It’s easy to memorize.

Well, we won’t go far, the function is immediately visible. Yaka function є ringing for show function? logarithm:

Our vipadku has a number to serve:

Such a logarithm (tobto a logarithm with a base) is called "natural", and for a new victoriously special meaning: replace what is written.

What is it for? Zychayno well,.

The look of the natural logarithm is also simpler:

put on:

  1. Know the lost function.
  2. Why do you have a lost function?

indications: The exponent and the natural logarithm are functions uniquely simple from the point of view. Show and logarithmic functions because I’m going to walk away from you, I’ll go out with you, in order to pass the rules of differentiation.

Differentiation rules

What rules? I know a new term, I know ?! ...

differentiation- the whole process of znakhozhennya pohіdnoї.

Only and everything. How can you name the process in one word? Chi is not a product ... The differential of mathematics is called those same primitive functions at. View the term in Latin differentia - development. Axis.

When all these rules are given, we will be victorious in two functions, for example, i. We also need to know the formulas for the best:

Total є 5 rules.

Constant to blame for the bad sign.

Якшо - as a post_ine number (constant), todі.

Obviously, the whole rule of thumb is for business :.

Brought to you. Come on, it's simpler.

Put it on.

Know the lost functions:

  1. in point;
  2. in point;
  3. in point;
  4. at point.

solution:

  1. (It is the same at all points, so yak tse line function, Pam'yataєsh?);

lost create

Everything is similar here: we introduce a new function and know it well:

kidnapped:

put on:

  1. Know the lost functions;
  2. Know the lost function at the point.

solution:

Go to the show function

Now your knowledge is to be finished, what if I know and I will go away if it is a demonstration function, and not just an exponential (not forgetting, how tse so?).

The same, de - the number.

If I already know the function is lost, then let's try to bring our function to a new basis:

For all speedy people, let's just use the rule: Todi:

Well, the axis, viyshlo. Now try to know it will go away, and do not forget that the function is foldable.

Has it gone?

Axis, change yourself:

The formula came out even more similar to the lost exponentiality: like bulo, so it has become redundant, it has appeared only a multiplier, like є just a number, but not a wicked one.

put on:
Know the lost functions:

indications:

It’s just a number, as it’s a shame to fuck without a calculator, it’s impossible to write it down in a simpler view. To that, in such a view, it is necessary.

    Amazingly, there is a part of two functions here, so the differentiation rule is very common:

    This application has two functions:

Logarithmic function

Here it is analogous: I also know the form of the natural logarithm:

That, if you know enough of the logarithm from the beginning, for example,:

It is necessary to bring the whole logarithm to the base. And how to remember the logarithm? I am encouraged to remember the formula:

Tilki now substitute for writing:

The denominator has just a constant (post-number, no change). It’s even simpler to go:

The current display and logarithmic functions cannot be used in ADI, otherwise we will not occupy the nobility.

It has a folding function.

So also "foldable function"? Ні, tse not a logarithm, і not an arctangent. These functions can be foldable for intelligence (if you want the logarithm to be foldable, read the topic "Logarithms" and go through everything), but from the point of view of mathematics, the word "foldable" does not mean "important".

Have a small conveyor: sit two people and prophesy like objects. For example, peeling the chocolate bar into a chunk, and the other wrapping around a line. To go through such folds of the object: a chocolate bar, lit up and tied with a line. To make a chocolate bar, you will need a victorious game in a ringing order.

Let’s solve the mathematical conveyor: first we will know the cosine of the number, and then we will deduct the number of squares. Also, they give us a number (chocolate), I know the cosine (obgortka), and then they put me in a square (tie it with a line). What happened? Functions. The whole butt has a folding function: if for the knowledge of the meaning, it is predicted by the first step without any change, and, at the same time, another step, which happened as a result of the first.

In other words, folding function - a whole function, the argument of which is the function: .

For our butt ,.

We can do it all in a zoomed order: the set of tees is squared, and then I whisper the cosine of the rejected number :. It’s awkward to ask if the result will be more likely to grow. The particularity of folding functions is important: when the order changes, the function changes.

Another butt: (the same). ...

Diya, yake robimo we will stay "Zovnishnoi" function, And dia, scho zdіysnyuєtsya first - apparently "Internal" function(Name it informally, I am implanting it only in order to explain the material in a simple way).

Try it yourself, as a function is called, and as an internal one:

indications: The added internal and external functions are even more similar to the replacement of the minions: for example, in the function

  1. Let's go to the first yaku diyu? I will make a sinus, and only we will increase the volume in cubic meters. It means, the internal function, but the call.
    And pochatkova function є їх composition :.
  2. Internal :; call :.
    Perevirka :.
  3. Internal :; call :.
    Perevirka :.
  4. Internal :; call :.
    Perevirka :.
  5. Internal :; call :.
    Perevirka :.

viroblyaєmo replace the winter and accept the function.

Well, well, now we will trim our chocolate bar - I will lose it. The order is dyy zvorotny: I will get lost new functions, We can multiply the result for the lost internal function. One hundred percent of the wiggle's application is like this:

Іnshy butt:

Otzhe, I will formulate, nareshty, official rule:

Algorithm for the knowledge of the obsolete folding function:

Everything is simple, right?

Convertible on butts:

solution:

1) Internal :;

Zovnishnya :;

2) Internal :;

(Do not think now to speed up!)

3) Internal :;

Zovnishnya :;

It is immediately evident that there is a three-fold folding function here: it’s also a folding function, and it’s a vivid root, so that it’s the third day (a chocolate bar in the back and with a line to put in a briefcase). There are a lot of reasons for this: all-one "unpacking" the function will be in the same order as before: from the end.

So let's differentiate the root, then the cosine, and only then the viraz in the arches. And then everything is multiplied.

Such vipadkas are manually numbered. Tobto, clearly, what we see. In what order will we be able to count the value of the viraz? Freeze on the butt:

In order to improve the performance of the action, the greater the “call” will be the function. Last day - yak and earlier:

Here the contribution is 4-ruble. Come on, the order is important.

1. root viraz. ...

2. Root. ...

3. Sinus. ...

4. Square. ...

5. Take everything together:

POKHIDNA. BRIEFLY ABOUT HEAD

Functions- the ratio of the increase in function to the increase in the argument with an indefinitely small increase in the argument:

Basic items:

Differentiation rules:

Constant to blame for the bad sign:

Pohidna Sumi:

Come on create:

Looks like private:

Suitable for folding function:

Algorithm for the knowledge of simple folding functions:

  1. Visnachaєmo "internal" function, it is known її will go.
  2. Visnachaєmo "zvnishnyu" function, it is known її will go away.
  3. The results of the first and other points are multiplied.

The formula for a funky fraction with two functions. Proof in two ways. Detailed selection of butt differentiation of private.

zm_st

Formula for funky fraction

Do not have a function and value in the deyakom near the point and may be in the point of origin. I don't care. Todi їkhnya is private I can go to the point where I start to follow the formula:
(1) .

Dovedennya

Introduced value:
;
.
Here and with the functions of the winter. For the sake of simplicity, we will omit the meaning of the arguments.

Dali pomichaumo, scho
;
.
For the wash function and may be lost at the points, like the next steps:
;
.
From the beginning of old viplivas, with functions and irreproachable points. Tom
;
.

The function y from the change x, as є as a fraction of the function i, is understandable:
.
The increase in the number of functions in the following points is clearly visible:
.
Multiplied by:

.
Zvidsi
.

Now I know I will go:

.

Otzhe,
.
The formula has been completed.

It is possible to substitute victorious if it’s like me. Significantly її yak x. Todi, if there is a lost one, and, moreover, it is a fraction, folded two functions, so start with the formula:
.
Anyway, in a big short record
(1) .

Proof in another way

put on

Here, it is easy to put on the numbered fraudulent fraction, and the formula of the obscene private (1). Amazingly, in more folding vipadkah, I know the fraction is simpler for the additional logarithmic obscene.

butt 1

Know the lost fraction
,
de ,,, - post_yni.

There is a rule of differentiation of sumi functions:
.
lost post
.
From the tables of older ones we know:
.
Todi
;
.

Replaceable by i on:
.

Now, I know I will go for the fraction for the formula
.

.

butt 2

Know the lost function from the change x
.

Zastosovuєmo differentiation rules, like in the front butt.
;
.

Zastosovuєmo rule of fraction differentiation
.


.

If you go for the value, then the function is lost at the point - the boundary of the increase in the function Δ y before the increase in the argument Δ x:

Nachebto everything zrozumіlo. Ale try porahuvati for the whole formula, say, I'll lose the function f(x) = x 2 + (2x+ 3) e x Sin x... As soon as all work is due to viznachennyam, then after a couple of parties, you will just fall asleep. To this, there are simple and effective ways.

For a cob, it is very important that you can see the so-called elementary functions from the great development of functions. Quite simply virazi, old ones, have long been calculated and entered in the table. These functions can be simply forgotten - all at once with those who are old.

Additional elementary functions

Elementary functions - all that is covered below. Pochіdnі these functions of the demand of the nobility to remind. Tim is more awkward to say they are awkward - for those stinks and elements.

Otzhe, lost elementary functions:

Name function lost
Constant f(x) = C, C ∈ R 0 (well, well, zero!)
Step with a rational indicator f(x) = x n n · x n − 1
sinus f(x) = Sin x cos x
cosine f(x) = Cos x - sin x(Minus sine)
tangent f(x) = Tg x 1 / cos 2 x
cotangent f(x) = Ctg x - 1 / sin 2 x
natural logarithm f(x) = Ln x 1/x
consistent logarithm f(x) = Log a x 1/(x Ln a)
show function f(x) = e x e x(nothing has changed)

If the elementary function is multiplied by a sufficient post-life, then the lost new function can be easily used:

(C · f)’ = C · f ’.

Zagalom, you can always be blamed for a bad sign. for example:

(2x 3) '= 2 ( x 3) '= 2 3 x 2 = 6x 2 .

Obviously, elementary functions can be folded one to one, multiplicity, delay - and a lot of things. So there are new functions, not particularly elementary, but also differentiated by singing rules... The rules are shown below.

Go to Sumi and Riznitsi

Let the function be given f(x) і g(x), What we see. For example, it is possible to take elementary functions, which can be seen. Today you can know the lost sum and difference of these functions:

  1. (f + g)’ = f ’ + g ’
  2. (f − g)’ = f ’ − g ’

Otzhe, the sum (growth) of the two functions of the house (growth) of the older ones is lost. Dodankiv can be bigger. For example, ( f + g + h)’ = f ’ + g ’ + h ’.

Strictly it seems, in algebra there is no understanding of the "vision". Є understanding the "negative element". Tom growth f − g you can rewrite yak sum f+ (-1) g, And to lose one formula - lost sumi.

f(x) = x 2 + sin x; g(x) = x 4 + 2x 2 − 3.

function f(x) - the price of two elementary functions, to that:

f ’(x) = (x 2 + sin x)’ = (x 2) '+ (sin x)’ = 2x+ Cos x;

Similarly to the world for the function g(x). Only there are three additional points (from the point of view of algebra):

g ’(x) = (x 4 + 2x 2 − 3)’ = (x 4 + 2x 2 + (−3))’ = (x 4)’ + (2x 2)’ + (−3)’ = 4x 3 + 4x + 0 = 4x · ( x 2 + 1).

as follows:
f ’(x) = 2x+ Cos x;
g ’(x) = 4x · ( x 2 + 1).

lost create

Mathematics is a logical science, that is so great that it is so important, that if the sum of the darlings of the old is lost, then it is lost. strike"> Dorіvnyu dobutku old ones. And the axis of figs is for you! Go ahead, create yourselves to vzsіm for іnshoy formula. And itself:

(f · g) ’ = f ’ · g + f · g ’

The formula is awkward, ale її is often forgotten. It’s not only schoolchildren, but first students. The result is a wrong version of the title.

Zavdannya. Know the lost functions: f(x) = x 3 cos x; g(x) = (x 2 + 7x- 7) e x .

function f(x) It is a bunch of two elementary functions, everything is simple:

f ’(x) = (x 3 cos x)’ = (x 3) 'cos x + x 3 (cos x)’ = 3x 2 cos x + x 3 (- sin x) = x 2 (3cos x − x Sin x)

The function g(x) The first multiplier of the trocha is foldable, the allegorical scheme does not change. Obviously, the first multiplier of the function g(x) It is a polynomial, and it is lost - the price is lost. maєmo:

g ’(x) = ((x 2 + 7x- 7) e x)’ = (x 2 + 7x- 7) ' e x + (x 2 + 7x- 7) ( e x)’ = (2x+ 7) e x + (x 2 + 7x- 7) e x = e x· (2 x + 7 + x 2 + 7x −7) = (x 2 + 9x) · e x = x(x+ 9) e x .

as follows:
f ’(x) = x 2 (3cos x − x Sin x);
g ’(x) = x(x+ 9) e x .

To brutalize respect, but at the end of the day it’s going to be multiplied. Formally, no work is required, however, most of the older ones are not counted by themselves, but rather by the function. And it means that far away will be primed to zero, there will be signs and so far. For such, please be more beautiful mother viraz, multiplier.

Yaksho є two functions f(x) і g(x), And g(x) ≠ 0 to whichever you want us to be without help, you can use a new function h(x) = f(x)/g(x). For such a function, you can also know the following:

Not weak, right? Are the stars taken as a minus? what g 2? And the axis is so! Tse one of the most sophisticated formulas - you will not work out without a dance. That is more beautiful than vivchati її on specific butts.

Zavdannya. Know the lost functions:

The number and the denominator of the skin fraction have elementary functions, for that everything that we need is the whole formula of the obscene private:


For tradition, the number is divided into multipliers - it is worthwhile to simplify the view:

Foldable function - the tse is not an obligatory formula with a pre-cut in a pivkilometer. For example, to finish taking a function f(x) = Sin x and replace the change x, Say, on x 2 + ln x... Wiide f(x) = Sin ( x 2 + ln x) - tse і є foldable function. She may have been lost, however, to know the rules that are visible to the eye, not to see.

Yak bootie? In such vipadkas additional help replaces the formula for the funky folding function:

f ’(x) = f ’(t) · t', Yaksho x substitute for t(x).

As a rule, with the reasoning of the formula, it is more correct to get it right, not just a private one. That її can be more beautifully explained on specific butts, z report description skin crocus.

Zavdannya. Know the lost functions: f(x) = e 2x + 3 ; g(x) = Sin ( x 2 + ln x)

Fascinating, which is in the function f(x) Replace viraz 2 x+ 3 just x, To see the elementary function f(x) = e x... I will robimo to replace him: hey 2 x + 3 = t, f(x) = f(t) = e t... I will go to the folding function for the formula:

f ’(x) = f ’(t) · t ’ = (e t)’ · t ’ = e t · t ’

And now - uvaga! Viconuєmo zvorotnu replace: t = 2x+ 3. Otrimaєmo:

f ’(x) = e t · t ’ = e 2x+ 3 (2 x + 3)’ = e 2x+ 3 2 = 2 e 2x + 3

Now we have removed the functions g(x). Obviously, you need to replace x 2 + ln x = t... maєmo:

g ’(x) = g ’(t) · t'= (Sin t)’ · t'= Cos t · t ’

Zvorotn_y substitute: t = x 2 + ln x... Todi:

g ’(x) = Cos ( x 2 + ln x) · ( x 2 + ln x) '= Cos ( x 2 + ln x) (2 x + 1/x).

From and everything! The yak can be seen from the last viraz, the whole task started to be counted up to the ridiculous sumi.

as follows:
f ’(x) = 2 e 2x + 3 ;
g ’(x) = (2x + 1/x) Cos ( x 2 + ln x).

Even more often in my lessons I substitute the term “lost” by the vicorist word “stroke”. For example, the stroke is from the sum of the doors. So smart? Well, good luck.

Such a rank, numbered obhidnoy, is raised to the extent of the elimination of the strokes themselves behind the rules that are visible. We'll turn the yak butt stock up to an awkward step with a rational indicator:

(x n)’ = n · x n − 1

Little hto know, scho in the role n a whole number of fractions. For example, root - tse x 0.5. And why, how can we stand by the root? I know the foldable function - such constructions are to be loved by control robots and sleep.

Zavdannya. Know the lost function:

For the cob, the root of the step viglyad is rewritten with a rational indicator:

f(x) = (x 2 + 8x − 7) 0,5 .

Now robimo I will change: hey x 2 + 8x − 7 = t... I know I will go for the formula:

f ’(x) = f ’(t) · t ’ = (t 0.5) ' t'= 0.5 t-0.5 t ’.

Robimo will ring a bell: t = x 2 + 8x- 7. Maєmo:

f ’(x) = 0.5 ( x 2 + 8x- 7) -0.5 x 2 + 8x- 7) '= 0.5 (2 x+ 8) ( x 2 + 8x − 7) −0,5 .

Nareshty, turning to roots:

Virishuvati physical knowledge or apply mathematics is absolutely unwise without knowledge about the origin and methods of calculation. Pochidna - one of the best to understand mathematical analysis... On the basis of the fundamental themes and we have decided to dedicate this year's statute. It is also lost, as її physical і geometric sense, How can I lose my function? All food can be combined into one: how will I lose sight?

Geometric and physical zmist

Nekhai є function f (x) , Set in the interval (A, b) ... Points х і х0 lie within the same interval. When changing, the function itself changes. A change to an argument - a difference in its meaning x-x0 ... Tsya reznitsya sign up yak delta ix and is called an argument gain. A change or an increase in function is called a difference in the meaning of a function in two points. Significant value:

The function is similar to the point - the boundary of the increase in the function in the given point to the increase in the argument, if it stops straight to zero.

The price can be written as follows:

What kind of sense is there in such a boundary? And the axis is yak:

The function is lost to the point to the tangent of the kuta from the point of view of the OX and is similar to the graph of the function in the given point.


The physical zmist of the obscene: It is lost to the way for an hour before the transport of the straightforward movement.

Great, every school hour, every day, how fast is a private way x = f (t) i hour t ... Average performance per song per hour:

Schob know the speed of the ruch at the moment of the hour t0 it is necessary to calculate the boundary:

Perche's rule: vinosimo constant

Constant can be blamed for the badass sign. Moreover, it is necessary to work. When applying mathematics, take a rule - if you can forgive the viraz, just say goodbye .

Butt. Numerously I will:

Rule to a friend: lost sumi functions

There are two functions available for transporting older functions. The same is true for the funniest functions.

We will not lead to the conclusion of the theorem, but a practical butt is more beautifully discernible.

Know the lost function:

Rule three: lost to additional functions

There is a need to add two differential functions to be calculated according to the formula:

Butt: know the lost function:

Decision:

Here it is important to say about the number of old folding functions. The folding function is similar to that of the obsolete function according to the industrial argument, to the similar argument from the independent change.

At the vischevkazannoy butts mi zustrichamo viraz:

In this vipadku, the intermediate argument is 8x in the fifth step. In order to count the lost of such a viraz, a collection of the most important functions from the intermediate argument, and then multiplying to the beginning of the most intermediate argument on the independent changes.

Rule four: private two functions are lost

Formula for assigning one type of private two functions:

We tried to tell you about the loss for teapots from scratch. The topic is not so simple, as to be built, this is in order: there are often pastes in the butts, so be respectful when numbered older ones.

You can turn to the student service if you want to eat from the center of those topics. For a short term, it is possible to send the best control and go to the staff, see if none of them have previously taken care of the number of the older ones.