Complex derivatives. Logarithmic derivative.
Derivative of a power-exponential function
We continue to improve our differentiation technique. In this lesson, we will consolidate the material we have covered, look at more complex derivatives, and also get acquainted with new techniques and tricks for finding a derivative, in particular, with the logarithmic derivative.
To those readers who have low level preparation, you should refer to the article How to find the derivative? Examples of solutions, which will allow you to raise your skills almost from scratch. Next, you need to carefully study the page Derivative of a complex function, understand and solve All the examples I gave. This lesson is logically the third in a row, and after mastering it you will confidently differentiate fairly complex functions. It is undesirable to take the position of “Where else? Yes, that’s enough! ”, since all examples and solutions are taken from real tests and are often encountered in practice.
Let's start with repetition. At the lesson Derivative of a complex function We looked at a number of examples with detailed comments. During the study of differential calculus and other sections mathematical analysis– you will have to differentiate very often, and it is not always convenient (and not always necessary) to describe examples in great detail. Therefore, we will practice finding derivatives orally. The most suitable “candidates” for this are derivatives of the simplest of complex functions, for example:
According to the rule of differentiation of complex functions 
:
When studying other matan topics in the future, such a detailed record is most often not required; it is assumed that the student knows how to find such derivatives on autopilot. Let's imagine that at 3 o'clock in the morning there was a phone call, And a pleasant voice asked: “What is the derivative of the tangent of two X’s?” This should be followed by an almost instantaneous and polite response: 
.
The first example will be immediately intended for independent decision.
Example 1
Find the following derivatives orally, in one action, for example: . To complete the task you only need to use table of derivatives of elementary functions(if you haven't remembered it yet). If you have any difficulties, I recommend re-reading the lesson Derivative of a complex function.
, , ,
, 
, , 
, , ,
, , 
,
, , 
,
, , ,
, 
,
Answers at the end of the lesson
Complex derivatives
After preliminary artillery preparation, examples with 3-4-5 nestings of functions will be less scary. Perhaps the following two examples will seem complicated to some, but if you understand them (someone will suffer), then almost everything else in differential calculus It will seem like a child's joke.
Example 2
Find the derivative of a function 
As already noted, when finding the derivative of a complex function, first of all, it is necessary Right UNDERSTAND your investments. In cases where there are doubts, I remind you useful trick: we take the experimental value of “x”, for example, and try (mentally or in a draft) to substitute given value into a "terrible expression".
1) First we need to calculate the expression, which means the sum is the deepest embedding.
2) Then you need to calculate the logarithm:
4) Then cube the cosine:
5) At the fifth step the difference:
6) And finally, the outermost function is the square root: 
Formula for differentiating a complex function 
will be used in reverse order, from the outermost function to the innermost. We decide:
There seem to be no errors...
(1) Take the derivative of the square root.
(2) We take the derivative of the difference using the rule 
(3) The derivative of a triple is zero. In the second term we take the derivative of the degree (cube).
(4) Take the derivative of the cosine.
(5) Take the derivative of the logarithm.
(6) And finally, we take the derivative of the deepest embedding.
It may seem too difficult, but this is not the most brutal example. Take, for example, Kuznetsov’s collection and you will appreciate all the beauty and simplicity of the analyzed derivative. I noticed that they like to give a similar thing in an exam to check whether a student understands how to find the derivative of a complex function or does not understand.
The following example is for you to solve on your own.
Example 3
Find the derivative of a function
Hint: First we apply the linearity rules and the product differentiation rule
Full solution and answer at the end of the lesson.
It's time to move on to something smaller and nicer.
It is not uncommon for an example to show the product of not two, but three functions. How to find the derivative of products of three multipliers?
Example 4
Find the derivative of a function 
First we look, is it possible to turn the product of three functions into the product of two functions? For example, if we had two polynomials in the product, then we could open the brackets. But in the example under consideration, all the functions are different: degree, exponent and logarithm.
In such cases it is necessary sequentially apply the product differentiation rule 
twice
The trick is that by “y” we denote the product of two functions: , and by “ve” we denote the logarithm: . Why can this be done? Is it really 
– this is not a product of two factors and the rule does not work?! There is nothing complicated:
Now it remains to apply the rule a second time 
to bracket:
You can still be perverted and take something out of brackets, but in in this case It is better to leave the answer in this form - it will be easier to check.
The considered example can be solved in the second way: 
Both solutions are absolutely equivalent.
Example 5
Find the derivative of a function
This is an example for an independent solution; in the sample it is solved using the first method.
Let's look at similar examples with fractions.
Example 6
Find the derivative of a function 
There are several ways you can go here:
Or like this:
But the solution will be written more compactly if we first use the rule of differentiation of the quotient 
, taking for the entire numerator:
In principle, the example is solved, and if it is left as is, it will not be an error. But if you have time, it is always advisable to check on a draft to see if the answer can be simplified? Let us reduce the expression of the numerator to common denominator And let's get rid of the three-story fraction:
The disadvantage of additional simplifications is that there is a risk of making a mistake not when finding the derivative, but during banal school transformations. On the other hand, teachers often reject the assignment and ask to “bring it to mind” the derivative.
A simpler example to solve on your own:
Example 7
Find the derivative of a function
We continue to master the methods of finding the derivative, and now we will consider a typical case when the “terrible” logarithm is proposed for differentiation
Example 8
Find the derivative of a function
Here you can go the long way, using the rule for differentiating a complex function:
But the very first step immediately plunges you into despondency - you have to take the unpleasant derivative of fractional power, and then also from the fraction.
That's why before how to take the derivative of a “sophisticated” logarithm, it is first simplified using well-known school properties:
! If you have a practice notebook at hand, copy these formulas directly there. If you don't have a notebook, copy them onto a piece of paper, since the remaining examples of the lesson will revolve around these formulas.
The solution itself can be written something like this:
Let's transform the function:
Finding the derivative: 
Pre-converting the function itself greatly simplified the solution. Thus, when a similar logarithm is proposed for differentiation, it is always advisable to “break it down”.
And now a couple of simple examples for you to solve on your own:
Example 9
Find the derivative of a function 
Example 10
Find the derivative of a function
All transformations and answers are at the end of the lesson.
Logarithmic derivative
If the derivative of logarithms is such sweet music, then the question arises: is it possible in some cases to organize the logarithm artificially? Can! And even necessary.
Example 11
Find the derivative of a function 
We recently looked at similar examples. What to do? You can sequentially apply the rule of differentiation of the quotient, and then the rule of differentiation of the product. The disadvantage of this method is that you end up with a huge three-story fraction, which you don’t want to deal with at all.
But in theory and practice there is such a wonderful thing as the logarithmic derivative. Logarithms can be organized artificially by “hanging” them on both sides:
Now you need to “disintegrate” the logarithm of the right side as much as possible (formulas before your eyes?). I will describe this process in great detail:
Let's start with differentiation.
We conclude both parts under the prime:
The derivative of the right-hand side is quite simple; I will not comment on it, because if you are reading this text, you should be able to handle it confidently.
What about the left side?
On the left side we have complex function. I foresee the question: “Why, is there one letter “Y” under the logarithm?”
The fact is that this “one letter game” - IS ITSELF A FUNCTION(if it is not very clear, refer to the article Derivative of a function specified implicitly). Therefore, the logarithm is an external function, and the “y” is internal function. And we use the rule for differentiating a complex function 
:
On the left side, as if by magic magic wand we have a derivative . Next, according to the rule of proportion, we transfer the “y” from the denominator of the left side to the top of the right side:
And now let’s remember what kind of “player”-function we talked about during differentiation? Let's look at the condition: 
Final answer: 
Example 12
Find the derivative of a function
This is an example for you to solve on your own. Example design example of this type at the end of the lesson.
Using the logarithmic derivative it was possible to solve any of the examples No. 4-7, another thing is that the functions there are simpler, and, perhaps, the use of the logarithmic derivative is not very justified.
Derivative of a power-exponential function
We have not considered this function yet. A power-exponential function is a function for which both the degree and the base depend on the “x”. Classic example, which will be given to you in any textbook or at any lecture:
How to find the derivative of a power-exponential function?
It is necessary to use the technique just discussed - the logarithmic derivative. We hang logarithms on both sides:
As a rule, on the right side the degree is taken out from under the logarithm:
As a result, on the right side we have the product of two functions, which will be differentiated by standard formula 
.
We find the derivative; to do this, we enclose both parts under strokes:
Further actions are simple:
Finally: 
If any conversion is not entirely clear, please re-read the explanations of Example #11 carefully.
IN practical tasks The power-exponential function will always be more complex than the example discussed in the lecture.
Example 13
Find the derivative of a function
We use the logarithmic derivative. 
On the right side we have a constant and the product of two factors - “x” and “logarithm of logarithm x” (another logarithm is nested under the logarithm). When differentiating, as we remember, it is better to immediately move the constant out of the derivative sign so that it does not get in the way; and, of course, we apply the familiar rule 
:
As you can see, the algorithm for using the logarithmic derivative does not contain any special tricks or tricks, and finding the derivative of a power-exponential function is usually not associated with “torment.”
Decide physical tasks or examples in mathematics is completely impossible without knowledge of the derivative and methods of calculating it. Derivative is one of the most important concepts mathematical analysis. This fundamental topic we decided to dedicate today's article. What is a derivative, what is its physical and geometric meaning how to calculate the derivative of a function? All these questions can be combined into one: how to understand the derivative?
Geometric and physical meaning of derivative
Let there be a function f(x) , specified in a certain interval (a, b) . Points x and x0 belong to this interval. When x changes, the function itself changes. Changing the argument - the difference in its values x-x0 . This difference is written as delta x and is called argument increment. A change or increment of a function is the difference between the values of a function at two points. Definition of derivative:
The derivative of a function at a point is the limit of the ratio of the increment 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's the point of finding such a limit? And here's what it is:
the derivative of a function at a point is equal to the tangent of the angle between the OX axis and the tangent to the graph of the function at a given point.
Physical meaning derivative: the derivative of the path with respect to time is equal to the speed of rectilinear motion.
Indeed, since school days everyone knows that speed is a particular path x=f(t) and time t . average speed for a certain period of time:
To find out the speed of movement at a moment in time t0 you need to calculate the limit:
Rule one: set a constant
The constant can be taken out of the derivative sign. Moreover, this must be done. When solving examples in mathematics, take it as a rule - If you can simplify an expression, be sure to simplify it .
Example. Let's calculate the derivative:
Rule two: derivative of the sum of functions
The derivative of the sum of 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 give a proof of this theorem, but rather consider a practical example.
Find the derivative of the function:
Rule three: derivative of the product of functions
The derivative of the product of two differentiable functions is calculated by the formula:
Example: find the derivative of a function:
Solution: 
It is important to talk about calculating derivatives of complex functions here. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument and the derivative of the intermediate argument with respect to the independent variable.
In the above example we come across the expression:
In this case, the intermediate argument is 8x to the fifth power. In order to calculate the derivative of such an expression, we first calculate the derivative of the external function with respect to the intermediate argument, and then multiply by the derivative of the intermediate argument itself with respect to the independent variable.
Rule four: derivative of the quotient of two functions
Formula for determining the derivative of the quotient of two functions:
We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it seems, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.
With any questions on this and other topics, you can contact the student service. Behind short term We will help you solve the most difficult tests and solve problems, even if you have never done derivative calculations before.
After preliminary artillery preparation, examples with 3-4-5 nestings of functions will be less scary. The following two examples may seem complicated to some, but if you understand them (someone will suffer), then almost everything else in differential calculus will seem like a child's joke.
Example 2
Find the derivative of a function
As already noted, when finding the derivative of a complex function, first of all, it is necessary Right UNDERSTAND your investments. In cases where there are doubts, I remind you of a useful technique: we take the experimental value of “x”, for example, and try (mentally or in a draft) to substitute this value into the “terrible expression”.
1) First we need to calculate the expression, which means the sum is the deepest embedding.
2) Then you need to calculate the logarithm:
4) Then cube the cosine:
5) At the fifth step the difference:
6) And finally, the outermost function is the square root: 
Formula for differentiating a complex function 
are applied in reverse order, from the outermost function to the innermost. We decide:
It seems without errors:
1) Take the derivative of the square root.
2) Take the derivative of the difference using the rule 
3) The derivative of a triple is zero. In the second term we take the derivative of the degree (cube).
4) Take the derivative of the cosine.
6) And finally, we take the derivative of the deepest embedding.
It may seem too difficult, but this is not the most brutal example. Take, for example, Kuznetsov’s collection and you will appreciate all the beauty and simplicity of the analyzed derivative. I noticed that they like to give a similar thing in an exam to check whether a student understands how to find the derivative of a complex function or does not understand.
The following example is for you to solve on your own.
Example 3
Find the derivative of a function
Hint: First we apply the linearity rules and the product differentiation rule
Full solution and answer at the end of the lesson.
It's time to move on to something smaller and nicer.
It is not uncommon for an example to show the product of not two, but three functions. How to find the derivative of the product of three factors?
Example 4
Find the derivative of a function 
First we look, is it possible to turn the product of three functions into the product of two functions? For example, if we had two polynomials in the product, then we could open the brackets. But in the example under consideration, all the functions are different: degree, exponent and logarithm.
In such cases it is necessary sequentially apply the product differentiation rule 
twice
The trick is that by “y” we denote the product of two functions: , and by “ve” we denote the logarithm: . Why can this be done? Is it really 
- this is not a product of two factors and the rule does not work?! There is nothing complicated:
Now it remains to apply the rule a second time to bracket:
You can also get twisted and put something out of brackets, but in this case it’s better to leave the answer exactly in this form - it will be easier to check.
The considered example can be solved in the second way:
Both solutions are absolutely equivalent.
Example 5
Find the derivative of a function
This is an example for an independent solution; in the sample it is solved using the first method.
Let's look at similar examples with fractions.
Example 6
Find the derivative of a function 
There are several ways you can go here:
Or like this:
But the solution will be written more compactly if we first use the rule of differentiation of the quotient 
, taking for the entire numerator:
In principle, the example is solved, and if it is left as is, it will not be an error. But if you have time, it is always advisable to check on a draft to see if the answer can be simplified?
Let's reduce the expression of the numerator to a common denominator and get rid of the three-story structure of the fraction:
The disadvantage of additional simplifications is that there is a risk of making a mistake not when finding the derivative, but during banal school transformations. On the other hand, teachers often reject the assignment and ask to “bring it to mind” the derivative.
A simpler example to solve on your own:
Example 7
Find the derivative of a function
We continue to master the methods of finding the derivative, and now we will consider a typical case when the “terrible” logarithm is proposed for differentiation
Examples are given of calculating derivatives using the formula for the derivative of a complex function.
Here we give examples of calculating derivatives of following functions:
;
;
;
;
.
If a function can be represented as complex function V the following form:
,
then its derivative is determined by the formula:
.
In the examples below, we will write this formula as follows:
.
Where .
Here, the subscripts or , located under the derivative sign, denote the variables by which differentiation is performed.
Usually, in tables of derivatives, derivatives of functions from the variable x are given. However, x is a formal parameter. The variable x can be replaced by any other variable. Therefore, when differentiating a function from a variable, we simply change, in the table of derivatives, the variable x to the variable u.
Simple examples
Example 1
Find the derivative of a complex function
.
Solution
Let's write it down given function in equivalent form:
.
In the table of derivatives we find:
;
.
According to the formula for the derivative of a complex function, we have:
.
Here .
Answer
Example 2
Find the derivative
.
Solution
We take the constant 5 out of the derivative sign and from the table of derivatives we find:
.
.
Here .
Answer
Example 3
Find the derivative
.
Solution
We take out a constant -1
for the sign of the derivative and from the table of derivatives we find:
;
From the table of derivatives we find:
.
We apply the formula for the derivative of a complex function:
.
Here .
Answer
More complex examples
In more complex examples we apply the rule for differentiating a complex function several times. In this case, we calculate the derivative from the end. That is, we break the function into its component parts and find the derivatives of the simplest parts using table of derivatives. We also use rules for differentiating sums, products and fractions. Then we make substitutions and apply the formula for the derivative of a complex function.
Example 4
Find the derivative
.
Solution
Let's highlight the most simple part formula and find its derivative. .
.
Here we have used the notation
.
We find the derivative of the next part of the original function using the results obtained. We apply the rule for differentiating the sum:
.
Once again we apply the rule of differentiation of complex functions.
.
Here .
Answer
Example 5
Find the derivative of the function
.
Solution
Let's select the simplest part of the formula and find its derivative from the table of derivatives. .
We apply the rule of differentiation of complex functions.
.
Here
.
In this article we will talk about such an important mathematical concept as a complex function, and learn how to find the derivative of a complex function.
Before learning to find the derivative of a complex function, let's understand the concept of a complex function, what it is, “what it is eaten with,” and “how to cook it correctly.”
Let's consider arbitrary function, for example, like this:
Note that the argument on the right and left sides of the function equation is the same number, or expression.
Instead of a variable, we can put, for example, the following expression: . And then we get the function
Let's call the expression an intermediate argument, and the function an outer function. It's not strict mathematical concepts, but they help to understand the meaning of the concept of a complex function.
A strict definition of the concept of a complex function sounds like this:
Let a function be defined on a set and be the set of values of this function. Let the set (or its subset) be the domain of definition of the function. Let's assign a number to each of them. Thus, the function will be defined on the set. It is called function composition or complex function.
In this definition, if we use our terminology, an external function is an intermediate argument.
The derivative of a complex function is found according to the following rule:
To make it more clear, I like to write this rule as follows:
In this expression, using denotes an intermediate function.
So. To find the derivative of a complex function, you need
1. Determine which function is external and find the corresponding derivative from the table of derivatives.
2. Define an intermediate argument.
In this procedure, the greatest difficulty is finding the external function. A simple algorithm is used for this:
A. Write down the equation of the function.
b. Imagine that you need to calculate the value of a function for some value of x. To do this, you substitute this value of x into the equation of the function and produce arithmetic operations. The last action you do is the external function.
For example, in the function
The last action is exponentiation.
Let's find the derivative of this function. To do this, we write an intermediate argument