Pokhіdna functions are given implicitly.
Pokhіdna parametrically given functions

At this article, we can see two more typical tasks, yakі often zustrіchayutsya in control robots on advanced mathematics. In order to successfully master the material, it is necessary to know similar things, even if on the middle level. You can learn how to know the good things practically from scratch in two basic lessons that Folding function. As for the beginners of differentiation, everyone is ready, they also went.

Pokhіdna functions, set implicitly

Abo shorter - similar implicit functions. What is an implicit function? Let's guess the most important function of one of the change:

The function of one change-Tse as a rule, for a skin significance of an independent change, it has one and only one significance of a function.

Change is called independent mine or argument.
Change is called fallow or function .

Dosi looked at the functions, tasks in explicit look. What does it mean? Powerful analysis of benefits on specific butts.

Let's look at the function

Mi Bachimo, that we have a left-handed self-made "grave", and right-handed - only "iksi". That function in obvious sight expressed through an independent change.

Let's take a look at our function:

Here change and roztashovani "vperemish". And why by no means impossible say "igrok" only through "iks". What are the methods? The transfer of additions from part to part of the change of the sign, the fault for the arms, the transfer of multiples according to the rule of proportion and others. Rewrite the equivalence and try to express the “gravity” in the look:. You can twist and turn like a year old, but you don’t see anything.

Let me know: the butt implicit functions.

In the course of mathematical analysis, it was brought to the attention that the function is implicit іsnuє(don’t start the program), she has a schedule (just like that, like a “normal” function). The implicit function has the same іsnuє the first is lost, the friend is lost, etc. As it seems, the sex men are right.

For the first time in my life, I know how to look like a function given implicitly. Tse is not so difficult! All the rules of differentiation, the table of similar elementary functions are overridden by the force. Retail in one peculiar moment, which we can see at once.

So, and let me tell you a good news - look at the bottom of the task to finish with a hard and clear algorithm without a stone in front of three tracks.

butt 1

1) At the first stage, we put strokes on the offending parts:

2) Vikoristovuemo rules of linearity of the future (the first two rules of the lesson How to know if I'm going to go? Apply solution):

3) Direct differentiation.
How to differentiate and understand. What work there, de under strokes є "Іgreki"?

- just to dissimilarity, good: .

How to differentiate
Here we have folding function. Why? There is only one letter “igrok” under the sine. Ale, rich to the one who has only one letter "igrok" - FOR ITSELF AND FUNCTION(Div. vyznachennya on the cob of the lesson). Also, the sine is an external function, an internal function. Vikoristovuєmo rule of differentiation of collapsible functions :

Dobutok differentially according to the ultimate rule :

Bring respect, which is also a folding function, be a "player with bells and whistles" - folding function:

The very design of the solution may look something like this:


If there are arches, then you can open them:

4) Dodanki are taken from the left part, in which there is a “player” with a stroke. In the right part - everything else is transferable:

5) At the left part, I’m sorry to go for the arms:

6) І according to the rule of proportion, we throw off the qi of the bow at the banner of the right part:

Pokhіdna is known. Ready.

It means that the implicit view can rewrite a function. For example, the function can be rewritten like this: . І differenziuvati її for a well-looked-for algorithm. Indeed, the phrases “function assigned to an implicit view” and “implicit function” are considered by one meaningful nuance. The phrase "function given in implicit visual" is more meaningful and correct, – this function is set for the implicit viewer, but here you can express “graves” and reveal the function for the explicit viewer. Under the words "implicit function" often understand the "classical" implicit function, if "igreek" is not possible.

It also means that "implicitly equal" can implicitly set two or more functions, so, for example, equal stakes implicitly set functions, , as they designate pivkola. the kind of nuances, the whole thing was just information for a global development.

Another way of seeing

Respect! In another way, you can get to know it only in the same way that you know private holidays. Pochatkіvtsі vivchati mathematical analysis that teapots, be kind, do not read and skip the paragraph Otherwise, there will be a mess in your head.

Let's know how the implicit functions work in another way.

We transfer all warehouses to the left part:

I consider the function of two variables:

You can know our way behind the formula
We know private trips:

In this manner:

Another way of solving is to allow the vikonati to re-verify. But it’s not necessary to draw up a clean version of the task beforehand, the shards of private holidays will master later, and the student, who has learned the topic “The function of one change”, the nobility of private holidays is not yet to blame.

Let's take a look at some sprats of applications.

butt 2

Know the exact type of a function given implicitly

We hang strokes on the offending parts:

Vikoristovuemo rules of linearity:

We know the following:

Opening all the arches:

We transfer all the additions to the left part, the rest - to the right part:

Residual evidence:

butt 3

Know the exact type of a function given implicitly

Outwardly, the solution is that the pattern is designed like a lesson.

It is not rare if fractions appear after differentiation. In such situations, shots need to be spared. Let's take a look at two more butts.

butt 4

Know the exact type of a function given implicitly

We lay down insulting parts of strokes and victorious rule of linearity:

Differentiation, vicorist rule of differentiation folding function that rule of differentiation of private :

Opening the temples:

Now we need to take a shot. It’s worth it to work and pіznіshe, but more rationally to zrobіt vіdraz. The banner officer knows the fraction. multiplying on the . As a matter of fact, let’s look at it like this:

Sometimes after differentiation, 2-3 fractions are settled. Yakby in us was only one drib, for example, then the operation would have to be repeated - multiply skin dodanok skin parts on the

At the left part, we blame the bow:

Residual evidence:

butt 5

Know the exact type of a function given implicitly

This is an example of an independent solution. Alone, in the dark, in front of him, like a shot, it will be necessary to get ahead of the tripartite surface of the shot itself. Outwardly, the solution is that it is similar to the lesson.

Pokhіdna parametrically given functions

Don’t stress, it’s easy to finish everything in this paragraph. You can write down the general formula of a parametrically given function, but in order to make it clear, I will immediately write down a specific example. In a parametric form, the function has two equals: . Often equals are recorded under curly arches, and in sequence:,.

Change is called a parameter You can take the value from minus indistinctness to plus indistinctness. Let's take a look, for example, the meaning and imagine them at the insult equal: . But in a human way: "as if it's a good one, then the player is a good one." On the coordinate plane, you can specify a point, and this point can match the value of the parameter. Similarly, you can know the point of any value of the parameter "te". As for the "singular" function, for the American Indians, the parametrically assigned function is also right to do: you can induce a schedule, know the times better. Before speech, as it is necessary to induce a graph of a parametrically assigned function, you can speed up my program.

In the simplest views, it is possible to reveal a function in a clear view. Virazimo from the first alignment parameter: - and conceivable to another alignment: . As a result, the ultimate cubic function is taken away.

In "important" vipadkas, such a trick does not work. But it’s not a big deal, because for the purpose of knowing a similar parametric function, there is a formula:

It is known that I will die of "grave for the change":

All the rules of differentiation and the table of similar ones are fair, naturally, and for letters, in such a manner, there is no such thing as novelty in the process itself. Just replace the thoughts in the table with all the "iksi" for the letter "te".

It is known that I will die in the form of "iks for change":

Now only a few have lost the knowledge of similarities to our formula:

Ready. Pokhіdna, like the function itself, also fall in the parameter .

As for the significance, then the formula for replacing the record can simply be written without a successive index, the shards are “singular” like “iks”. But in the literature, there is always a variant, so I do not agree with the standard.

butt 6

Vikoristovuemo formula

At to this particular type:

In this manner:

The peculiarity of the significance of a similar parametric function is the fact that on the skin, the result is visible as much as possible. So, at the looked-out butt, when I know it, I opened the arches under the root (I don’t want to work for a moment). There is a great chance that, with the support of that formula, a lot of speeches will go well. If you want to chime, obviously, apply it with kostrubatymi vіdpovіdyami.

butt 7

Know the exact type of function given parametrically

This is an example of an independent solution.

at the stati The simplest types of tasks for the funeral we looked at butts, in which it was necessary for a friend to know the similar functions. For a parametrically given function, it is also possible to know a friend better, and out of the offensive formula: . It is quite obvious that in order to know a friend's bad, it is necessary to know first a little.

butt 8

To know first and friend similar functions given parametrically

I know I'm going to lose my head.
Vikoristovuemo formula

In this view:

We know similar functions that are implicitly assigned, so that they are assigned by equal equals, which change between them xі y. Apply functions that are given implicitly:

,

Pokhіdnі funktsіy, zadakh implicitly, but pokhіdnі implicit funktsіy, it's easy to finish. At once we will analyze the rule and the butt, and then we will understand why it is necessary.

In order to know the exact function, given implicitly, it is necessary to differentiate the offending parts of the equation by x. Those dodanki, in some kind of present only x, turn up to the greatest pokhіdnoї funktsії vіd iksu. And the addendums with the Greek need to be differentiated, crusty with the rule of differentiation of the folding function, the shards of the player are the same function as the xx. It’s even simpler, then in the absence of the worst addendum, it’s possible to enter: the worse function of the Greek is multiplied by the worse of the Greek. For example, it’s better to sign up as a dodanka, it’s better to sign up as a dodanka. Further, it is necessary to use the "engraving stroke" and it will be possible to remove the appropriate function, given implicitly. Let's take a look at the butt.

example 1.

Solution. Differentially insulting parts of alignment by x, considering that gravel is a function of x:

Zvіdsi otrimuєmo pokhіdnu, yak require zavdannya:

Now it's about the ambiguous power of functions, given implicitly, and why you need special rules of differentiation. At a part of the vipadkiv it is possible to reconsider, which is a substitution in task equalization(Div. Put more) the Greek yogo viraz through x to bring it to the point where it turns into the sameness. So. hover more alignment implicitly assign the following functions:

After the substitution of the wording of the Greek in the square through the iks, the equalness is taken to be the same:

.

Virazi, as they imagined us, weaved with a path of rozvyazannya ryvnyanna schodo engraving.

Yakby mi began to differentiate a distinct explicit function

then we took away the input as in the application 1 - in the form of a function, given implicitly:

But don’t be like a function, given implicitly, you can see it in your eyes y = f(x) . So, for example, the implicitly assigned functions

do not go through elementary functions, so equalization cannot be allowed to be engraved. For this reason, the rule of differentiation of a function that is implicitly specified, as we have already learned and given, is sequentially fixed in other applications.

butt 2. Know the exact function given implicitly:

.

We can see the gravel stroke and - at the exit - a random function, given implicitly:

example 3. Know the exact function given implicitly:

.

Solution. Differentially insulting parts of the equalization of x iks:

.

butt 4. Know the exact function given implicitly:

.

Solution. Differentially insulting parts of the equalization of x iks:

.

It turns out that we will lose it:

.

Example 5. Know the exact function given implicitly:

Solution. It is transferable dodanki in the right part equal to the left part and the right hand is filled with zero. Differentially offended parts of equal s iksa.

Let's take a look at the function y(x), as it can be written in an implicit way in the form $F(x, y(x)) = 0$. There are two ways to know implicit functions:

1. Differentiation of both parts equal
2. For help using the finished formula $ y" = - \frac(F"_x)(F"_y) $

How to know?

Method 1

It is not necessary to point the function to a clear view. It is necessary to immediately proceed to the differentiation of the left and right parts of the equal $ x $. For example, $(y^2)"_x = 2yy"$. at the left side.

Method 2

You can use the formula to figure out in the numeral book and the banner of similar private implicit functions $ F(x,y(x)) = 0 $. For the znakhodzhennya numeralist, I will take $ x $, and the znamennik I will take $ y $.

Another similar implicit function can be known for the help of repeated differentiation of the first implicit similar function.

Apply solution

Let's take a practical look at the calculation solution for a random implicitly given function.