Formulas for differentiating logarithmic and exponential functions. Logarithmic derivative. Differentiation of exponential function. The case of negative y values

Lesson topic: “Differentiation of exponential and logarithmic functions. antiderivative exponential function» in UNT tasks

Target : to develop students' skills in applying theoretical knowledge on the topic “Differentiation of exponential and logarithmic functions. An antiderivative of an exponential function” for solving UNT problems.

Tasks

Educational: to systematize the theoretical knowledge of students, to consolidate the skills of solving problems on this topic.

Developing: develop memory, observation, logical thinking, mathematical speech of students, attention, self-esteem and self-control skills.

Educational: promote:

the formation of students' responsible attitude to learning;

development of a sustainable interest in mathematics;

creating positive intrinsic motivation to study mathematics.

Teaching methods: verbal, visual, practical.

Forms of work: individual, frontal, in pairs.

During the classes

Epigraph: "Mind consists not only in knowledge, but also in the ability to apply knowledge in practice" Aristotle (slide 2)

I. Organizing time.

II. Solving the crossword puzzle. (slide 3-21)

The 17th-century French mathematician Pierre Fermat defined this line as "the straight line closest to the curve in a small neighborhood of a point."

Tangent

The function that is given by the formula y = log a x.

logarithmic

The function that is given by the formula y = a X.

Demonstration

In mathematics, this concept is used to find the speed of movement material point and the slope of the tangent to the graph of the function at a given point.

Derivative

What is the name of the function F (x) for the function f (x), if the condition F "(x) \u003d f (x) is satisfied for any point from the interval I.

antiderivative

What is the name of the relationship between X and Y, in which each element of X is associated with a single element of Y.

Derivative of displacement

Speed

A function that is given by the formula y \u003d e x.

Exhibitor

If the function f(x) can be represented as f(x)=g(t(x)), then this function is called…

III. Mathematical dictation. (Slide 22)

1. Write down the formula for the derivative of the exponential function. ( a x)" = a x ln a

2. Write down the formula for the derivative of the exponent. (e x)" = e x

3. Write down the formula for the derivative of the natural logarithm. (lnx)"=

4. Write down the formula for the derivative of the logarithmic function. (log a x)"=

5. Write down the general form of antiderivatives for the function f(x) = a X. F(x)=

6. Write down the general form of antiderivatives for the function f(x) =, x≠0. F(x)=ln|x|+C

Check the work (answers on slide 23).

IV. Problem solving UNT (simulator)

A) No. 1,2,3,6,10,36 on the board and in the notebook (slide 24)

B) Work in pairs No. 19.28 (simulator) (slide 25-26)

V. 1. Find errors: (slide 27)

1) f (x) \u003d 5 e - 3x, f "(x) \u003d - 3 e - 3x

2) f (x) \u003d 17 2x, f "(x) \u003d 17 2x ln17

3) f(x)= log 5 (7x+1),f "(x)=

4) f (x) \u003d ln (9 - 4x), f "(x) \u003d
.

VI. Student presentation.

Epigraph: “Knowledge is such a precious thing that it is not shameful to get it from any source” Thomas Aquinas (slide 28)

VII. Homework No. 19,20 p.116

VIII. Test (reserve task) (slide 29-32)

IX. Summary of the lesson.

“If you want to participate in the big life, fill your head with math while you can. She will then provide you with great help throughout your life ”M. Kalinin (slide 33)

Let be
(1)
is a differentiable function of x . First, we will consider it on the set of x values ​​for which y takes positive values: . In what follows, we will show that all the results obtained are also applicable for negative values ​​of .

In some cases, to find the derivative of the function (1), it is convenient to preliminarily take the logarithm
,
and then calculate the derivative. Then, according to the rule of differentiation of a complex function,
.
From here
(2) .

The derivative of the logarithm of a function is called the logarithmic derivative:
.

The logarithmic derivative of the function y = f(x) is the derivative of the natural logarithm of this function: (log f(x))′.

The case of negative y values

Now consider the case when the variable can take both positive and negative values. In this case, take the logarithm of the modulus and find its derivative:
.
From here
(3) .
That is, in the general case, you need to find the derivative of the logarithm of the modulus of the function.

Comparing (2) and (3) we have:
.
That is, the formal result of calculating the logarithmic derivative does not depend on whether we took modulo or not. Therefore, when calculating the logarithmic derivative, we do not have to worry about what sign the function has.

This situation can be clarified with the help of complex numbers. Let, for some values ​​of x , be negative: . If we consider only real numbers, then the function is not defined. However, if we introduce complex numbers into consideration, we get the following:
.
That is, the functions and differ by a complex constant:
.
Since the derivative of a constant is zero, then
.

Property of the logarithmic derivative

From such consideration it follows that the logarithmic derivative does not change if the function is multiplied by an arbitrary constant :
.
Indeed, applying logarithm properties, formulas derivative sum and derivative of a constant, we have:

.

Application of the logarithmic derivative

It is convenient to use the logarithmic derivative in cases where the original function consists of a product of power or exponential functions. In this case, the logarithm operation turns the product of functions into their sum. This simplifies the calculation of the derivative.

Example 1

Find the derivative of a function:
.

Decision

We take the logarithm of the original function:
.

Differentiate with respect to x .
In the table of derivatives we find:
.
We apply the rule of differentiation of a complex function.
;
;
;
;
(P1.1) .
Let's multiply by:

.

So, we found the logarithmic derivative:
.
From here we find the derivative of the original function:
.

Note

If we want to use only real numbers, then we should take the logarithm of the modulus of the original function:
.
Then
;
.
And we got the formula (A1.1). Therefore, the result has not changed.

Answer

Example 2

Using the logarithmic derivative, find the derivative of a function
.

Decision

Logarithm:
(P2.1) .
Differentiate with respect to x :
;
;

;
;
;
.

Let's multiply by:
.
From here we get the logarithmic derivative:
.

Derivative of the original function:
.

Note

Here the original function is non-negative: . It is defined at . If we do not assume that the logarithm can be determined for negative values ​​of the argument, then formula (A2.1) should be written as follows:
.
Insofar as

and
,
it will not affect the final result.

Answer

Example 3

Find the derivative
.

Decision

Differentiation is performed using the logarithmic derivative. Logarithm, given that:
(P3.1) .

By differentiating, we get the logarithmic derivative.
;
;
;
(P3.2) .

Because , then

.

Note

Let's do the calculations without assuming that the logarithm can be defined for negative values ​​of the argument. To do this, take the logarithm of the modulus of the original function:
.
Then instead of (A3.1) we have:
;

.
Comparing with (A3.2) we see that the result has not changed.

When differentiating, it is indicative power function or bulky fractional expressions It is convenient to use the logarithmic derivative. In this article, we will look at examples of its application with detailed solutions.

Further presentation implies the ability to use the table of derivatives, the rules of differentiation and knowledge of the formula for the derivative of a complex function.

Derivation of the formula for the logarithmic derivative.

First, we take the logarithm to the base e, simplify the form of the function using the properties of the logarithm, and then find the derivative of the implicitly given function:

For example, let's find the derivative of the exponential power function x to the power of x.

Logarithm gives . According to the properties of the logarithm. Differentiating both parts of the equality leads to the result:

Answer: .

The same example can be solved without using the logarithmic derivative. You can make some transformations and go from differentiating an exponential power function to finding the derivative of a complex function:

Example.

Find the derivative of a function .

Decision.

In this example, the function is a fraction and its derivative can be found using the rules of differentiation. But due to the cumbersome expression, this will require many transformations. In such cases, it is more reasonable to use the formula for the logarithmic derivative . Why? You will understand now.

Let's find it first. In transformations, we will use the properties of the logarithm (the logarithm of the fraction is equal to the difference logarithms, and the logarithm of the product is equal to the sum of logarithms, and the degree of the expression under the sign of the logarithm can also be taken out as a coefficient in front of the logarithm):

These transformations have led us to a fairly simple expression, the derivative of which is easy to find:

We substitute the result obtained into the formula for the logarithmic derivative and get the answer:

To consolidate the material, we give a couple more examples without detailed explanations.

Example.

Find the derivative of an exponential power function

Algebra and beginning of mathematical analysis

Differentiation of the exponential and logarithmic function

Compiled by:

mathematics teacher MOU secondary school №203 CHETs

Novosibirsk city

Vidutova T.V.

Number e. Function y=e x, its properties, graph, differentiation

1. Let's build graphs for various bases a: 1. y = 2 x 3. y = 10 x 2. y = 3 x (Option 2) (Option 1) "width="640"

Consider the exponential function y = a x, where a 1.

Let's build for different bases a charts:

1. y=2 x

3. y=10 x

2. y=3 x

(Option 2)

(1 option)

1) All graphs pass through the point (0; 1);

2) All charts have horizontal asymptote y = 0

at X  ∞;

3) All of them are turned with a bulge down;

4) They all have tangents at all their points.

Draw a tangent to the graph of the function y=2 x at the point X= 0 and measure the angle formed by the tangent to the axis X

With the help of exact constructions of tangents to graphs, it can be seen that if the base a exponential function y = a x the base gradually increases from 2 to 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'.

Therefore, there is a basis a, for which the corresponding angle is 45'. And this meaning a concluded between 2 and 3, because at a= 2 the angle is 35’, with a= 3 it is equal to 48'.

In the course of mathematical analysis, it is proved that this base exists, it is usually denoted by the letter e.

Determined that e - an irrational number, that is, it is an infinite non-periodic decimal fraction:

e = 2.7182818284590… ;

In practice, it is usually assumed that e ≈ 2,7.

Graph and function properties y = e x :

1) D(f) = (- ∞; + ∞);

3) increases;

4) not limited from above, limited from below

5) has neither the largest nor the smallest

values;

6) continuous;

7) E(f) = (0; + ∞);

8) convex down;

9) is differentiable.

Function y = e x called exhibitor .

In the course of mathematical analysis, it was proved that the function y = e x has a derivative at any point X :

(e x ) = e x

(e 5x )" = 5e 5x

(e x-3 )" = e x-3

(e -4x+1 )" = -4e -4x-1

Example 1 . Draw a tangent to the graph of the function at the point x=1.

2) f()=f(1)=e

4) y=e+e(x-1); y = ex

Answer:

Example 2 .

x = 3.

Example 3 .

Investigate a function for an extremum

x=0 and x=-2

X= -2 - maximum point

X= 0 – minimum point

If the base of the logarithm is the number e, then they say that given natural logarithm . For natural logarithms, a special notation has been introduced ln (l - logarithm, n - natural).

Graph and properties of the function y = ln x

Function properties y = lnx:

1) D(f) = (0; + ∞);

2) is neither even nor odd;

3) increases by (0; + ∞);

4) not limited;

5) has neither the largest nor the smallest values;

6) continuous;

7) E (f) = (- ∞; + ∞);

8) convex top;

9) is differentiable.

0 the differentiation formula "width="640" is valid

In the course of mathematical analysis, it was proved that for any value x0 the differentiation formula is valid

Example 4:

Calculate the value of the derivative of a function at a point x = -1.

For example:

Internet resources:

• http://egemaximum.ru/pokazatelnaya-funktsiya/
• http://or-gr2005.narod.ru/grafik/sod/gr-3.html
• http://en.wikipedia.org/wiki/
• http://900igr.net/prezentatsii
• http://ppt4web.ru/algebra/proizvodnaja-pokazatelnojj-funkcii.html

Differentiation of exponential and logarithmic functions

1. Number e. Function y \u003d e x, its properties, graph, differentiation

Consider an exponential function y \u003d a x, where a\u003e 1. For different bases a we get different graphs (Fig. 232-234), but you can see that they all pass through the point (0; 1), they all have a horizontal asymptote y \u003d 0 at , they are all convex downwards and, finally, they all have tangents at all their points. For example, let's draw a tangent to graphics functions y \u003d 2x at the point x \u003d 0 (Fig. 232). If you make precise constructions and measurements, you can make sure that this tangent forms an angle of 35 ° (approximately) with the x-axis.

Now let's draw a tangent to the graph of the function y \u003d 3 x, also at the point x \u003d 0 (Fig. 233). Here the angle between the tangent and the x-axis will be greater - 48°. And for the exponential function y \u003d 10 x in a similar
situation, we get an angle of 66.5 ° (Fig. 234).

So, if the base a of the exponential function y \u003d ax gradually increases from 2 to 10, then the angle between the tangent to the graph of the function at the point x \u003d 0 and the x-axis gradually increases from 35 ° to 66.5 °. It is logical to assume that there is a base a for which the corresponding angle is 45°. This base must be enclosed between the numbers 2 and 3, since for the function y-2x the angle of interest to us is 35 °, which is less than 45 °, and for the function y \u003d 3 x it is 48 °, which is already a little more than 45 °. The basis of interest to us is usually denoted by the letter e. It is established that the number e is irrational, i.e. is an infinite decimal non-periodic fraction:

e = 2.7182818284590...;

in practice it is usually assumed that e=2.7.

Comment(not very serious). It is clear that L.N. Tolstoy has nothing to do with the number e, nevertheless, in writing the number e, please note that the number 1828 is repeated twice in a row - the year of birth of L.N. Tolstoy.

The graph of the function y \u003d e x is shown in Fig. 235. This is an exponent that differs from other exponents (graphs of exponential functions with other bases) in that the angle between the tangent to the graph at x=0 and the x-axis is 45°.

Properties of the function y \u003d e x:

1)
2) is neither even nor odd;
3) increases;
4) not limited from above, limited from below;
5) has neither the largest nor the smallest values;
6) continuous;
7)
8) convex down;
9) is differentiable.

Go back to § 45, take a look at the list of properties of the exponential function y \u003d a x for a > 1. You will find the same properties 1-8 (which is quite natural), and the ninth property associated with
differentiability of the function, we did not mention then. Let's discuss it now.

Let us derive a formula for finding the derivative y-ex. In doing so, we will not use the usual algorithm, which was developed in § 32 and which has been successfully applied more than once. In this algorithm, at the final stage, it is necessary to calculate the limit, and our knowledge of the theory of limits is still very, very limited. Therefore, we will rely on geometric premises, considering, in particular, the very fact of the existence of a tangent to the graph of the exponential function beyond doubt (that is why we so confidently wrote down the ninth property in the above list of properties - the differentiability of the function y \u003d e x).

1. Note that for the function y = f(x), where f(x) = ex, we already know the value of the derivative at the point x = 0: f / = tg45°=1.

2. Let us introduce the function y=g(x), where g(x) -f(x-a), i.e. g(x)-ex "a. Fig. 236 shows the graph of the function y \u003d g (x): it is obtained from the graph of the function y - fx) by shifting along the x axis by |a| scale units. The tangent to the graph of the function y \u003d g (x) in point x-a is parallel to the tangent to the graph of the function y \u003d f (x) at the point x -0 (see Fig. 236), which means that it forms an angle of 45 ° with the x-axis. Using the geometric meaning of the derivative, we can write that g(а) =tg45°;=1.

3. Let's return to the function y = f(x). We have:

4. We have established that for any value of a, the relation is true. Instead of the letter a, one can, of course, use the letter x; then we get

From this formula, the corresponding integration formula is obtained:

A.G. Mordkovich Algebra Grade 10

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