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The yak knows and I will go from the root. It has a folding function. The formulation of the eponymous degree function

Foldable functions do not need to go by design folding function... If the function is of the form y = sin x - (2 - 3) a r c t g x x 5 7 x 10 - 17 x 3 + x - 11, then it cannot be folded into the view as y = sin 2 x.

The article is given to show the understanding of the folding function and manifestation. Corrected with the formulas of the zakhodzhennya pohidnoy with the butts of the rishen in the link. The stagnation of tables of the old and the rules of differentiation can be changed to change the hour for the old ones.

Main values

value 1

The folding function is used for such a function, in which the argument is also used.

It is denoted by such a rank: f (g (x)). Mum, that the function g (x) is involved in the argument f (g (x)).

value 2

Also, the function is f and the cotangent function, and g (x) = ln x is the whole function of the natural logarithm. We recognize that the folding function f (g (x)) is written as arctan (lnx). For the function f, which is a function of 4 steps, de g (x) = x 2 + 2 x - 3 to be used as a whole rational function, we can deny that f (g (x)) = (x 2 + 2 x - 3) 4 .

Obviously, g (x) can be foldable. From the butt y = sin 2 x + 1 x 3 - 5 it can be seen that the value of g is a cubic root with a fraction. Tsey viraz is allowed to mean yak y = f (f 1 (f 2 (x))). The stars are maєmo, f is a sine function, and f 1 is a function that can be transformed into a square root, f 2 (x) = 2 x + 1 x 3 - 5 is another rational function.

value 3

The step of contribution is meant to be a natural number and write as y = f (f 1 (f 2 (f 3 (... (F n (x)))))).

value 4

The understanding of the composition of the function is related to the number of contributions to the brain tasks. For the first time, the formula for the knowledge of the lean folding function is considered

(F (g (x))) "= f" (g (x)) g "(x)

put on

butt 1

Know the lost folding function in the form y = (2 x + 1) 2.

Decision

Behind the wash it can be seen that f is the squared function, and g (x) = 2 x + 1 is used for the linear function.

It’s a very common formula for a folding function and can be written down:

f "(g (x)) = ((g (x)) 2)" = 2 · (g (x)) 2 - 1 = 2 · g (x) = 2 · (2 x + 1); g "(x) = (2 x + 1)" = (2 x) "+ 1" = 2 x "+ 0 = 2 1 x 1 - 1 = 2 ⇒ (f (g (x))) "= f" (g (x)) g "(x) = 2 (2 x + 1) 2 = 8 x + 4

It is necessary to know the lost with the forgiveness of the outward kind of function. otrimuєmo:

y = (2 x + 1) 2 = 4 x 2 + 4 x + 1

Zvidsi maєmo, scho

y "= (4 x 2 + 4 x + 1)" = (4 x 2) "+ (4 x)" + 1 "= 4 · (x 2)" + 4 · (x) "+ 0 = = 4 2 x 2 - 1 + 4 1 x 1 - 1 = 8 x + 4

The results were zbіglisya.

When viewing this type, it is important to think about it, because the function of the type f і g (x) will grow.

butt 2

Slide to know the old folding functions of the form y = sin 2 x і y = sin x 2.

Decision

The first to write down the function is to say about those where f is a squaring function, and g (x) is a sine function. Todi otrimaєmo, scho

y "= (sin 2 x)" = 2 sin 2 - 1 x (sin x) "= 2 sin x cos x

Another notation shows that f is a sine function, and g (x) = x 2 is a step function. Zvidsy viplya, which is a folding function that can be written down yak

y "= (sin x 2)" = cos (x 2) (x 2) "= cos (x 2) 2 x 2 - 1 = 2 x cos (x 2)

The formula for an ordinary y = f (f 1 (f 2 (f 3 (... (Fn (x)))))) is written as y "= f" (f 1 (f 2 (f 3 (... ( fn (x)))))) f 1 "(f 2 (f 3 (... (fn (x))))) f 2" (f 3 (... (fn (x))) ) ·. ... ... · F n "(x)

butt 3

Know the lost function y = sin (ln 3 a r c t g (2 x)).

Decision

The Danish butt will show the foldability of the recording and the introduction of the functions. Todi y = f (f 1 (f 2 (f 3 (f 4 (x))))) meaningfully, de f, f 1, f 2, f 3, f 4 (x) is the sine function, the function of reduction in 3 steps, the function of the logarithm and the step, the function of arctangent and line.

3 formulas for the value of the folding function maєmo, scho

y "= f" (f 1 (f 2 (f 3 (f 4 (x))))) f 1 "(f 2 (f 3 (f 4 (x)))) f 2" (f 3 (f 4 (x))) f 3 "(f 4 (x)) f 4" (x)

Otrimuєmo, how to know

f "(f 1 (f 2 (f 3 (f 4 (x))))) in the capacity of an obscure sine according to the table of older ones, todi f" (f 1 (f 2 (f 3 (f 4 (x)))) ) = cos (ln 3 arctan (2 x)).
f 1 "(f 2 (f 3 (f 4 (x)))) as a similar power function, to f 1" (f 2 (f 3 (f 4 (x)))) = 3 ln 3 - 1 arctan (2 x) = 3 ln 2 arctan (2 x).
f 2 "(f 3 (f 4 (x))) in the sense of an ordinary logarithmic, to f 2" (f 3 (f 4 (x))) = 1 a r c t g (2 x).
f 3 "(f 4 (x)) in the quality of the primitive arctangent, so f 3" (f 4 (x)) = 1 + 1 + (2 x) 2 = 1 + 1 + 4 x 2.
With a known bad f 4 (x) = 2 x death 2 for a bad sign due to the lack of a formula of a bad power function with an exponent, like a road 1, so f 4 "(x) = (2 x)" = 2 x "= 2 1 x 1 - 1 = 2.

Viroblyaєmo ob'dnannya of intermediate results and і іtrimumo, scho

y "= f" (f 1 (f 2 (f 3 (f 4 (x))))) f 1 "(f 2 (f 3 (f 4 (x)))) f 2" (f 3 (f 4 (x))) f 3 "(f 4 (x)) f 4" (x) = = cos (ln 3 arctan (2 x)) 3 ln 2 arctan (2 x) 1 arctan (2 x) 1 1 + 4 x 2 2 = = 6 cos (ln 3 arctan (2 x)) ln 2 arctan (2 x) arctan (2 x) (1 + 4 x 2)

Selection of such functions of nagadu matrioshka. The rules of differentiation cannot be expected to be stuck in an explicit view behind the additional tables of the older ones. Most often, it is necessary to establish a formula for the knowledge of old folding functions.

Find out how the foldable looks like the foldable functions. With the obvious cleverness of development, the knowledge of the older ones is especially easy.

butt 4

It is necessary to look at the reduced buttstock. If the function is of the form y = tan 2 x + 3 tanx + 1, then it can be seen in folding form g (x) = tanx, f (g) = g 2 + 3 g + 1. Obviously, it is necessary to fix the formula for folding ruddy:

f "(g (x)) = (g 2 (x) + 3 g (x) + 1)" = (g 2 (x)) "+ (3 g (x))" + 1 "= = 2 · g 2 - 1 (x) + 3 g "(x) + 0 = 2 g (x) + 3 1 g 1 - 1 (x) = = 2 g (x) + 3 = 2 tgx + 3; g "(x) = (tgx)" = 1 cos 2 x ⇒ y "= (f (g (x)))" = f "(g (x)) g" (x) = (2 tgx + 3 ) 1 cos 2 x = 2 tgx + 3 cos 2 x

The function of the form y = tgx 2 + 3 tgx +1 does not fit into a folding function, since the maximum amount tgx 2, 3 tgx і 1. However, tgx 2 fits into a folding function, then we can recognize the step function of the form g (x) = x 2 і f, the tangent function. For ts'go slid differentiate for the bag. Otrimuєmo, scho

y "= (tgx 2 + 3 tgx + 1)" = (tgx 2) "+ (3 tgx)" + 1 "= = (tgx 2)" + 3 · (tgx) "+ 0 = (tgx 2)" + 3 cos 2 x

We pass to the familiar rudimentary folding function (t g x 2) ":

f "(g (x)) = (tan (g (x)))" = 1 cos 2 g (x) = 1 cos 2 (x 2) g "(x) = (x 2)" = 2 x 2 - 1 = 2 x ⇒ (tgx 2) "= f" (g (x)) g "(x) = 2 x cos 2 (x 2)

We can recognize that y "= (t g x 2 + 3 t g x + 1)" = (t g x 2) "+ 3 cos 2 x = 2 x cos 2 (x 2) + 3 cos 2 x

The folding functions can be included before the folding functions warehouse, and the folding functions themselves can be folding folding functions.

butt 5

For the butt, the function is clearly foldable y = log 3 x 2 + 3 cos 3 (2 x + 1) + 7 e x 2 + 3 3 + ln 2 x (x 2 + 1)

This function can be represented in the view y = f (g (x)), meaning f is the function of the logarithm behind the base 3, and g (x) is used as a sum of two functions of the form h (x) = x 2 + 3 cos 3 (2 x + 1) + 7 ex 2 + 3 3 і k (x) = ln 2 x (x 2 + 1). Obviously, y = f (h (x) + k (x)).

The function h (x) is understandable. Price l (x) = x 2 + 3 cos 3 (2 x + 1) + 7 to m (x) = e x 2 + 3 3

Mahmo, l (x) = x 2 + 3 cos 2 (2 x + 1) + 7 = n (x) + p (x) є a sum of two functions n (x) = x 2 + 7 and p (x) P 2 cosine function, p 3 (x) = 2 x + 1 - linear function.

Were off that m (x) = ex 2 + 3 3 = q (x) + r (x) є the sum of two functions q (x) = ex 2 і r (x) = 3 3, de q (x) = q 1 (q 2 (x)) is a folding function, q 1 is an exponential function, q 2 (x) = x 2 is a state function.

It can be seen that h (x) = l (x) m (x) = n (x) + p (x) q (x) + r (x) = n (x) + 3 p 1 (p 2 ( p 3 (x))) q 1 (q 2 (x)) + r (x)

When going to a turn of the form k (x) = ln 2 x (x 2 + 1) = s (x) 1 (s 2 (x)) with an integral rational t (x) = x 2 + 1, de s 1 is a squaring function, and s 2 (x) = ln x is a logarithmic base e.

Looks like a wink, so you can see k (x) = s (x) t (x) = s 1 (s 2 (x)) t (x).

Todi otrimaєmo, scho

y = log 3 x 2 + 3 cos 3 (2 x + 1) + 7 ex 2 + 3 3 + ln 2 x (x 2 + 1) = = fn (x) + 3 p 1 (p 2 (p 3 (x))) q 1 (q 2 (x)) = r (x) + s 1 (s 2 (x)) t (x)

Behind the structures of the function it became obvious, as if the formulas were necessary to be fixed in order to forgive viraz when differentiating them. In order to know the other buildings and for the understanding of their transmission, it is necessary to turn to the point of differentiation of the function, in order to know the funniest one.

As soon as you have noted a pardon in the text, be weasel, see and natisnit Ctrl + Enter

Instructions

Before Tim, the yak knows and disappears from the root, to be brutal with respect for this function, which is present in the virgin butt. If you are in the problem of a lot of pidkorenevik viraziv, then speed up the offensive rule of knowing the obsolete square root:

(√x) "= 1 / 2√x.

And for the knowledge of the obsolete cubic root, use the formula:

(³√x) "= 1/3 (³√x) ²,

de through ³√x values the cubic root of x.

If, for the purpose of differentiation, it is intended to change in others, then to translate the root into a step function with a general indicator. For the square root, the steps will be ½, and for the cube root - ⅓:

√x = x ^ 1,
³√x = x ^ ⅓,

de ^ the sign of the elevation in the steps.

For the knowledge of the obscene power function in the zagala і x ^ ½, x ^ ⅓, zokrem, speed up the next rule:

(X ^ n) "= n * x ^ (n-1).

For an obscene root from the tsyo spivvidnoshennya viplyaє:

(X ^ ½) "= ½ x ^ (-1) і
(X ^ ⅓) "= ⅓ x ^ (-⅔).

Differentiating everything, respectfully marvel at one of the parts of the butt. As soon as you have seen a twist, it can be forgiven melodiously. Most of the school butts are stacked in such a rank, but as a result, there is a small number of or more compact virases.

At the same time, tasks on the basis of common, basic (square and cubic) tasks are developed at once with the other functions. In order to know I will go to the root in a whole vipad, fix the rules:
a constant (post-number, C) is lost to zero: C "= 0;
the constant multiplier to blame for the bad sign: (k * f) "= k * (f)" (f is a good function);
abducted sumy dekilkoh functiy for dorіvnyu sumi old: (f + g) "= (f)" + (g) ";
two functions are lost to the door ... ni, not to the creature of the old, but to the offensive viraz: (fg) "= (f)" g + f (g) ";
a private one is also not a private one, but is known to be an offensive rule: (f / g) "= ((f)" g - f (g) ") / g².

Beast to respect

At the same time, you can calculate the lost functions online from the deduction of the report solution of the problem. The decision of old functions is carried out according to the rules of differentiation, as students vivchayut at the course mathematical analysis in the Institute. In order to know the lost function, it is necessary to enter a function for differentiating the rules for entering data in the "Functions" field.

Korisna is happy

The transitional function is called the boundary of the increase in the function to the increase in the argument, if the argument is straight to zero: The mathematical difference in the value of the intelligence is not even simpler, although in the school course of algebra it is not even easier to understand the boundaries. Ale in order to see if it is known and lost new functions, Tse and not obov'yazkovo.

dzherela:

lost root from ix

Zagalny vypadok of the formula of the old root of the pre-stage- drib, for the number one is one, and in the denominator the number, equal to the step of the root, for which one was counted as lost, multiplied by the root of the same world, the rooted virase of the same - is the serpent in the steps of the root, for which one was counted as abducted
Like a square root- є Let's add a drop in front of the front of the formula. Take the square root of x- tse drіb, the number of which is a dvіyka odinitsі, and the denominator is dvіyka, multiplied by the square root x
Looks like a cubic root, Also okremiy vipadok foreign formulas... Pochіdna cubic root - a single unit, subdivided into three cubic roots from a square.

Bottom guided re-implementation, to explain why the formulas of the deplorable square and cubic root are the same, as they are aimed at the little one.

Zrozumіlo, given the formula, it is possible not to remember it, as if taking it up to respect, it’s the knight of the root of the age-old step - the very same, which is brought into the steps of the fraction, the standard of what the world is darling. That is, the knowledge of the lean root is raised to the point where the formula for the knowledge of the lean step of the same fraction is set.

Looks like a wicked square root

(√x) "= 1 / (2√x) or 1/2 x -1/2

clarified:
(√x) "= (x 1/2)"

The square root is exactly the same as the work, which is raised in 1/2 steps,to mean for the knowledge of an obscure root, it is possible to fix the formula for the rules of the knowledge of an obsolete from the evil in the most advanced stage:

(X 1/2) "= 1/2 x -1/2 = 1 / (2√x)

Abandoned the cubic root (abducted the root of the third degree)

Like a cubic root, it is exactly the same principle as a square one.

Clearly so cubic root yak step 1/3 and I know I will walk along by the rules differentiation. short formula You can wonder at the picture, and below is an explanation of why it is so.

Step -2/3 go to the inheritance of a single one from 1/3

The operation of making a difference is called differentiation.

As a result, the solution of problems about the introduction of old in the simplest (and not even simpler) functions by the value of the old one as between the increase in the argument before the increase in the argument, appeared the tables of the old and exactly the same rules of differentiation. Isaac Newton (1643-1727) and Gotfrid Vilgelm Leibnits (1646-1716) worked as the first in the field of knowledge of the elderly.

To that in our hour, to know if I’m going to lose any function, I don’t need to count the guesswork as the boundary of the increase in function to the increase in the argument, but it’s necessary to use the table of old ones and the rules of differentiation. For the knowledge of the poorest, the offensive algorithm is used.

To know I will go, Treba viraz with a stroke sign return to warehouses with simple functions and by value, in other words (Tvir, soum, private) knitted and function. The distance of the old elementary functions is known in the tables of the old ones, and the formulas of the old ones create, sum and private - in the rules of differentiation. Table of old and rules for differentiating data from the first two butts.

Butt 1. Know the lost function

Decision. The three rules of differentiation are based on what functions are lost, i.e.

From the tables of the old ones, the "xy" is the old one, and the old sine is the cosine. For the money in the bag of the old, and of course I need to get it:

Butt 2. Know the lost function

Decision. Differentiated yak I will go Sumi, in some other dodanok with a permanent multiplier, which can be blamed for a bad sign:

As soon as the food is found out, the sounds are taken, the stench, as a rule, will become clearer by reading the table of old and simple rules of differentiation. Before them, we pass directly at a time.

Table of legacy simple functions

1. Looks like a constant (number). Be it a number (1, 2, 5, 200 ...), like є in a rotated function. Set the door to zero. It is even more important to remember, so it is necessary even more often
2. Pochіdnaya nezalezhnaya zminnoї. Most of the "ixi". Set up a door unit. Tse tezh importantly to remember
3. Pochіdna step. In the steps, when solving tasks, it is necessary to re-create non-square roots.
4. Wandering in step -1
5. Like a square root
6. Sinus deceleration
7. On the cosine
8. On the tangent
9. Cotangent is the same
10. The arc-sine waveform
11. Walking arccosine
12. It is similar to the arctangent
13. Looks like arc cotangent
14. Similar to the natural logarithm
15. Logarithmic function
16. The exponential is coming
17. Go to the show function

Differentiation rules

1. Pochіdna sumi abo rіznitsі
2. Go ahead create
2a. The turn, multiplied by the constant multiplier
3. Looks like a private
4. Ideal folding function

Rule 1.what functions

differentiated in deyak_y points, then in the same point may be similar functions

whereby

so that the algebraic sum of the functions of the old algebraic sum of the older functions is lost.

Slidstvo. Whenever two differentiates go back to the old ones, then the old ones, Tobto

Rule 2.what functions

differentiated in deyak_y points, then at the same point differentiated і їх tvir

whereby

so that two functions are lost to the end of the day.

Slidstvo 1. A permanent multiplier can be blamed for a bad sign:

Slidstvo 2. There is a way to create some differentiation in the amount of creations in the skinny and in the multipliers for all of them.

For example, for three multipliers:

Rule 3.what functions

differentiated in deyakiy points і , then at the point of the differentiable і їхnya is privateu / v, moreover

so that the private two functions are lost to the road fraction, the number of which є.

De scho shukati on the other sides

When there are known obscene additions and parts in real tasks, it is necessary to establish a number of differentiation rules at once, and to the same time, it is more important to apply them to the current tasks - in the statute"Go to add and part of functions".

Respect. Do not cheat a constant (tobto, number) as a sum of money and as a constant multiplier! If the donation is old, it will be zero, and if it is a constant multiplier, it will be blamed for the sign of the old. tse typical pardon, Yaka study on the cob stage of the vivchennya of the elderly, but in the world of richennya even decilkoh one- two-storey applied to the middle student of the price of pardon no longer to rob.

And when you differentiate, create something that you have a private donation u"v in which u- a number, for example, 2 or 5, that is a constant, then the lost number will be zero and, again, all the additional ones are zero (this kind of drop-down in the butt is 10).

Іnsha frequent pardon is a mechanical solution of a simple folding function as a simple simple function. Tom lean folding function the statute was assigned. We will be able to read a collection of old simple functions.

Along the way, you can not do without a revision of viraz. For whoever can see the criterion in the new windows, the posibniki Diy with steps and rootsі Diy with fractions .

Yaksho Vi shukєte the solution of old fractions in steps and roots, tobto, if the function is ma viglyad nachebto Then go to the busy "Go to the sum of the fractions in steps and roots".

Yaksho well in front of you is a zabdannya Then you are busy with "Pochіdnі simple trigonometric functions".

Pokrokovy lay down - how to know I will go

Butt 3. Know the lost function

Decision. This is because of the part of the viraz of the function: all viraz are tvir, and those multipliers are sumi, in other of which one of the warehouses is to revenge the constant multiplier. There is a very common rule for differentiating create: two additional functions are lost for the transport of the skin from the same functions for the lost:

Given the established rule of differentiation of sumi: lost algebraic sumi functions of previous algebraic sumi of old functions. Our vipadku in the skin of the sum has another dodanok with a minus sign. In the case of leather goods, bachimo and square change, which is lost as a single road unit, and a constant (number), which is lost as a road unit is zero. Otzhe, "ix" is converted to one, and minus 5 - to zero. The other has a twisted "x" multiplication by 2, so the two can be multiplied by the same unit as "xi" will go. Otrimuєmo offensive meanings of the older ones:

It is necessary for the mind to lose all functions:

And it is possible to reconsider the solution of tasks to the end.

Butt 4. Know the lost function

Decision. From us you will be able to know the disappearance of a private one. There is a fixed formula for differentiating a private one: a private two functions are lost to a road fraction, the number of a banner’s є is a square of a common number. otrimuєmo:

I will go to the number of factors in the number, but I already knew in the butt 2. It is also not forgotten that the tvir, that є another multiplier in the number in the flow butt is taken with the minus sign:

Yaksho Vi shukєte the solution of such tasks, in which it is necessary to know the lost functions, de facto a pile of roots and steps, such as, for example, , Then kindly ask for busy "Go to sumy fractions in steps and roots" .

Would you like to know more about lost sines, cosines, tangents and others trigonometric functions, Tobto, if the function is maє viglyad nachebto , Then for you a lesson "Similar simple trigonometric functions" .

Butt 5. Know the lost function

Decision. In the given function of the Bachimo Tvir, one of the multipliers of which is a square root with an independent winter, with the old ones, which are known in the tables of the older ones. According to the rule of differentiation, create and the tabular value of an obscure square root will be recognized:

You can revise the solution of tasks for the end of the day. online calculator .

Butt 6. Know the lost function

Decision. In a given function, bachimo is private, and in a dilene there is a square root of an independent landscape. According to the rule of differentiation of the private, they repeated and stuck in the butt 4, and the tabular value of the ingenious square root is recognized:

Schob pozbutis as a fraction in a numeral, multiplying a numeral and a denominator on.

A draft of the formula of an elementary degree function (x in step a). Viewed from the roots of x. The formula of the eerie degree function of the higher order. Add the number of the older ones.

zm_st

Div. also: Degree function and root, formulas and graph
Statistical function graphs

Basic formulas

Looks from x in step a of road a, multiplied by x in step a minus one:
(1) .

Looks from the root of step n from x in step m roads:
(2) .

The formulas of the funeral power function

Vipadoc x> 0

The step function is clear from the change x with the step a indicator:
(3) .
Here a є is a fairly valid number. A collection of vipadoks is available.

To know the lost function (3), speedy by the authorities of the statical function and re-created before the onset of the watch:
.

Now I know I’ll go, stuck:
;
.
Here.

Formula (1) has been completed.

The formulas are drawn from the root of step n from x in step m

Now the function is discernible, like the root of the offensive mind:
(4) .

To know I’ll go, I’ll re-create the root to the statical function:
.
Rivnyuchi with the formula (3) mi bachimo, scho
.
Todi
.

For the formula (1) I will go:
(1) ;
;
(2) .

In practice, it is not necessary to memorize formula (2). There is a great deal of quick re-adaptation of the root to the state functions, and then there is also the old, stagnant formula (1) (Div. Put in the corner of the hand).

Vipadoc x = 0

Well, then the state function is assigned if the value is changed x = 0 ... We know, I will lose function (3) for x = 0 ... For many speedy people, the following are important:
.

Pidstavami x = 0 :
.
At the same time, there is a right-handed border, for someone.

Otzhe, we knew:
.
Zvidsey can be seen at,.
At,.
At,.
The result should be followed by the formula (1):
(1) .
Therefore, formula (1) is valid і for x = 0 .

vipadok x< 0

I know the function (3):
(3) .
In case of deyak values of the post-a, there is a value and in case of negative values of the change x. But itself, let it be a rational number. That one can be represented in the view of a non-short fraction:
,
de m і n - whole numbers, as they do not seem to be out of the ordinary.

If n is unpaired, then the state function is assigned if negative values of the change x. For example, for n = 3 i m = 1 mi maєmo cubic root z x:
.
Vіn vizvalues i in case of negative values of the change x.

We know, I will lose the statical function (3) for and for rational values post_ynoї a, for those who are assigned. For tsiogo representable x in the offensive view:
.
Todi,
.
It is known that I will lose the wine afterwards for the sign of the obscene and stagnant rule of differentiation of the folding function:

.
Here. ale
.
Oskilki, then
.
Todi
.
So formula (1) is valid if:
(1) .

Changing orders

Nowadays we know the old orders of the statistic functions
(3) .
I'll go first to order, we already knew:
.

Winosyachi post_ine a for the sign of the obscene, it is known I will go in a different order:
.
The similar rank is known to be of the third and fourth order:
;

.

You can see the stars lost to n-th order maє nasty viglyad:
.

Dear, scho where a is a natural number,, That n -a is lost є postynoyu:
.
Todi all advances come back to zero:
,
at.

Put the number of the older ones

butt

Know the lost function:
.

The root can be reworked up to the steps:
;
.
Todi vyhіdna funktsіya nabuv viglyadu:
.

Known similar steps:
;
.
Goes post-mortem to zero:
.