Looks like lnx. Folding old. Logarithmic is lost. Looks like a statically-displaying function. The power of the logarithmic

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Looks like the natural logarithm of x from the road unit, subordinate to x:
(1) (Ln x) '=.

The logarithm for the base a is taken from the base unit, subdivided by the change x, multiplied by the natural logarithm from a:
(2) (Log a x) '=.

Dovedennya

Nekhai є deyake is a positive number, not equal to one. The function is easy to understand, I will lay it down from the change x, like є by the logarithm according to the present:
.
Qia function is assigned at. I know I will go by the winter x. For viznachennyam, lost є we will step on the border:
(3) .

It is possible to re-create a viraz, to bring it to the view of mathematical powers and rules. For all of us we need the nobility of the following facts:
A) The power of the logarithm. We need to know the next formula:
(4) ;
(5) ;
(6) ;
B) The continuity of the logarithm and the power between for the continuous function:
(7) .
Here is a deyaka of function, in the case of a clear boundary and a positive boundary.
V) The meaning of another wonderland:
(8) .

Zastosovuєmo ci facts to our border. A little bit of re-imagining the twist of algebra
.
For those who are stasis power (4) and (5).

.

We are quick to power (7) and other monstrous bridges (8):
.

I, nareshti, you can have power (6):
.
Logarithm e be called natural logarithm... Win is meant like this:
.
todі;
.

Tim ourselves have rejected formula (2) as an obscene logarithm.

Similar to the natural logarithm

Once again, we will write the formula of the obnoxious logarithm behind the base a:
.
The qia formula is the simplest view for the natural logarithm, for yak ,. Todi
(1) .

Through such simplicity, the natural logarithm is widely used in mathematical analysis and in other branches of mathematics, linked to differential numbers. logarithmic functions With the largest numbers, you can use the natural logarithm, the vicious power (6):
.

I will take the logarithm behind the base, you can know from the formula (1), as I blame the difference for the sign of differentiation:
.

Інші ways of proving an obscure logarithm

Here we are letting go, but we see the formula of an obscene exponential:
(9) .
Todi we can introduce the formula of an obscene natural logarithm, vrahoyuchi, but the logarithm є verbal function to the exponent.

The formula for the erotic natural logarithm is deducible, stuck with the formula for the funky funcion:
.
Our vipadku. sound function to natural logarithm є of the exponent:
.
We should start with the formula (9). Zminny can be understood to be like a letter. In formula (9), replace x by y:
.
Oskilki, then
.
Todi
.
The formula has been completed.

Now we have brought the formula of the obscene natural logarithm behind the aid differentiation rules folding function ... Oscillations of function and є sound one to one, then
.
Differentiating prices for changes x:
(10) .
Shown as one of the following:
.
There is a rule for differentiating folding functions:
.
Here. Pidstavami in (10):
.
Zvidsi
.

butt

Know the lost from ln 2x, ln 3xі ln nx.

Decision

Visible functions may have a similar view. To that we know, I will go from the function y = ln nx... Sometimes n = 2 and n = 3.I, by ourselves, we can accept the formulas for older types ln 2xі ln 3x .

Otzhe, shukaєmo go away from the function
y = ln nx .
Clearly the function will be folded into two functions:
1) Functions, scho to deposit from a change :;
2) Functions, scho to deposit from zminnoy :.
The current function is folded into the following functions:
.

I know I will go from the function to change x:
.
I know I will go from the function to change:
.
Zastosovuєmo formula of obsolete folding function.
.
Here we were presented.

Otzhe, we knew:
(11) .
Mi bachimo, it’s lost to n. The result is a whole lot of natural things, like recreating the specific function, the stasis formula of the logarithm of the creature:
.
- price post_yna. It is lost to zero. Todi for the rule of differentiation sumi maєmo:
.

vidpovid

; ; .

Looks like the logarithm of the modulus x

Of course, I will lose from one more important function - the natural logarithm of the module x:
(12) .

Vipadok is visible. Todi and function of ma viglyad:
.
It should start with the formula (1):
.

Now vipadok is visible. Todi and function of ma viglyad:
,
de.
I will also go to the end of the functionalities in the guided butt. Don't get stale from n і dorіvnyu
.
Todi
.

Combined, there are two drops into one formula:
.

Apparently, for the logarithm behind the base a, we can say:
.

Other orders of the natural logarithm

Functionality is visible
.
We knew її I will go to the first order:
(13) .

I know, I'll go in a different order:
.
I know I will go to the third order:
.
I know I'll go to the fourth order:
.

You can take note of what the n-th order of magnitude is missing:
(14) .
It is achieved by the method of mathematical induction.

Dovedennya

By adding to the formula (14) the value n = 1:
.
Oskilki, then for n = 1 , Formula (14) is valid.

It is admissible that formula (14) is valid for n = k. Probably, like a vaping, but the formula is valid for n = k + 1 .

Indeed, for n = k the value is:
.
Differentiating by change x:

.
Otzhe, we otrima:
.
The formula is based on formula (14) for n = k + 1 ... Thus, if we let it down, then formula (14) is valid for n = k, but formula (14) is valid for n = k + 1 .

Therefore, formula (14), for the primitive n-th order, is valid for any n.

Similar orders in the logarithm behind the base a

To know I will lose the n-th order of the logarithm behind the base a, it is necessary to see it through the natural logarithm:
.
Zastosovyuchi formula (14), we know the n-th source:
.

When differentiating, showing statical functions for the bulky shotguns twists and turns with a handy logarithmic obscenity. At the tsіy statty mi podladynemu put її zasosuvannya z lecturer decisions.

The submitted viclade is able to respect the table of the old, the rules of differentiation and knowledge of the formula of the old folding function.

Visnovok formulas are logarithmic and obscene.

With a reference, the logarithm is carried out according to the representation e, simply the type of function, the power of the logarithm, and it is known from the implicitly given function:

For the butt, I know I will go to show the statical function x in step x.

Logarithm yesє. Behind the power of the logarithm. Differentiation of both parts of equality to produce the result:

as follows: .

The whole butt can be viral and without a victorious logarithmic obscenity. You can carry out redevelopment actions and go from the differentiation of the demonstration statical function to the familiar folding function:

Butt.

Know the lost function .

Decision.

Have a wide application function It is also possible to learn from the rules of differentiation. Ale, due to the bulkiness, the price is thirsty for a helpless re-revolution. In such cases, the smallest vicoristovuvati the formula is logarithmic, obscene ... To what? At the same time, you should be aware of it.

We know a lot. The re-incarnations will have the power of the logarithm (the logarithm of the fraction is the leading difference between the logarithms, and the logarithm dorіvnyu sumi logarithms, if there are more steps in the rotation under the sign of the logarithm, it is possible to win as the coefficient in front of the logarithm):

The reconfigurations brought us to the end of the simple line, which is lost, which is easy to find:

The result is presented as a logarithmic, simple, and recognizable result:

For fastening to the material, guided by a chain of butts without reports, explanations.

Butt.

Know the lost display statical function

Folding old. Logarithmic is lost.
Looks like a statically-displaying function

Prodovzhuєmo develop its differentiation technology. At this level, the passage of material is closed, more foldable old, and also knowable with new tricks and tricks of the familiar, old age, with a logarithmic primitive.

Tim chitacham, who low rіven preparation, then turn up to the stat How can I go? put on the solution, Yaka, allowing you to take your navichki practically from scratch. Anything you need to respectfully pick up Suitable for folding function, Intelligence and resolution all aim me butt. The Danish lesson is logically the third behind the rakhunk, and if you master it, you will be able to differentiate and complete the folding functions. Negligently hitrimuvatisya position "Kudi shche? That and so whistle! " control robots and often take part in practice.

Possibly a repetition. At the level Suitable for folding function we looked at a number of attachments with lecture notes. In the course of the differentiation of the differential calculation mathematical analysis- Differentiation will happen even more often, and do not wait manually (that is not necessary). Tom will be consumed by the old ones. By the adjunct "candidates" for the most lost and simplest folding functions, for example:

For the rule of differentiation of folding functions :

In case of vivchennih topics mat in the future, such a report is most often not necessary, it is not necessary to transfer it, but the student knows at least some of the same on the autopilot automaton. Uyavimo, about 3 years old, having blown through the telephone call, and fed a receptive voice: "Why did you get the tangent two ix?" The blame for the price is the following: .

The first butt will be one of the signs for independent solution.

butt 1

Knowing such old asleep, in one day, for example :. For vikonannya zavdannya need vikoristovuvati tilki table of old elementary functions(I haven't forgotten myself yet). It's hard to find out, I recommend rereading the lesson Suitable for folding function.

, , ,
, , ,
, , ,

, , ,

, , ,

, , ,

, ,

Vidpovidі in kіntsі lesson

folding old

Pislya ahead of the front artillery preparation will be less terrible butt, with 3-4-5 nested functions. It is possible, both of these two butts to be built as foldable figures, if they are visceral (or suffer), then everything else in differential calculus will be childish.

butt 2

Know the lost function

It was already meant, with the familiar rudimentary folding function, the first for everything, it is necessary right ROZIBRATISYA at deposits. At quiet vipads, if I’m guessing, I’m guessing a cinnamon priyom: we take the value "x", for example, and try (thoughts or in black) to present the value in "terrible viraz".

1) We need to count the number of viraz, which means the sum is the best investment.

2) Then it is necessary to calculate the logarithm:

4) Let's convert the cosine into a cube:

5) At the bottom of the bottom line:

6) І, nareshty, the most significant function is the square root:

Formula of differentiation of folding function stagnate in vortex order, From the very first function, to the very inner. virishuєmo:

Nachebto without pardons ...

(1) I take it from the square root.

(2) I will take away from the price, the vicarist rule

(3) There are zero trips. From another dodanku I take away from the step (cube).

(4) I take it from the cosine.

(5) I will take the logarithm.

(6) I, nareshty, I will take all the most important investment.

You can get busy with a big butt. See, for example, the book of Kuznotsov, and appreciate all the beauty and simplicity of the rosy childish. Having thought, I’m going to love to give it to sleep, to reconsider, mind a student, as I know and I’ll lose the folded function, for it’s not mind.

An offensive buttstock for an independent solution.

butt 3

Know the lost function

Pidkazka: A collection of stasis rules of linearity and the rule of differentiation create

Outside the decision and see in the end of the lesson.

Having come the hour to go to something more compact and cute.
It’s not a normal situation, if in the butt there’s not two, but three functions... How do I know I will go from three multipliers?

butt 4

Know the lost function

I am astonished, but why is it impossible to transform three functions into two functions? For example, if we have two polynomials in our creation, then it is possible to create critical arcs. All the functions of the development are in the opened application: steps, exponent and logarithm.

In such cases, it is necessary lastly zasosuvati rule of differentiation create two times

The focus of the polarity is in the fact that for "y" mi two functions are meaningfully twir, and for "ve" - ​​logarithm :. Why is it so possible to zrobiti? A hiba - tse not tvir two multipliers and the rule is not practical ?! Nichogo folding mute:

Now it has become a rule to the bow:

You can still turn around and blame it on the bows, but in this case you can look more beautifully in such a view - it will be easier to overturn.

The butt can be viewed in another way:

The insult of the way of verishennya is absolutely equitable.

butt 5

Know the lost function

Tse butt for an independent solution, in a clever way, in the first way.

Clearly analogous butt with a fraction.

butt 6

Know the lost function

Here you can drink the decilkom with nobles:

For this:

Alle the solution to be recorded more compactly, as in Pershu chergu keruvatisya by the rule of differentiation of private , Accepted for the entire date:

As a matter of principle, the butt is true, and if you put it in such a view, then there will be no mercy. If the hour is obvious, it’s bazed to translate it into Chernets, but why can’t it be explained? Pointing the number to the spіlny znamennyk i awake three-top shot:

Minus dodatkovyh forgive the pole in that, when є risik to get pardon already not with the familiar old, but with banal school re-incarnations. On the other hand, it’s not easy to deny the employee and ask him to “bring it to the point” when I’m lost.

Bigger simple buttstock for self-help solution:

butt 7

Know the lost function

Prodovzhumo master the priyomy and knowledge of the obscene, and at once the typical vipadoks are discernible, if for the differentiation of the proponations there is a "terrible" logarithm

butt 8

Know the lost function

Here you can take a good walk, the rule of differentiation of folding functions:

But the first croc will immediately become agonized - to trace the recognition of an unacceptable fraction of the step, and then more from the fraction.

Tom in front of tim as my brothers go away from the “cool” logarithm, I’ll say goodbye in front of them, and I’ll see them from the school of power:

! It’s easy to practice, rewrite the formulas right there. I don't know how to zoshiti, grind them into a piece of paper, the fragments are lost, but I will wrap myself around the formulas.

The solution itself can be arranged approximately like this:

Retrofit function:

I know I will go:

The earlier re-implementation of the function itself significantly simplified the solution. In such a rank, if there is a logarithm for the differentiation of proponations, then it should be completely "razvaliti".

And at once a couple of non-folding butts for an independent solution:

butt 9

Know the lost function

butt 10

Know the lost function

All revisions and suggestions in the end of the lesson.

logarithmic

How much is lost from logarithms - tse such licorice is music, then wine is food, but why is it not possible to organize a logarithm piece by piece? You can, you can! I need to navigate.

butt 11

Know the lost function

We recently saw similar butts. How robiti? You can lastly fix the rule of differentiation of the private, and then create the rule of differentiation. It is not like the method of the field in the fact that the tripoverkhova drib is great, for which the mother does not want to help.

But in theory and practice, such a miraculous rich, like a logarithmic old one. Logarithms can be organized by piece, "hanging" on the offense part:

Note : Since the function can accept negative meanings, then, apparently, it is necessary to use the following modules: , I know the results of the differentiation. However, it is permissible and accurate registration, de for the changes they take up to respect complex value. Alle, with a strong severity, then in that and in the last fall, it was.

Now it is necessary to maximize the "value" of the logarithm of the right part (the formulas before ochima?). I will write down the process in two reports:

Vlasne attack before differentiation.
Laying out a stroke of offense to a part:

Going to the right part is simple to finish, I won't comment, if you read the text, then you are guilty of imprisoning it.

Yak bootie with lіvoyu partna?

We have a live part with us folding function... I transmit the power supply: "Why, there is also one letter" іgrek "in the logarithm?".

On the right in that, scho tsya "one letter іgrek" - BY YOURSELF Є FUNCTIONS(It’s not even smarter, it’s going to go up to the statty. It’s like a function, implicitly set). To that, the logarithm is the whole function, and "іgrek" is internal function... The first rule of differentiation of folding functions :

At the left part of the yak behind the wave of a charming wand, we "namalyuvalasya" is obsolete. Following the rule of proportion, the "igrek" is thrown from the banner of the left part to the top of the right part:

And now it’s hidden, about what kind of "Greek" -functions were used during the differentiation? Wondering at the mind:

Residual view:

butt 12

Know the lost function

Tse butt for an independent solution. Zrazok formalized the butt of this type in the end of the lesson.

Behind the additional logarithmic obscenity, it is possible to look like from butts No. 4-7, on the right, where the function is simpler, and maybe it’s not even true.

Looks like a statically-displaying function

They did not look at this function. The streak-showing function - the function, at the yak і steps і before laying down at the "ix"... A classic butt, how can you bring it to a be-like handler, or for a lecture:

How do I know I will go from the statistic-display function?

It is necessary to vikoristovuvati only when looking at it - I will logarithmically. Navishuєmo logarithms on offense parts:

As a rule, in the right part of the logarithm there are steps:

As a result, in the right part, we have two functions, as we will differentiate for the standard formula .

Of course, I’ll lose it, for a lot of robustness, offending parts of the stroke:

Some of the awkward ones:

residual:

As soon as the re-creation is not invisible, be weasel, respectfully re-read the explanations of Example No. 11.

V practical knowledge statical-demonstration function will be folded, not displayed lekts_yny butt.

butt 13

Know the lost function

Vikoristovuєmo logarithm will be lost.

In the right part we have a constant і twir of two multipliers - "ix" і "logarithm of the logarithm ix" (the logarithm of contributions is more than one logarithm). When differentiating a constant, like a memory, it is more beautiful to blame for a bad sign at once; і, slyly, there is a rule :

hey
(1)
є Differentiated function of change x. On the cob, we can see the value x, for those y, the same value is:. It can be shown that all results are rejected and for negative meanings.

For kids, you should know the lost function (1),
,
and then count the lost. Todi for the rule of differentiation of folding functions,
.
Zvidsi
(2) .

The type of the logarithm of the function is called the logarithmic common:
.

Logarithmic function y = f (x) - the same as the natural logarithm of the function: (Ln f (x)) '.

Variety of negative values ​​y

It is now clear, if you can accept both positive and negative meanings. At the same time, the logarithm of the module and I know it will be lost:
.
Zvidsi
(3) .
So, in a zealous way, it is necessary to know the origin of the logarithm of the module of the function.

Porіvnyuchi (2) and (3) mi maєmo:
.
So that the formal result of calculating the logarithmic commonplace does not lie in the one taken by the modulo chi ni. Moreover, when calculated in a logarithmic, commonplace, we can not worry about those, which sign of a function.

This situation can be clarified with the help of complex numbers. Nekhai, with deyak values ​​x, is negative :. As long as the numbers are displayed, the function is not meaningful. However, if you enter complex numbers into the view, then it will be possible to start:
.
To function and adapt to a complex post:
.
Oscillations are lost as a result of the post-mortem, then
.

The power of the logarithmic

With a look of vipliv, scho logarithmic does not change, if you multiply the function by a sufficient amount of time :
.
Dyisno, stuck the power of the logarithm, formulas lonely sumiі lonely post, Maєmo:

.

Stagnant logarithmic

I will save logarithmically I will go by hand in quiet vipads, if the function is based on the creation of the state. displays of their functions... In general, the operation of the logarithm converts the twir functions into the sum. I will forgive the calculation of the old.

butt 1

Know the lost function:
.

Decision

Logarithms to the output function:
.

Differentiating by change x.
The old tables have:
.
Zastosovuєmo rule of differentiation of folding functions.
;
;
;
;
(A1.1) .
Multiplied by:

.

Otzhe, we knew the logarithm will go away:
.
The sound is known to be lost from the following functions:
.

Note

If I want to pick up only valid numbers, then take the logarithm of the module of the outgoing function:
.
Todi
;
.
I we rejected formula (A1.1). The result is unchanged.

vidpovid

butt 2

Behind the additional logarithmic obscene, know the lost functions
.

Decision

logarithms:
(A2.1) .
Differentiating by change x:
;
;

;
;
;
.

Multiplied by:
.
I’ll be able to recognize it logarithmically:
.

The following functions are available:
.

Note

Here the function is not clear :. Vaughn is designated for. If you do not allow, but the logarithm can be used for negative values ​​of the argument, then the formula (A2.1) should be written as follows:
.
scraps

і
,
then it does not fit into the residual result.

vidpovid

butt 3

Know I will go
.

Decision

Differentiation of the viconuєmo behind the help of the logarithmic obscene. Logarithms, vrahoyuchi scho:
(A3.1) .

Differentiated, I will take a logarithmic loss.
;
;
;
(A3.2) .

Oskilki, then

.

Note

Probably calculating without prejudice, but the logarithm can be used for negative values ​​of the argument. For the whole logarithm of the module of the output function:
.
Todi substitute (A3.1) maєmo:
;

.
It is necessary to change the result (A3.2), but the result does not change.