How to find a derivative of a complex function Examples of solution. Derivative complex function
It is absolutely impossible to solve physical tasks or examples of mathematics without knowledge about the derivative and methods of its calculation. The derivative is one of the most important concepts of mathematical analysis. We decided to devote this fundamental topic to the current article. What is a derivative, what is its physical and geometric meaning, how to calculate the derived function? All these questions can be combined into one: how to understand the derivative?
Geometric and physical meaning derivative
Let there be a function f (x) set in a certain interval (A, B) . Points x and x0 belong to this interval. When changing X changes the function itself. Change of argument - the difference of its values x-x0. . This difference is written as delta X. and is called the increment of the argument. Changing or incrementing the function is the difference of function values \u200b\u200bat two points. Definition of the derivative:
The derivative function at the point is the limit of the function of the function of the function at a given point to the increment of the argument when the latter tends to zero.
Otherwise it can be written like this:
What is the point in finding such a limit? But what:
The derivative of the function at the point is equal to the angle tangent between the OX axis and tangent to the graph of the function at this point.
Physical meaning derivative: The time derivative is equal to the rate of straight movement.
Indeed, since school times everyone knows that speed is a private path x \u003d f (t) and time t. . Average speed for some time interval:
To find out the speed of movement at time t0. You need to calculate the limit:
Rule first: we take a constant
The constant can be taken out of the sign of the derivative. Moreover, it needs to be done. When solving examples in mathematics, take a rule - if you can simplify the expression, be sure to simplify .
Example. Calculate the derivative:
Rule Second: Derivative Functions
The derivative of the two functions is equal to the sum of the derivatives of these functions. The same is true for the derivative of the difference of functions.
We will not lead the proof of this theorem, and it is better to consider a practical example.
Find a derivative function:
Rule Third: derived works of functions
The derivative of the work of two differentiable functions is calculated by the formula:
Example: Find a derivative function:
Decision: 
It is important to say about the calculation of derivatives of complex functions. The derivative of the complex function is equal to the product of the derivative of this function by an intermediate argument on the derivative of the intermediate argument on an independent variable.
In the above example, we encounter an expression:
In this case, the intermediate argument is 8x to the fifth degree. In order to calculate the derivative of such an expression, we first consider the derivative of the external function by an intermediate argument, and then multiply on the derivative directly the very intermediate argument on an independent variable.
Rule fourth: Derivative of private two functions
Formula for determining the derivative of private two functions:
We tried to talk about derivatives for teapots from scratch. This topic is not so simple, as it seems, so I warn it: in the examples there are often traps, so be careful when calculating derivatives.
With any question on this and other topics you can contact the student service. In a short time, we will help solve the most difficult control and deal with the tasks, even if you have never been involved in the calculation of derivatives.
The functions of a complex species are not entirely correctly called the term "complex function". For example, it looks very impressive, but this feature is not difficult, unlike.
In this article, we will deal with the concept of a complex function, learn to identify it in the composition of elementary functions, we will give a formula for finding it a derivative and consider in detail the solution of the characteristic examples.
When solving examples, we will constantly use a table of derivatives and differentiation rules, so keep them before your eyes.
Complex function - This is a function, the argument of which is also a function.
From our point of view, this definition is most clear. Conditionally can be denoted as f (G (x)). That is, G (x), as it were, the argument function f (G (x)).
For example, let F be the function of the Arctangent, and G (x) \u003d lnx is the function of the natural logarithm, then the complex function f (G (x)) is ARCTG (LNX). Another example: F - the function of the fourth degree, and 
- whole rational function (see), then 
.
In turn, G (x) can also be a complex function. For example, 
. Conditionally such an expression can be designated as 
. Here f - the sinus function - the function of extraction of the square root, 
- Fractional rational function. It is logical to assume that the degree of immunity nesting can be any finite natural number.
Often you can hear that the complex function is called composition of functions.
The formula for finding a derivative complex function.
Example.
Find a derivative complex function.
Decision.
In this example, f - the construction function of the square, and G (x) \u003d 2x + 1 is a linear function.
Here is a detailed solution using the formula of a derivative complex function: 
Let's find this derivative, pre-simplifying the type of source function.
Hence,
As you can see, the results coincide.
Try not to be confused, what function is F, and which G (x).
Let us explain the example for attentiveness.
Example.
Find derivatives of complex functions and.
Decision.
In the first case f - this is the function of the construction of the square, and G (x) is the sinus function, so
.
In the second case, f is the function of sinus, and the power function. Consequently, by the formula of a complex function we have
The formula derivative for a function has the form 
Example.
Differentiate function 
.
Decision.
In this example, a complex function can be conditionally written as 
, where - the sinus function, the construction function to the third degree, the logarithming function for the base E, the function of the capture of the Arctgennes and the linear function, respectively.
By the formula of a derivative complex function 
Now found
We collect together the intermediate results: 
There is nothing terrible, disassemble the complex functions as matryoshki.
It would be possible to finish this article if it were not ...
It is advisable to clearly understand when applying differentiation rules and a derivative table, and when the formula of the derivative complex function.
Now be especially careful. We will talk about the differences between the functions of a complex view from complex functions. From how much you see this distinction, and success will be dependent when the derivatives are found.
Let's start with simple examples. Function 
can be considered as complex: G (x) \u003d TGX, 
. Therefore, you can immediately apply the formula of the derivative complex function 
But the function 
Difficult to already name cannot be called.
This feature is the sum of three functions, 3TGX and 1. Although it is a complex function: - a power function (quadratic parabola), and F is a function of the tangent. Therefore, first apply the amount of differentiation formula: 
It remains to find a derivative complex function: 
Therefore .
We hope that the essence you caught.
If you look more widely, it can be argued that the functions of a complex species can be included in complex functions and complex functions can be composed parts of the functions of complex species.
As an example, we will analyze the component parts function 
.
Firstly, this is a complex function that can be represented in the form where f is the logarithming function based on 3, and G (x) is the sum of two functions 
and 
. I.e, 
.
Secondly, Take a function h (x). It is attitude to 
.
This is the sum of two functions and 
where 
- A complex function with a numerical coefficient of 3. - The construction function in the cube is the cosine function, - linear function.
This is the sum of two functions and where 
- complex function, - the function of exponential, is a power function.
In this way, .
Thirdly, go to, which is a piece of complex function 
and the whole rational function
The runoff function is the logarithming function based on E.
Hence, .
Let's summarize: 
Now the structure structure is clear and has become visible which formulas and in which sequence to apply during its differentiation.
In the Differentiation section of the function (finding a derivative), you can familiarize yourself with the solution of such tasks.
After preliminary art preparation, examples will be less terrible, with 3-4-5 attachments of functions. Perhaps the next two examples will seem some complicated, but if they understand them (someone and peels), then almost everything else in differential calculus will seem like a children's joke.
Example 2.
Find a derivative function
As noted, when finding a derivative complex function, first of all, it is necessary rightUnderstand investings. In cases where there are doubts, I remind a useful reception: we take the experimental meaning of "X", for example, and try (mentally or on a draft) to substitute this value in the "terrible expression".
1) First, we need to calculate the expression, it means that the amount is the deepest investment.
2) Then it is necessary to calculate the logarithm:
4) Then cosine to build into a cube:
5) In the fifth step, the difference:
6) and, finally, the most external function is a square root: 
Differentiation formula complex function 
It will be applied in the reverse order, from the external function itself, to the innermost. We decide:
It seems without errors:
1) Take a derivative of square root.
2) take a derivative of the difference using the rule 
3) Troika derivative is zero. In the second term, we take a derivative on degree (Cuba).
4) We take a cosine derivative.
6) and finally take a derivative of the deepest investment.
It may seem too hard, but this is not the most brutal example. Take, for example, the Kuznetsov Collection and you will appreciate the beauty and simplicity of the disassembled derivative. I noticed that I like to give a similar thing to give on the exam to check, understands a student how to find a derivative of a complex function, or does not understand.
The following example is for an independent solution.
Example 3.
Find a derivative function
Tip: First apply linearity rules and a derivation of the work
Complete solution and answer at the end of the lesson.
It is time to move to anything more compact and pretty.
The situation is not rare when the example is given a product of not two, but three functions. How to find a derivative from the work of three multipliers?
Example 4.
Find a derivative function 
First, look, and whether it is impossible to turn the work of three functions into the work of two functions? For example, if we had two polynomials in the work, it would be possible to reveal brackets. But in this example, all the functions are different: degree, exhibitor and logarithm.
In such cases it is necessary sequenceapply Rule Differentiation Production 
twice
The focus is that for "y" we denote the product of two functions:, and for "VE" - logarithm :. Why can this be done? And not 
- This is not a work of two multipliers and the rule does not work?! There is nothing complicated:
Now it remains the second time to apply the rule To bracket:
You can still play and take something behind the brackets, but in this case the answer is better to leave in this form - it will be easier to check.
The considered example can be solved in the second way:
Both solutions are absolutely equal.
Example 5.
Find a derivative function
This is an example for an independent solution, in the sample it is resolved in the first way.
Consider similar examples with fractions.
Example 6.
Find a derivative function 
Here you can go a few ways:
Or so:
But the solution will be written more compact if it first to use a private differentiation rule 
, Accepting for the entire Numerator:
In principle, an example is solved, and if you leave it in this form, it will not be an error. But in the presence of time it is always advisable to check on the draft, and is it possible to simplify the answer?
We present the expression of the numerator to the general denominator and get rid of three-story:
The minus of additional simplifications is that there is a risk to allow an error no longer when the derivative is already founding, but when banal school transformations. On the other hand, teachers often remember the task and ask to "bring to mind" derivative.
Simpler example for self solutions:
Example 7.
Find a derivative function
We continue to learn the receptions of the derivative, and now we will consider a typical case when "scary" logarithm is proposed for differentiation
Remember very easy.
Well, let's not go far, we will immediately consider the reverse function. What function is the reverse for an indicative function? Logarithm:
In our case, the basis is the number:
Such a logarithm (that is, a logarithm with a base) is called "natural", and for it we use a special designation: instead of writing.
What is equal to? Of course, .
The derivative of the natural logarithm is also very simple:
Examples:
- Find derived function.
- What is the derived function equal?
Answers: Exhibitor and natural logarithm - the functions are uniquely simple from the point of view of the derivative. Exchange and logarithmic functions with any other base will have another derivative, which we will analyze later with you, after passing the differentiation rules.
Differentiation rules
Rules What? Again the new term, again?! ...
Differentiation - This is the process of finding a derivative.
Only and everything. And how else to name this process in one word? Not a production of ... The differential of mathematics is called the most increment of the function at. This term is happening from Latin Differentia - a difference. Here.
When displaying all these rules, we will use two functions, for example, and. We will also need formulas for their increments:
Total there are 5 rules.
The constant is made out of the sign of the derivative.
If - some kind of constant number (constant), then.
Obviously, this rule works for difference :.
We prove. Let, or easier.
Examples.
Find derived functions:
- at the point;
- at the point;
- at the point;
- at point.
Solutions:
- (the derivative is the same in all points, since this is a linear function, remember?);
Derived work
Here everything is similar: we introduce a new function and find its increment:
Derivative:
Examples:
- Find derivatives of functions and;
- Find the function derivative at the point.
Solutions:
Derivative indicative function
Now your knowledge is enough to learn how to find a derivative of any indicative function, and not just exhibitors (not forgotten what it is?).
So, where is some number.
We already know the derivative function, so let's try to bring our function to a new base:
To do this, we use a simple rule :. Then:
Well, it turned out. Now try to find a derivative, and do not forget that this feature is complex.
Happened?
Here, check yourself:
The formula turned out to be very similar to the derivative exhibit: as it was, it remained, only a multiplier appeared, which is just a number, but not a variable.
Examples:
Find derived functions:
Answers:
This is just a number that cannot be counted without a calculator, that is, not to record in a simpler form. Therefore, in response in this form and leave.
Note that there are private two functions here, therefore apply the appropriate differentiation rule:
In this example, the product of two functions:
Derivative logarithmic function
Here is similar: you already know the derivative from the natural logarithm:
Therefore, to find an arbitrary from logarithm with another reason, for example:
You need to bring this logarithm to the base. And how to change the basis of the logarithm? I hope you remember this formula:
Only now instead we will write:
In the denominator, it turned out just a constant (constant number, without a variable). The derivative is very simple:
The derivatives of the indicative and logarithmic functions are almost not found in the exam, but it will not be superfluous to know them.
Derivative complex function.
What is a "complex function"? No, it is not a logarithm, and not Arcthangence. These functions can be complex for understanding (although if the logarithm seems to you difficult, read the topic "Logarithms" and everything will pass), but from the point of view of mathematics the word "complex" does not mean "difficult".
Imagine a small conveyor: two people are sitting and have some kind of actions with some objects. For example, the first wraps a chocolate in the wrapper, and the second implies it with a ribbon. It turns out such an integral object: a chocolate, wrapped and lined with a ribbon. To eat a chocolate, you need to do reverse action in reverse order.
Let's create a similar mathematical conveyor: first we will find a cosine of the number, and then the resulting number to be erected into a square. So, we give a number (chocolate), I find his cosine (wrap), and then you will be erected by what I did, in a square (tie to the ribbon). What happened? Function. This is an example of a complex function: when to find its meanings we do the first action directly with the variable, and then another action with what happened as a result of the first one.
In other words, a complex function is a function, the argument of which is another feature.: .
For our example,.
We can completely do the same actions and in the reverse order: first you will be erected into a square, and then I'm looking for a cosine of the resulting number :. It is easy to guess that the result will be almost always different. An important feature of complex functions: when the procedure change, the function changes.
The second example: (the same). .
Action that we do the latter will call "External" function, and the action performed first - respectively "Internal" function (These are informal names, I use them only to explain the material in simple language).
Try to determine myself what function is external, and which is internal:
Answers:The separation of internal and external functions is very similar to replacement of variables: for example, in function
- First we will perform what action? First, consider sinus, but only then erected into the cube. So, the internal function, and the external one.
And the initial function is their composition :. - Inner:; External :.
Check :. - Inner:; External :.
Check :. - Inner:; External :.
Check :. - Inner:; External :.
Check :.
we produce a replacement of variables and get a function.
Well, now we will extract our chocolate chocolate - search for a derivative. The procedure is always reverse: first we are looking for an external function derivative, then multiply the result on the derivative of the internal function. With regard to the original example, it looks like this:
Another example:
So, we finally formulate the official rule:
The algorithm for finding a derivative complex function:
It seems that everything is simple, yes?
Check on the examples:
Solutions:
1) internal :;
External:;
2) internal :;
(Only do not think now to cut on! From under the cosine, nothing is done, remember?)
3) internal :;
External:;
It is immediately clear that here a three-level complex function: after all, it is already the complex function itself, and it is still removing the root from it, that is, we carry out the third action (chocolate in the wrapper and with a ribbon put into the portfolio). But there are no reason to be afraid: all the same "unpack" this function will be in the same order as usual: from the end.
That is, first use the root, then cosine, and only then expression in brackets. And then all this variables.
In such cases, it is convenient to numbered actions. That is, imagine that we are known. What order are we going to perform actions to calculate the value of this expression? We will examine on the example:
The later the action takes place, the more the "external" will be the corresponding function. Sequence of actions - as before:
Here the nesting is generally 4-level. Let's determine the procedure.
1. Forced expression. .
2. Root. .
3. Sinus. .
4. Square. .
5. We collect everything in a bunch:
DERIVATIVE. Briefly about the main thing
Derived function - The ratio of the increment of the function to the increment of the argument with an infinitely small increment of the argument:
Basic derivatives:
Differentiation Rules:
The constant is made for the sign of the derivative:
Derived amount:
Production work:
Private derivative:
Derivative complex function:
Algorithm for finding a derivative of complex function:
- We define the "internal" function, we find its derivative.
- We define the "external" function, we find its derivative.
- Multiply the results of the first and second items.
In this lesson, we will learn to find derivative complex function. The lesson is a logical continuation of the classes How to find a derivative?where we disassemble the simplest derivatives, and also got acquainted with the rules of differentiation and some technical techniques of finding derivatives. Thus, if you are not very clear with derivatives of functions, you will not be completely clear, then first read the above lesson. Please set up to a serious way - the material is not simple, but I still try to set it out simply and accessible.
In practice, a derivative of a complex function has to face very often, I would even say, almost always when you tasks to find derivatives.
We look at the table for a rule (No. 5) of differentiation of a complex function:
We understand. First of all, pay attention to the record. Here we have two functions - and, moreover, the function, figuratively speaking, is invested in the function. The function of this type (when one function is embedded in another) and is called a complex function.
I will call the function external function, and function - internal (or nested) function.
! These definitions are not theoretical and should not appear in the piston design of tasks. I use informal expressions "External Function", "Internal" function only to make it easier for you to understand the material.
In order to clarify the situation, consider:
Example 1.
Find a derivative function
Under the sinus, we are not just the letter "X", but an integer expression, so it will not be possible to find a derivative immediately on the table. We also notice that here it is impossible to apply the first four rules, it seems there is a difference, but the fact is that the sinus is not "separated into parts":
In this example, from my explanations, it is intuitive that the function is a complex function, and the polynomial is an internal function (attachment), and is an external function.
First stepto perform when finding a derivative complex function is to figure out what function is internal and what is the external.
In the case of simple examples, it seems it seems that a polynomial is invested under sine. But what if everything is not obvious? How to determine exactly what function is external, and what is the inner? To do this, I propose to use the next reception, which can be carried out mentally or on the draft.
Imagine that we need to calculate the value of an expression value on the calculator (instead of a unit there may be any number).
What do we calculate first? First of all You will need to perform the following:, Therefore, the polynomial and will be internal function: 
Secondly It will be necessary to find, so sinus - it will be an external function: 
After we Have figured out With internal and external functions, it's time to apply the differentiation rule of a complex function.
We start to solve. From lesson How to find a derivative? We remember that the decoration of the solution of any derivative always begins so - we conclude an expression in the brackets and put on the right at the top of the barcode:
First We find the external function derivative (sinus), we look at the table of derivative elementary functions and notice that. All tabular formulas are applicable and in the case, if "X" is replaced by a complex expression, in this case:
Note that the internal function did not change, we do not touch her.
Well, it is quite obvious that
The result of the application of the formula in the piston design looks like this:
A permanent multiplier usually endure expressions:
If any misunderstanding remains, rewrite the decision on paper and read the explanations again.
Example 2.
Find a derivative function
Example 3.
Find a derivative function
As always, write: 
We understand where we have an external function, and where is the inner. To do this, try (mentally or on a draft) to calculate the value of the expression at. What needs to be performed first? First of all, it is necessary to count what is equal to the base:, it means that the polynomial is internal function: 
And, only then the exercise is carried out into the extent, therefore, the power function is an external function: 
According to the formula, you first need to find a derivative from the external function, in this case, on the extent. We wanted the necessary formula in the table :. We repeat again: any tabular formula is valid not only for "X", but also for complex expression. Thus, the result of applying the differentiation runt of a complex function is as follows:
I emphasize again that when we take a derivative of an external function, the internal function does not change with us: 
Now it remains to find a completely simple derivative from the internal function and a little "combing" the result:
Example 4.
Find a derivative function
This is an example for an independent decision (answer at the end of the lesson).
To secure an understanding of the derivative complex function, I will give an example without comment, try to figure it out yourself, paint, where external and where is the internal function, why are the tasks solved this way?
Example 5.
a) find a derivative function
b) find a derivative function
Example 6.
Find a derivative function 
Here we have a root, and in order to indifferentiate the root, it must be represented in the form of a degree. Thus, first give the function to the proper form:
Analyzing the function, we conclude that the sum of the three terms is an internal function, and the external function is the external function. Apply the differentiation rule of a complex function:
The degree again represent in the form of a radical (root), and for the derivative of the internal function, use a simple rule of differentiation amount:
Ready. You can also put the expression to the general denominator and write down with one fraction in brackets. Beautiful, of course, but when bulky long derivatives are obtained - it is better not to do this (it's easy to get confused, to allow an unnecessary error, and the teacher will inconveniently check).
Example 7.
Find a derivative function
This is an example for an independent decision (answer at the end of the lesson).
It is interesting to note that sometimes instead of the procedure for differentiation of a complex function, you can use the proportion differentiation rule 
, But such a solution will look like a perversion fun. Here is a characteristic example:
Example 8.
Find a derivative function
Here you can use the proportion differentiation rule But it is much more profitable to find a derivative through a differentiation rule of a complex function:
We prepare the function for differentiation - we take a minus per sign of the derivative, and the cosine raise into the numerator:
Cosine is an internal function, the external function is an external function.
We use our rule:
We find the derivative of the internal function, the cosine is discarding back down:
Ready. In the examined example, it is important not to get confused in signs. By the way, try to solve it using the rule. The answers must match.
Example 9.
Find a derivative function
This is an example for an independent decision (answer at the end of the lesson).
So far, we have considered cases when only one investment was in our complex function. In the practical tasks, it is often possible to meet derivatives, where, as matryoshki, one to another, are embedded at once 3, or even 4-5 functions.
Example 10.
Find a derivative function
We understand in the investments of this function. We try to calculate the expression using the experimental value. How would we believe on the calculator?
First you need to find, it means, Arksinus is the deepest investment:
Then this arxinus units should be built into the square:
And finally, the seven is erected into a degree: 
That is, in this example, we have three different functions and two attachments, while the inner function is arxinus, and the external function itself is an indicative function.
We begin to decide
According to the rule, you first need to take a derivative from the external function. We look at the table of derivatives and find a derivative of the indicative function: the only difference is instead of "X" we have a difficult expression that does not cancel the validity of this formula. So, the result of applying the differentiation runt of a complex function is as follows:
Under the stroke we have a complicated function again! But it is easier. It is easy to make sure that the internal function is arxinus, the external function is a degree. According to the differentiation of a complex function, you first need to take a derivative.