathematics, it is established that the number e- irrational, i.e. is an infinite decimal non-periodic fraction.
e = 2.7182818284590…
Note (not very serious). slide 6.
On the next slide 7 there are portraits of great mathematicians - John Napier, Leonard Euler and a brief note about them.
- Consider the properties of the function y=e x
- Proof of Theorem 1. Slide 8.
- Proof of Theorem 2. Slide 9.
4. Dynamic pause or discharge for the eyes.
(Starting position - sitting, each exercise is repeated 3-4 times):
1. Leaning back, take a deep breath, then leaning forward, exhale.
2. Leaning back in a chair, close your eyelids, close your eyes tightly without opening your eyelids.
3. Hands along the body, circular movements of the shoulders back and forth.
5. Consolidation of the studied material.
5.1 Solution of exercises No. 538, No. 540, No. 544c.
5.2 Independent application of knowledge, skills and abilities. Verification work in the form of a test. Time to complete the task - 5 minutes.
Criteria for evaluation:
"5" - 3 points
"4" - 2 points
“3” - 1 point
6. Summing up the results and results of the work in the lesson.
- Reflection.
- Grading.
- Submission of test tasks.
7. Homework: p. 41 (1, 2); No. 539 (a, b, d); 540 (c, d), 544 (a, b).
“Closed segment” No. 1950, 2142.
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Slides captions:
DERIVATIVE OF THE EXPONENTIAL FUNCTION Number e Grade 11
REPETITION is the mother of learning!
Definition of an exponential function The function given by the formula y \u003d a x (where a > 0, a ≠ 1) is called an exponential function with base a.
Properties of the exponential function y \u003d a x a > 1 0
Determination of the derivative of the function at the point x 0. as Δ → 0. The derivative of the function f at the point x 0 is the number to which the difference relation tends as Δx → 0.
The geometric meaning of the derivative x ₀ α A y \u003d f (x) 0 x y k \u003d tg α \u003d f "(x ₀) The slope to the tangent to the graph of the function f (x) at the point (x 0; f (x 0) is equal to the derivative functions f "(x ₀). f(x 0)
Game: "Find pairs" (u + v) "cos x e (u v)" n xⁿ ⁻" p (u / v)" - 1 / (sin² x) a (x ⁿ)" - sin x n C "u" v +u v" to (C u)" 1 / (cos ² x) t (sin x)" (u" v - u v") / v² c (cos x)" 0 o (tg x)" u "+v" u (ctg x) "C u" n
Check yourself! (u + v)" u" + v" e (u v)" u" v + u v "to (u / v)" (u' v –u v") / v² c (x ⁿ)" n x ⁿ ⁻¹ p C" 0 o (Cu)" C u "n (sin x)" Cos x e (cos x)" - sin x n (tg x)" 1 / (cos² x) t (ctg x)" - 1 / (sin² x) a
The exponent is a power function. The exponent is a function where e is the base of natural logarithms.
1 y \u003d e x 45 ° The function y \u003d e x is called the "exponent" x ₀ \u003d 0; tg 45° = 1 At the point (0;1) the slope to the tangent to the graph of the function k = tg 45° = 1 - the geometric meaning of the derivative of the exponent Exponent y = e x
Theorem 1. The function y \u003d e is differentiable at each point of the domain of definition, and (e)" \u003d e x x x The natural logarithm (ln) is the logarithm to the base e: ln x \u003d log x e and (a)" = a ∙ ln a x x Theorem 2 .
Formulas for differentiating the exponential function (e)" = e ; (e)" = k e ; (a)" = a ∙ ln a ; (a)" = k a ∙ ln a . x kx + b x x x kx + b kx + b kx + b F(a x) = + C; F(e x) = e x +C.
"Exercise creates mastery." Tacitus Publius Cornelius - ancient Roman historian
Examples: Find derivatives of functions: 1. = 3 e. 2. (e)" = (5x)" e = 5 e. 3. (4) "= 4 ln 4. 4. (2)" = (-7 x) "2 ∙ ln 2 = -7 ∙ 2 ∙ ln 2. 5 x 5 x x (3 e)" 5 x - 7 x x x -7 x -7 x x
Interesting nearby
Leonhard Euler 1707 -1783 Russian scientist - mathematician, physicist, mechanic, astronomer ... Introduced the designation of the number e. Proved that the number e ≈ 2, 718281 ... is irrational. John Napier 1550 - 1617 Scottish mathematician, inventor of logarithms. In his honor, the number e is called the "non-peer number".
The growth and decay of a function at the rate of an exponent is called exponential
When deriving the very first formula of the table, we will proceed from the definition of the derivative of a function at a point. Let's take where x- any real number, that is, x– any number from the function definition area . Let us write the limit of the ratio of the function increment to the argument increment at : 
It should be noted that under the sign of the limit, an expression is obtained, which is not the uncertainty of zero divided by zero, since the numerator contains not an infinitesimal value, but precisely zero. In other words, the increment of a constant function is always zero.
In this way, derivative of a constant functionis equal to zero on the entire domain of definition.
Derivative of a power function.
The formula for the derivative of a power function has the form 
, where the exponent p is any real number.
Let us first prove the formula for the natural exponent, that is, for p = 1, 2, 3, ...
We will use the definition of a derivative. Let us write the limit of the ratio of the increment of the power function to the increment of the argument: 
To simplify the expression in the numerator, we turn to Newton's binomial formula: 
Consequently, 
This proves the formula for the derivative of a power function for a natural exponent.
Derivative of exponential function.
We derive the derivative formula based on the definition:
Came to uncertainty. To expand it, we introduce a new variable , and for . Then . In the last transition, we used the formula for the transition to a new base of the logarithm.
Let's perform a substitution in the original limit: 
If we recall the second remarkable limit, then we come to the formula for the derivative of the exponential function: 
Derivative of a logarithmic function.
Let us prove the formula for the derivative of the logarithmic function for all x from the scope and all valid base values a logarithm. By definition of the derivative, we have: 
As you noticed, in the proof, the transformations were carried out using the properties of the logarithm. Equality 
is valid due to the second remarkable limit.
Derivatives of trigonometric functions.
To derive formulas for derivatives of trigonometric functions, we will have to recall some trigonometry formulas, as well as the first remarkable limit.
By definition of the derivative for the sine function, we have 
.
We use the formula for the difference of sines: 
It remains to turn to the first remarkable limit: 
So the derivative of the function sin x there is cos x.
The formula for the cosine derivative is proved in exactly the same way. 
Therefore, the derivative of the function cos x there is –sin x.
The derivation of formulas for the table of derivatives for the tangent and cotangent will be carried out using the proven rules of differentiation (derivative of a fraction). 
Derivatives of hyperbolic functions.
The rules of differentiation and the formula for the derivative of the exponential function from the table of derivatives allow us to derive formulas for the derivatives of the hyperbolic sine, cosine, tangent and cotangent. 
Derivative of the inverse function.
So that there is no confusion in the presentation, let's denote in the lower index the argument of the function by which differentiation is performed, that is, it is the derivative of the function f(x) on x.
Now we formulate rule for finding the derivative of the inverse function.
Let the functions y = f(x) and x = g(y) mutually inverse, defined on the intervals and respectively. If at a point there exists a finite non-zero derivative of the function f(x), then at the point there exists a finite derivative of the inverse function g(y), and 
. In another entry 
.
This rule can be reformulated for any x from the interval , then we get 
.
Let's check the validity of these formulas.
Let's find the inverse function for the natural logarithm 
(here y is a function, and x- argument). Solving this equation for x, we get (here x is a function, and y her argument). That is, 
and mutually inverse functions.
From the table of derivatives, we see that 
and 
.
Let's make sure that the formulas for finding derivatives of the inverse function lead us to the same results: 
Mathematics teacher MOU
"Multanovskaya secondary school"
Makhanova Samiga Galimzhanovna
With. M u l t a n o v o
February 2011
Lesson topic:"The number e. Derivative of the exponential function".
Target: Introduce the concept of "exponent", "natural logarithm", form the concept of the derivative of the exponential function y \u003d e x, the antiderivative exponential function.
Educational:
Repeat and deepen knowledge on the topic “Exponential function. Properties of the exponential function”;
Repeat the rules for differentiating a function;
Introduce students to the concept of "exponent" (the number e);
To acquaint students with the formulas for the derivative of the exponential function y \u003d a X and y = a kx +b ;
Introduce the formula of the antiderivative exponential function;
To form the skills of calculating the derivative of an exponential function, using the rules and formulas of differentiation.
Developing:
Develop and improve the application of differentiation rules
for the exponential function;
To teach students to apply electronic information technologies in teaching and preparing for mathematics lessons.
To improve the graphic culture of students;
To promote the development of skills to carry out self-assessment of educational activities.
Educational:
To create positive motivation for students for the lesson of mathematics by involving everyone in active activities;
To educate the need to evaluate their own activities and the work of comrades;
To help realize the values of teamwork;
To educate students in accuracy, culture of mathematical speech.
Equipment for the lesson:
Computer class (8 laptops + 1 laptop for demonstration), projector, presentation, handouts.
During the classes:
Organization of the lesson, announcement of the topic and purpose of the lesson:
Today in the lesson we are studying a new topic "The derivative of the exponential function". Our goal: (Slide 2.) to get acquainted with the concept of "exponent", "natural logarithm", with the theorem on differentiation of the exponential function and learn how to differentiate the exponential function.
As an epigraph to our lesson, I chose the verses of B. Slutsky: (slide 3.)
…Exponential function
It was not by chance that she was born
Organically integrated into life
And took up the movement of progress.
B. Slutsky
I.Updating of basic knowledge:
Oral frontal work with the class:
Formulate the definition of an exponential function (Slide 5.)
List the main properties of the exponential function according to the graph.
(Slide 6)
Properties of the exponential function:(slide 4)
Function scope
Range of the exponential function
The graph of the function with the y-axis intersects at the point (0;1) and does not intersect with the x-axis.
The exponential function takes positive values on the entire number line.
List the properties of the exponential function for a 1.
List the properties of the exponential function at 0 .
Define the derivative of a function at a point x 0 . (slide 7)
Formulate the geometric meaning of the derivative. (slide 8)
And now we recall the rules for differentiating functions:
2) The game "Find the pairs." (slide 9.)
For the formulas in the first column, find the correct answers in the second column and read the word in the third column. Orally, with commentary.
| (u+v)" | cos x | E |
| (u v)" | n x n-1 | P |
| (u/v)" | -1/sin2x | BUT |
| (xn)" | Sin x | H |
| C" | u"v + uv" | To |
| (Cu)" | 1 / cos 2 x | T |
| (sinx)" | (u "v - u v") / v 2 | FROM |
| (cosx)" | 0 | O |
| (tgx)" | u"+v" | E |
| (ctgx)" | C u" | H |
| E | u"+v" | (u+v)" |
| To | u"v + uv" | (u v)" |
| FROM | (u "v - u v") / v 2 | (u/v)" |
| P | n x n-1 | (xn)" |
| O | 0 | C" |
| H | C u" | (Cu)" |
| E | Cos x | (sinx)" |
| H | -Sin x | (cosx)" |
| T | 1 / cos 2 x | (tgx)" |
| BUT | -1/sin2x | (ctgx)" |
Check your answer against the table :( slide 10)
II.Learning a new topic:
1) Research work using ESM resources for laptops. Pair work.
Open the Internet Digital Educational Resources for Algebra and the Beginnings of Analysis Grade 11 topic: "Derivatives of the exponential function, the number e and the natural logarithm." module I1
Carefully read each element of the Module, write down the main formulas in your notebooks, read their proofs.
Complete tasks for self-control. Check the summary of your work in "Statistics" (C).
Module work plan:
An exponential function with base e. - (Introduction to the exponent)
The formula for the derivative of the exponential function. - (Derivation of the formula for the derivative of the function y \u003d e x)
A task for self-control. – (test with a choice of answers)
Definition of the natural logarithm ln. – (ln x = log e x)
The formula for the derivative of the exponential function. - (derivation of the formula of the derivative of the exponential formula)
A task for self-control. - (Short answer task)
The antiderivative of the exponential function - (derivation of the formula for the derivative of the exponential function)
Task for self-control - (test with a choice of answers)
2)
Cl. 15-18 Frontal survey, based on the studied material. Primary fixation of the material. Application of formulas for the derivative of an exponential function.
(e X )" = e X ;
(e kx + b )" = ke kx + b ;
(a x )" = a x ∙ lna ;
(a kx + b )" = ka Kx+b ∙ lna
F(a x ) =
The student works independently at the blackboard:
Solution: f(x) = x 2 * 2 –x; D(f) = R; f "= 2x * 2 -x - x 2 * 2 -x ln2, D (f) \u003d R,
2x * 2 –x – x 2 * 2 –x ln2 = 0;
X * 2 -x (2 - x * ln 2) = 0; - min + max - f "(x)
X * 2 –x = 0; 2 – x * ln x = 0 2 – x > 0, x = 0; 2 – x * ln2 = 0 0 2/ln2 f(x)
Answer: x max = 2 / ln2; x min = 0
Independent work of a teaching nature:
Independent work in pairs at laptops. Interactive module P1 “derivative of exponential function. Number e. Natural logarithm. - test of 5 tasks. When you open the module, different tasks appear on each computer.
V. Lesson summary: What new did you learn in the lesson?
What parts of the lesson were the most interesting for you?
Who is satisfied with their work in the classroom?
VI. Homework: item 41; No. 539(a,b,d); 540(c); 542(a,b); 544(b).
Interactive test with a computer. Properties of the exponential function K1.
On the desktop of each computer, open the Word Module. eleven
"Properties of the exponential function K1". Click "mouse" on "play module". You will be given a test of 5 tasks.
Complete the 1st task of the Module, click the mouse on the number of the correct answer or write down the answer in the test. Click "mouse" on "answer" and move on to another task.
If you completed the task incorrectly, open a hint,
find the error in your solution.
Check the summary of your work in "Statistics" (C).
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