Let
(1)
is a differentiable function of the variable x. First we'll look at it for the set of values of x for which y takes positive values: . In the following, we will show that all the results obtained are also applicable for negative values of .

In some cases, in order to find the derivative of function (1), it is convenient to pre-logarithm it
,
and then calculate the derivative. Then, according to the rule of differentiation of a complex function,
.
From here
(2) .

The derivative of the logarithm of a function is called the logarithmic derivative:
.

Logarithmic derivative of the function y = f(x) is a derivative natural logarithm this function: (ln f(x))′.

The case of negative y values

Now consider the case when the variable can take both positive and negative values. In this case, take the logarithm of the modulus and find its derivative:
.
From here
(3) .
That is, in the general case, you need to find the derivative of the logarithm of the modulus of the function.

Comparing (2) and (3) we have:
.
That is, the formal result of calculating the logarithmic derivative does not depend on whether we took the modulo or not. Therefore, when calculating the logarithmic derivative, we do not have to worry about what sign the function has.

This situation can be clarified using complex numbers. Let, for some values of x, be negative: . If we only consider real numbers, then the function is not defined. However, if we introduce into consideration complex numbers, then we get the following:
.
That is, the functions and differ by a complex constant:
.
Since the derivative of a constant is zero, then
.

Property of the logarithmic derivative

From such a consideration it follows that the logarithmic derivative will not change if you multiply the function by an arbitrary constant :
.
Indeed, using properties of logarithm, formulas derivative sum And derivative of a constant, we have:

.

Application of logarithmic derivative

It is convenient to use the logarithmic derivative in cases where the original function consists of a product of power or exponential functions. In this case, the logarithm operation turns the product of functions into their sum. This simplifies the calculation of the derivative.

Example 1

Find the derivative of the function:
.

Solution

Let's logarithm the original function:
.

Let's differentiate with respect to the variable x.
In the table of derivatives we find:
.
We apply the rule of differentiation of complex functions.
;
;
;
;
(A1.1) .
Multiply by:

.

So, we found the logarithmic derivative:
.
From here we find the derivative of the original function:
.

Note

If we want to use only real numbers, then we should take the logarithm of the modulus of the original function:
.
Then
;
.
And we got formula (A1.1). Therefore the result has not changed.

Answer

Example 2

Using the logarithmic derivative, find the derivative of the function
.

Solution

Let's take logarithms:
(A2.1) .
Differentiate with respect to the variable x:
;
;

;
;
;
.

Multiply by:
.
From here we get the logarithmic derivative:
.

Derivative of the original function:
.

Note

Here the original function is non-negative: . It is defined at . If we do not assume that the logarithm can be defined for negative values of the argument, then formula (A2.1) should be written as follows:
.
Since

And
,
this will not affect the final result.

Answer

Example 3

Find the derivative
.

Solution

We perform differentiation using the logarithmic derivative. Let's take a logarithm, taking into account that:
(A3.1) .

By differentiating, we obtain the logarithmic derivative.
;
;
;
(A3.2) .

Since then

.

Note

Let us carry out the calculations without the assumption that the logarithm can be defined for negative values of the argument. To do this, take the logarithm of the modulus of the original function:
.
Then instead of (A3.1) we have:
;

.
Comparing with (A3.2) we see that the result has not changed.

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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 improve your skills almost from scratch. Next, you need to carefully study the page Derivative of a complex function, understand and resolve All the examples I gave. This lesson is logically the third, 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. In class Derivative of a complex function We looked at a number of examples with detailed comments. In the course of studying differential calculus and other branches of 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 complex function :

When studying other matan topics in the future, such a detailed recording 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 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. 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 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 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 the 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

Complete solution and the 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, let’s see if it’s possible to turn the product of three functions into the product of two functions? For example, if we had two polynomials in the product, 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 take 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 a 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 a “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 “break up” the logarithm of the right side as much as possible (the formulas in front of 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. 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

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”. A classic example that will be given to you in any textbook or 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 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 :

The derivative of the natural logarithm of x is equal to one divided by x:
(1) (ln x)′ =.

The derivative of the logarithm to base a is equal to one divided by the variable x multiplied by the natural logarithm of a:
(2) (log a x)′ =.

Proof

Let there be some positive number not equal to one. Consider a function depending on a variable x, which is a logarithm to the base:
.
This function is defined at . Let's find its derivative with respect to the variable x. By definition, the derivative is the following limit:
(3) .

Let's transform this expression to reduce it to known mathematical properties and rules. To do this we need to know the following facts:
A) Properties of the logarithm. We will need the following formulas:
(4) ;
(5) ;
(6) ;
B) Continuity of the logarithm and the property of limits for a continuous function:
(7) .
Here is a function that has a limit and this limit is positive.
IN) The meaning of the second remarkable limit:
(8) .

Let's apply these facts to our limit. First we transform the algebraic expression
.
To do this, we apply properties (4) and (5).

.

Let us use property (7) and the second remarkable limit (8):
.

And finally, we apply property (6):
.
Logarithm to base e called natural logarithm. It is designated as follows:
.
Then ;
.

Thus, we obtained formula (2) for the derivative of the logarithm.

Derivative of the natural logarithm

Once again we write out the formula for the derivative of the logarithm to base a:
.
This formula has the simplest form for the natural logarithm, for which , . Then
(1) .

Because of this simplicity, the natural logarithm is very widely used in mathematical analysis and in other branches of mathematics related to differential calculus. Logarithmic functions with other bases can be expressed through the natural logarithm using property (6):
.

The derivative of the logarithm with respect to the base can be found from formula (1), if you take the constant out of the differentiation sign:
.

Other ways to prove the derivative of a logarithm

Here we assume that we know the formula for the derivative of the exponential:
(9) .
Then we can derive the formula for the derivative of the natural logarithm, given that the logarithm is the inverse function of the exponential.

Let us prove the formula for the derivative of the natural logarithm, applying the formula for the derivative of the inverse function:
.
In our case. Inverse function the exponential to the natural logarithm is:
.
Its derivative is determined by formula (9). Variables can be designated by any letter. In formula (9), replace the variable x with y:
.
Since then
.
Then
.
The formula is proven.

Now we prove the formula for the derivative of the natural logarithm using rules for differentiating complex functions. Since the functions and are inverse to each other, then
.
Let's differentiate this equation with respect to the variable x:
(10) .
The derivative of x is equal to one:
.
We apply the rule of differentiation of complex functions:
.
Here . Let's substitute in (10):
.
From here
.

Example

Find derivatives of ln 2x, ln 3x And lnnx.

Solution

The original functions have a similar form. Therefore we will find the derivative of the function y = log nx. Then we substitute n = 2 and n = 3. And, thus, we obtain formulas for the derivatives of ln 2x And ln 3x .

So, we are looking for the derivative of the function
y = log nx .
Let's imagine this function as a complex function consisting of two functions:
1) Functions depending on a variable: ;
2) Functions depending on a variable: .
Then the original function is composed of the functions and :
.

Let's find the derivative of the function with respect to the variable x:
.
Let's find the derivative of the function with respect to the variable:
.
We apply the formula for the derivative of a complex function.
.
Here we set it up.

So we found:
(11) .
We see that the derivative does not depend on n. This result is quite natural if we transform the original function using the formula for the logarithm of the product:
.
- this is a constant. Its derivative is zero. Then, according to the rule of differentiation of the sum, we have:
.

Answer

; ; .

Derivative of the logarithm of modulus x

Let's find the derivative of another very important function - the natural logarithm of modulus x:
(12) .

Let's consider the case. Then the function looks like:
.
Its derivative is determined by formula (1):
.

Now let's consider the case. Then the function looks like:
,
Where .
But we also found the derivative of this function in the example above. It does not depend on n and is equal to
.
Then
.

We combine these two cases into one formula:
.

Accordingly, for the logarithm to base a, we have:
.

Derivatives of higher orders of the natural logarithm

Consider the function
.
We found its first-order derivative:
(13) .

Let's find the second order derivative:
.
Let's find the third order derivative:
.
Let's find the fourth order derivative:
.

You can notice that the nth order derivative has the form:
(14) .
Let us prove this by mathematical induction.

Proof

Let us substitute the value n = 1 into formula (14):
.
Since , then when n = 1 , formula (14) is valid.

Let us assume that formula (14) is satisfied for n = k. Let us prove that this implies that the formula is valid for n = k + 1 .

Indeed, for n = k we have:
.
Differentiate with respect to the variable x:

.
So we got:
.
This formula coincides with formula (14) for n = k + 1 . Thus, from the assumption that formula (14) is valid for n = k, it follows that formula (14) is valid for n = k + 1 .

Therefore, formula (14), for the nth order derivative, is valid for any n.

Derivatives of higher orders of the logarithm to base a

To find the nth order derivative of a logarithm to base a, you need to express it in terms of the natural logarithm:
.
Applying formula (14), we find the nth derivative:
.

Do you feel like there is still a lot of time before the exam? Is this a month? Two? Year? Practice shows that a student copes best with an exam if he begins to prepare for it in advance. There are a lot of difficult tasks, which stand in the way of schoolchildren and future applicants to the highest scores. You need to learn to overcome these obstacles, and besides, it’s not difficult to do. You need to understand the principle of working with various tasks from tickets. Then there will be no problems with the new ones.

Logarithms at first glance seem incredibly complex, but with a detailed analysis the situation becomes much simpler. If you want to take the Unified State Exam highest score, you should understand the concept in question, which is what we propose to do in this article.

First, let's separate these definitions. What is a logarithm (log)? This is an indicator of the power to which the base must be raised to obtain the specified number. If it’s not clear, let’s look at an elementary example.

In this case, the base at the bottom must be raised to the second power to get the number 4.

Now let's look at the second concept. The derivative of a function in any form is a concept that characterizes the change of a function at a given point. However, this school curriculum, and if you have problems with these concepts individually, it is worth repeating the topic.

Derivative of logarithm

IN Unified State Exam assignments On this topic, several problems can be given as examples. To begin with, the simplest logarithmic derivative. It is necessary to find the derivative of the following function.

We need to find the next derivative

There is a special formula.

In this case x=u, log3x=v. We substitute the values from our function into the formula.

The derivative of x will be equal to one. The logarithm is a little more difficult. But you will understand the principle if you simply substitute the values. Recall that the derivative of lg x is the derivative decimal logarithm, and the derivative ln x is the derivative of the natural logarithm (to base e).

Now just plug the resulting values into the formula. Try it yourself, then we’ll check the answer.

What could be the problem here for some? We introduced the concept of natural logarithm. Let's talk about it, and at the same time figure out how to solve problems with it. You won’t see anything complicated, especially when you understand the principle of its operation. You should get used to it, since it is often used in mathematics (in higher educational institutions especially).

Derivative of the natural logarithm

At its core, it is the derivative of the logarithm to base e (this is irrational number, which is approximately 2.7). In fact, ln is very simple, so it is often used in mathematics in general. Actually, solving the problem with it will not be a problem either. It is worth remembering that the derivative of the natural logarithm to the base e will be equal to one divided by x. The solution to the following example will be the most revealing.