When differentiating, it is indicative power function or cumbersome fractional expressions, it is convenient to use the logarithmic derivative. In this article we will look at examples of its application with detailed solutions.
Further presentation assumes the ability to use the table of derivatives, differentiation rules and knowledge of the formula for the derivative of a complex function.
Derivation of the formula for the logarithmic derivative.
First, we take logarithms to the base e, simplify the form of the function using the properties of the logarithm, and then find the derivative of the implicitly specified function: 
For example, let's find the derivative of an exponential power function x to the power x.
Taking logarithms gives . According to the properties of the logarithm. Differentiating both sides of the equality leads to the result: 
Answer: 
.
The same example can be solved without using the logarithmic derivative. You can carry out some transformations and move from differentiating an exponential power function to finding the derivative complex function:
Example.
Find the derivative of a function 
.
Solution.
In this example the function 
is a fraction and its derivative can be found using the rules of differentiation. But due to the cumbersomeness of the expression, this will require many transformations. In such cases, it is more reasonable to use the logarithmic derivative formula 
. Why? You will understand now.
Let's find it first. In transformations we will use the properties of the logarithm (the logarithm of a fraction is equal to the difference of logarithms, and the logarithm of the product equal to the sum logarithms, and also the degree of the expression under the logarithm sign can be taken out as a coefficient in front of the logarithm): 
These transformations led us to a fairly simple expression, the derivative of which is easy to find: 
We substitute the result obtained into the formula for the logarithmic derivative and get the answer:
To consolidate the material, we will give a couple more examples without detailed explanations.
Example.
Find the derivative of an exponential power function 
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.
Those readers who have a low level of preparation 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 the phone rang and a pleasant voice asked: “What is the derivative of the tangent of two X’s?” This should be followed by an almost instant 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. 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 most external function is Square root: 
Formula for differentiating a complex function 
are applied 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 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 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 from a fractional power, and then also from a 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:
Note
: because a function can take negative values, then, generally speaking, you need to use modules: 
, which will disappear as a result of differentiation. However, the current design is also acceptable, where by default it is taken into account complex meanings. But if in all rigor, then in both cases a reservation should be made that.
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, 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. A sample design of an example of this type is at the end of the lesson.
Using the logarithmic derivative it was possible to solve any of 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
This function We haven't looked at it 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 according to the 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 No. 11 carefully.
In practical tasks, the power-exponential function will always be more complicated than the lecture example considered.
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 
:
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