The derivative of an exponent is equal to the exponent itself (the derivative of e to the x power is equal to e to the x power):
(1) (e x )′ = e x.

Derivative exponential function with a base of power a is equal to the function itself multiplied by the natural logarithm of a:
(2) .

Derivation of the formula for the derivative of the exponential, e to the x power

An exponential is an exponential function whose power base is equal to the number e, which is the following limit:
.
Here it can be both natural and real number. Next, we derive formula (1) for the derivative of the exponential.

Derivation of the exponential derivative formula

Consider the exponential, e to the x power:
y = e x .
This function is defined for everyone. Let's find its derivative with respect to the variable x. By definition, the derivative is the following limit:
(3) .

Let's transform this expression to reduce it to known mathematical properties and rules. To do this we need the following facts:
A) Exponent property:
(4) ;
B) Property of logarithm:
(5) ;
IN) Continuity of the logarithm and the property of limits for a continuous function:
(6) .
Here is some function that has a limit and this limit is positive.
G) The meaning of the second remarkable limit:
(7) .

Let's apply these facts to our limit (3). We use property (4):
;
.

Let's make a substitution. Then ; .
Due to the continuity of the exponential,
.
Therefore, when , . As a result we get:
.

Let's make a substitution. Then . At , . And we have:
.

Let's apply the logarithm property (5):
. Then
.

Let us apply property (6). Since there is a positive limit and the logarithm is continuous, then:
.
Here we also used the second remarkable limit(7). Then
.

Thus, we obtained formula (1) for the derivative of the exponential.

Derivation of the formula for the derivative of an exponential function

Now we derive formula (2) for the derivative of the exponential function with a base of degree a. We believe that and . Then the exponential function
(8)
Defined for everyone.

Let's transform formula (8). To do this, we will use the properties of the exponential function and logarithm.
;
.
So, we transformed formula (8) to the following form:
.

Higher order derivatives of e to the x power

Now let's find derivatives of higher orders. Let's look at the exponent first:
(14) .
(1) .

We see that the derivative of function (14) is equal to function (14) itself. Differentiating (1), we obtain derivatives of the second and third order:
;
.

This shows that the nth order derivative is also equal to the original function:
.

Higher order derivatives of the exponential function

Now consider an exponential function with a base of degree a:
.
We found its first-order derivative:
(15) .

Differentiating (15), we obtain derivatives of the second and third order:
;
.

We see that each differentiation leads to the multiplication of the original function by . Therefore, the nth order derivative has the following form:
.

Lesson and presentation on the topic: "Number e. Function. Graph. Properties"

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Guys, today we will study a special number. It occupies a separate place in “adult” mathematics and has many remarkable properties, some of which we will consider.

Let's return to the exponential functions $y=a^x$, where $a>1$. We can plot many different function graphs for different bases.
But it should be noted that:

  • all functions pass through the point (0;1),
  • for $x→-∞$, the graph has a horizontal asymptote $y=0$,
  • all functions increase and convex downwards,
  • and also continuous, which in turn means that they are differentiable.
If functions are differentiable everywhere, then tangents can be constructed to them at each point. If all functions pass through the point (0;1), then it is of particular interest. Let's build several tangents sequentially.

Let's consider the function $y=2^x$ and construct a tangent to it.
Having carefully plotted our graphs, you can see that the angle of inclination of the tangent is 35°.
Now let's plot the function $y=3^x$ and also plot the tangent line:
This time the tangent angle is approximately 48°. In general, it is worth noting: the larger the base of the exponential function, the greater the angle of inclination.
Of particular interest is the tangent with an angle of inclination equal to 45°. To the graph of which exponential function can such a tangent be drawn at the point (0;1)?
The base of the exponential function must be greater than 2 but less than 3, since the required tangent angle is achieved somewhere between the functions $y=2^x$ and $y=3^x$. Such a number was found and it turned out to be quite unique.

An exponential function in which the tangent passing through the point (0;1) has an angle of inclination equal to 45° is usually denoted by: $y=e^x$ .
The basis of our function is irrational number. Mathematicians have derived the approximate value of this number $e=2.7182818284590…$.
In school mathematics courses, it is customary to round to the nearest tenth, that is, $e=2.7$.
Let's build a graph of the function $y=e^x$ and a tangent to this graph.
Our function is usually called exponential.
Properties of the function $y=e^x$.
1. $D(f)=(-∞;+∞)$.
2. Is neither even nor odd.
3. Increases throughout the entire domain of definition.
4. Not limited from above, limited from below.
5. Greatest value No, lowest value No.
6. Continuous.
7. $E(f)=(0; +∞)$.
8. Convex down.
In higher mathematics it has been proven that an exponential function is differentiable everywhere, and its derivative is equal to the function itself: $(e^x)"=e^x$.
Our function is widely used in many areas of mathematics (in mathematical analysis, in probability theory, in programming), and many real objects are associated with this number.

Example.
Find the tangent to the graph of the function $y=e^x$ at the point $x=2$.
Solution.
The tangent equation is described by the formula: $y=f(a)+f"(a)(x-a)$.
We sequentially find the required values:
1. $f(a)=f(2)=e^2$.
2. $f"(a)=e^a$.
3. $f"(2)=e^2$.
4. $y=f(a)+f"(a)(x-a)=e^2+e^2(x-2)=e^2*x-e^2$.
Answer: $y=e^2*x-e^2$

Example.
Find the value of the derivative of the function $y=e^(3x-15)$ at the point $x=5$.
Solution.
Let's remember the rule for differentiating a function of the form $y=f(kx+m)$.
$y"=k*f"(kx+m)$.
In our case $f(kx+m)=e^(3x-15)$.
Let's find the derivative:
$y"=(e^(3x-15))"=3*e^(3x-15)$.
$y"(5)=3*e^(15-15)=3*e^0=3$.
Answer: 3.

Example.
Examine the function $y=x^3*e^x$ for extrema.
Solution.
Let's find the derivative of our function $y"=(x^3*e^x)"=(x^3)"*e^x+x^3(e^x)"=3x^2*e^x+x^ 3*e^x=x^2*e^x(x+3)$.
The function has no critical points, since the derivative exists for any x.
Equating the derivative to 0, we get two roots: $x_1=0$ and $x_2=-3$.
Let's mark our points on the number line:

Problems to solve independently

1. Find the tangent to the graph of the function $y=e^(2x)$ at the point $x=2$.
2. Find the value of the derivative of the function $y=e^(4x-36)$ at the point $x=9$.
3. Examine the function $y=x^4*e^(2x)$ for extrema.

Lesson objectives: form an idea of ​​number e; prove differentiability of a function at any point X;consider the proof of the theorem on the differentiability of a function; checking the maturity of skills and abilities when solving examples of 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, develop the ability to find the derivative of an exponential function, test knowledge using a verification task and a test.

Developmental: promote the development of attention, the development of logical thinking, mathematical intuition, the ability to analyze, and apply knowledge in non-standard situations.

Educational: cultivate information culture, develop skills of working in a group and individually.

Teaching methods: verbal, visual, active.

Forms of training: collective, individual, group.

Equipment : textbook “Algebra and the beginnings of analysis” (edited by Kolmogorov), all tasks of group B “Closed segment” edited by A.L. Semenova, I.V. Yashchenko, multimedia projector.

Lesson steps:

  1. Statement of the topic, purpose, and objectives of the lesson (2 min.).
  2. Preparation for learning new material by repeating previously learned material (15 min.).
  3. Introduction to new material (10 min.)
  4. Initial comprehension and consolidation of new knowledge (15 min.).
  5. Homework assignment (1 min.).
  6. Summing up (2 min.).

Lesson progress

1. Organizational moment.

The topic of the lesson is announced: “Derivative of an exponential function. Number e.”, goals, objectives. Slide 1. Presentation

2. Activation of supporting knowledge.

To do this, at the first stage of the lesson we will answer questions and solve repetition problems. Slide 2.

At the board, two students work on cards, completing tasks like B8 Unified State Examination.

Assignment for the first student:

Assignment for the second student:

The rest of the students do independent work according to the following options:

Option 1 Option 2
1. 1.
2. 2.
3.
3.
4.
4.
5. 5.

Pairs exchange solutions and check each other's work, checking the answers on slide 3.

The solutions and answers of students working at the board are considered.

Checking homework No. 1904. Slide 4 is shown.

3. Updating the topic of the lesson, creating a problem situation.

The teacher asks to define an exponential function and list the properties of the function y = 2 x. Graphs of exponential functions are depicted as smooth lines, to which a tangent can be drawn at each point. But the existence of a tangent to the graph of a function at a point with the abscissa x 0 is equivalent to its differentiability at x 0.

For the graphs of the function y = 2 x and y = 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 abscissa 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 has been proven that there is a number greater than 2 and less than 3. It is usually denoted by the letter e. In mathematics it is established that the number e– irrational, i.e. represents an infinite decimal non-periodic fraction.

e = 2.7182818284590…

Note (not very serious). Slide 6.

On the next slide 7, portraits of great mathematicians appear - John Napier, Leonhard Euler and brief information 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 relaxation 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 the chair, close your eyelids, close your eyes tightly without opening your eyelids.

3. Arms 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. Test work in the form of a test. Task completion time – 5 minutes.

Evaluation criteria:

“5” – 3 points

“4” – 2 points

“3” - 1 point

6. Summing up the results of the work in the lesson.

  1. Reflection.
  2. Grading.
  3. Submission of test tasks.

7. Homework: paragraph 41 (1, 2); No. 539 (a, b, d); 540 (c, d), 544 (a, b).

“Closed segment” No. 1950, 2142.

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 a tangent can be drawn, then the function will be differentiable at each point of its domain of definition.

We will display in some coordinate axes several graphs of the function y = x a, For a = 2; a = 2.3; a = 3; a = 3.4.

At a point with coordinates (0;1). The 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 equal to 45 degrees.

Let us give a precise formulation of this statement: there is a number greater than 2 and less than 3, denoted by the letter e, such that the exponential function y = e x at point 0 has a derivative equal to 1. That is: (e ∆x -1) / ∆x tends to 1 as ∆x tends to zero.

This number e is irrational and is written as an infinite non-periodic decimal fraction:

e = 2.7182818284…

Since e is positive and non-zero, there is a logarithm to base e. This logarithm is called natural logarithm. Denoted by ln(x) = log e (x).

Derivative of an exponential function

Theorem: The function e x is differentiable at each point of its domain of definition, and (e x)’ = e x .

The exponential function a x is differentiable at each point of its domain of definition, and (a x)’ = (a x)*ln(a).
A corollary 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.

Using the formula for the derivative of the exponential function, we obtain:

(2 x)’ = (2 x)*ln(2).

Answer: (2 x)*ln(2).

Antiderivative of the exponential function

For an exponential function a x defined 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 proven this theorem.

Let's consider an example of finding the antiderivative of the exponential function.

Example: find the antiderivative of the function f(x) = 5 x. Let's use the formula given above and the rules for finding antiderivatives. We get: F(x) = (5 x) / (ln(5)) +C.

Topic: Derivative of an exponential function. Number .

Didactic purpose: form an idea of ​​the number e, prove the differentiability of the function at any point , differentiation of function . Give the concept natural logarithm.

Developmental goal: develop the ability to quickly and correctly carry out calculations using a personal computer.

Educational goal: continue to develop the ability to correctly perceive and actively remember new information what is the most important quality future specialist.

Visual aids: posters.

Handout: task cards for individual work. Equipment: teacher’s computer, multimedia projector, screen. Motivation cognitive activity students. Tell me which important role play logarithms in mathematics courses, as well as in general technical and special disciplines, while emphasizing the importance of the number e and the natural logarithm.

Progress of the lesson.

I. Organizational moment.

II. Explanation of new material.

1) Graphs of exponential functions.

3) Number .

4) Number calculation .

5) Formula for the derivative of an exponential function.

6) Calculate the natural logarithm usingMSExcel.

7) Antiderivative of the exponential function.

8) 3 meaning of the number .

III. Solving examples.

IV. Lesson summary.

V. Homework.

Explanation. The graphs of the exponential function were depicted as smooth lines (i.e., without kinks), to which a tangent can be drawn at each point. But the existence of a tangent to the graph of a function at the point with the abscissa is equivalent to its differentiability in x 0 . Therefore, it is natural to assume that it is differentiable at all points of the domain of definition. Let's draw several graphs of the function y=a X for y=2 X , y=Z X , y=2,З X (Appendix No. 1)

Let's draw tangents to them at the point with the abscissa . The tangents located to the graphs are different. We measure the angles of inclination of each of them to the abscissa axis and make sure that the angles of inclination of these tangents are approximately equal to 35°...51°, i.e. with increasing a slope to the graph at point M(0;1) gradually increases fromtg35 totg51.

There is a number greater than 2 and less than 3 such that the exponential function y = a X at point 0 has a derivative equal to 1. The base of this function is usually denoted by the letter e. The number e is irrational, and therefore is written as an infinite decimal

e ≈ 2.7182818284…

Using a computer, more than 2 thousand decimal places of the number e were found. The first numbers are 2.718288182459045 ~ 2.7.

Function often called an exponent. The resulting number plays a huge role in higher mathematics, just like the famous number 3.14. Formula for the derivative of an exponential function.

Theorem 1. Function .

Proof. Finding the increment of the function

at .

By definition of derivative , i.e. for any .

Prove that on one's own.

Example.

I give a definition: A natural logarithm is a logarithm to the base :

Theorem 2. Exponential function is differentiable at every point in the domain of definition, and .

Examples. , . Find derivatives of functions.

Calculating the natural logarithm usingMSExcel.

Example. Let's explore the function for increase (decrease) and extremum and build its graph.

Because for any , then the sign coincides with the sign . Hence on , - increases

on , - decreases.

To build a graph we use the programMSExcel.

Antiderivative of the exponential function.

Theorem 3. Antiderivative for the function onRis a function . Proof:

Examples:

A) ,

b) ,

V) , .

d) Calculate the area of ​​the figure, limited by lines , , , .

The meaning of e.

The resulting number plays a huge role in mathematics, physics, astronomy, biology and other sciences. Here are some:

This is glorious

It helps quite a lot

Make it clear to you and me

Year of birth of Tolstoy L.N. 2.71828

Euler's formula.

Leonhard Euler (1707-1783) Famous mathematician of the 18th century. Euler established the dependence of the friction force on the number of revolutions of the rope around the pile.

, -the force against which our effort is directed ; e;

The coefficient of friction between the rope and the pile, - winding angle, i.e. the ratio of the length of the arc covered by the rope to the radius of this arc. In everyday life, without even knowing it, we often take advantage of the benefits that Euler’s formula shows us.

What is a node? This is a string wound around a roller. How larger number turns of the rope, the greater the friction. The rule for increasing friction is such that, as the number of revolutions increases in an arithmetic progression, friction increases in a geometric progression.

The tailor unconsciously takes advantage of the same circumstance when sewing on a button. He wraps the thread many times around the area of ​​​​material captured by the stitch and then breaks it off, unless the thread is strong, the button will not come off. Here the rule already familiar to us applies: with an increase in the number of thread turns in an arithmetic progression, the sewing strength increases in a geometric progression. If there were no friction, we could not use buttons: the threads would unravel under their weight and the buttons would fall off. , -Ludwig Boltzmann (1844-1906), an Austrian physicist who discovered the fundamental law of nature, which determines the direction of all physical processes tending to equilibrium as the most probable state. -entropy, i.e. measure of achievement equilibrium system, -probability of the system state.

Lesson summary. Homework: No. 538, No. 542

Appendix No. 1