Derivative of the number e. “The number e. Derivative of the exponential function. Derivative of a power function

31.12.2019

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.

Let us display several graphs of the function y = x a in the same coordinate axes. 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.

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.

The derivative of an exponential function with a base 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 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.
(3) .

Let's find its derivative with respect to the variable x.
By definition, the derivative is the following limit: Let's transform this expression to reduce it to known mathematical properties and rules. To do this we need the following facts:
(4) ;
A) Exponent property:
(5) ;
B) Property of logarithm:
(6) .
IN)
Continuity of the logarithm and the property of limits for a continuous function: Here is a function that has a limit and this limit is positive.
(7) .

G)
;
.

The meaning of the second remarkable limit:
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:
(8)
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
(14) .
(1) .

Now we derive formula (2) for the derivative of the exponential function with a base of degree a.
;
.

We believe that and .
.

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 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 for their application.

Lesson objectives. Educational: repeat the definition of derivative, differentiation rules, derivative elementary functions , remember the graph and properties of the exponential function, develop the ability to find the derivative of the exponential function, control knowledge using test task

and dough.

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

Summing up (2 min.).

During the classes

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

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

Option 2

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. Examination homework

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 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 has been proven that there is a number greater than 2 and less than 3. It is usually denoted by the letter form an idea of number. In mathematics it is established that the number form an idea of number– 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. Verification work in the form of a test. Task completion time – 5 minutes.

Criteria for evaluation:

“5” – 3 points

“4” – 2 points

“3” - 1 point

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

Reflection.
Grading.
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.

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Slide captions:

DERIVATIVE OF AN EXPONENTIAL FUNCTION Number e Grade 11

REPETITION is the mother of learning!

Definition of an exponential function The function given by the formula y = a x (where a > 0, a ≠ 1) is called an exponential function with base a.

Properties of the exponential function y = a x a>1 0

Determination of the derivative of a function at point x 0. as Δ → 0. The derivative of the function f at the point x 0 is the number to which the difference ratio tends as Δx → 0.

Geometric meaning of the derivative x ₀ α A y = f(x) 0 x y к = tan α = f "(x ₀) The angle coefficient 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 the 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 " e (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 s (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 exponential is a power function. An exponent is a function where e is the base of natural logarithms.

1 y= e x 45° The function y= e x is called “exponent” x ₀ =0; tg 45° = 1 At point (0;1) the slope to the tangent to the graph of the function k = tg 45° = 1 - geometric meaning derivative of the exponent Exponent y = e x

Theorem 1. The function y = e is differentiable at each point of the domain of definition, and (e)" = e x x x The natural logarithm (ln) is the logarithm to the base e: ln x = log x e The exponential function is differentiable at each point of the domain of definition, 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 breeds mastery.” Tacitus Publius Cornelius - ancient Roman historian

Examples: Find the derivatives of the 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 things nearby

Leonhard Euler 1707 -1783 Russian scientist - mathematician, physicist, mechanic, astronomer... Introduced the designation for 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 “Neper number.”

The growth and decay of a function at an exponential rate is called exponential