Derived from the natural logarithm of x. Complex derivatives

Complex derivatives. Logarithmic derivative.
The derivative of the exponential function

We continue to improve our differentiation technique. In this lesson, we will consolidate the material covered, consider more complex derivatives, and also get acquainted with new techniques and tricks for finding the derivative, in particular, with the logarithmic derivative.

Those readers with a low level of training should refer to the article How do I 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 rather complex functions. It is undesirable to adhere to the position “Where else? And that's enough! ”, Because all examples and solutions are taken from real tests and are often found in practice.

Let's start with repetition. At the lesson Derivative of a complex function we have 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 write 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 a complex function :

When studying other topics of matan in the future, such a detailed record is often not required, it is assumed that the student is able to find similar derivatives on the automatic autopilot. Imagine that at 3 am the phone rang, and a pleasant voice asked: "What is the derivative of the tangent of two Xs?" This should be followed by an almost instant and polite response: .

The first example will be immediately intended for an independent solution.

Example 1

Find the following derivatives orally, in one step, for example:. To complete the task, you need to use only table of derivatives of elementary functions(if it is not remembered yet). If you have any difficulties, I recommend rereading 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 function attachments will be less scary. Perhaps the following two examples will seem difficult to some, but if you understand them (someone will suffer), then almost everything else in the 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 the attachments. In cases where there are doubts, I recall a useful technique: we take the experimental value "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, which means that the amount is the deepest investment.

2) Then you need to calculate the logarithm:

4) Then raise the cosine to a cube:

5) At the fifth step, the difference:

6) And finally, the outermost function is the square root:

Complex function differentiation formula are applied in reverse order, from the outermost function to the innermost. We decide:

It seems without mistakes….

(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) We take the derivative of the cosine.

(5) We take the derivative of the logarithm.

(6) Finally, we take the derivative of the deepest nesting.

It may sound too difficult, but this is not yet the most brutal example. Take, for example, Kuznetsov's collection and you will appreciate all the charm and simplicity of the analyzed derivative. I noticed that they like to give a similar thing on the exam to check whether the student understands how to find the derivative of a complex function, or does not understand.

The next example is for a do-it-yourself solution.

Example 3

Find the derivative of a function

Hint: First, apply the linearity rules and the product differentiation rule

Complete solution and answer at the end of the tutorial.

Now is the time to move on to something more compact and cute.
It is not uncommon for an example to give a 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 is 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 expand the brackets. But in this example, all functions are different: degree, exponent and logarithm.

In such cases, it is necessary consistently apply product differentiation rule twice

The trick is that for "y" we denote the product of two functions:, and for "ve" - the logarithm:. Why can this be done? Is it - this is not a product of two factors and the rule does not work ?! There is nothing complicated:

Now it remains for the second time to apply the rule to the parenthesis:

You can still be perverted and put something outside the brackets, but 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 in the first way.

Let's look at similar examples with fractions.

Example 6

Find the derivative of a function

There are several ways to go here:

Or like this:

But the solution will be written more compactly if, first of all, we use the rule for differentiating the quotient , taking for the entire numerator:

In principle, the example is solved, and if you leave it as it is, it will not be an error. But if you have time, it is always advisable to check on a draft, but is it possible to simplify the answer? Let us reduce the expression of the numerator to a common denominator and get rid of the three-story fraction:

The disadvantage of additional simplifications is that there is a risk of making a mistake not in finding the derivative, but in the case of banal school transformations. On the other hand, teachers often reject the assignment and ask to "bring to mind" the derivative.

A simpler example for a do-it-yourself solution:

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 a long way, using the rule of differentiating a complex function:

But the very first step immediately plunges you into despondency - you have to take an unpleasant derivative from a fractional degree, and then also from a fraction.

So before how to take the derivative of the "fancy" logarithm, it is preliminarily simplified using the well-known school properties:

! If you have a practice notebook on hand, copy these formulas right there. If you don't have a notebook, redraw them on a piece of paper, as the rest of the lesson examples will revolve around these formulas.

The solution itself can be structured like this:

Let's transform the function:

Find the derivative:

Preconfiguring the function itself has greatly simplified the solution. Thus, when a similar logarithm is proposed for differentiation, it is always advisable to "break up" it.

And now a couple of simple examples for an independent solution:

Example 9

Find the derivative of a function

Example 10

Find the derivative of a function

All transformations and answers 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 have seen similar examples recently. What to do? You can consistently apply the rule for differentiating the quotient, and then the rule for differentiating the work. The disadvantage of this method is that you get 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 : since the function can take negative values, then, generally speaking, you need to use modules: that will disappear as a result of differentiation. However, the current design is also acceptable, where the defaults are taken into account complex values. But if with all the severity, then in both cases, a reservation should be made that.

Now you need to maximally "destroy" the logarithm of the right side (formulas in front of your eyes?). I will describe this process in great detail:

Actually, we proceed to differentiation.
We enclose both parts under the stroke:

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 confidently cope with it.

What about the left side?

On the left we have complex function... I foresee the question: "Why, there is also one letter" ygrek "under the logarithm?"

The fact is that this "one letter igrek" - ITSELF IS A FUNCTION(if not very clear, refer to the article Derived from an Implicit Function). Therefore, the logarithm is an external function, and the "game" is an internal function. And we use the rule of differentiating a complex function :

On the left side, as if by magic, we have a derivative. Further, according to the rule of proportion, we throw the "game" from the denominator of the left side to the top of the right side:

And now we recall what kind of “game” -function we discussed in differentiation? We look at the condition:

Final answer:

Example 12

Find the derivative of a function

This is an example for a do-it-yourself solution. A sample of the design of an example of this type at the end of the lesson.

With the help of the logarithmic derivative it was possible to solve any of examples 4-7, but the other thing is that the functions there are simpler, and, perhaps, the use of the logarithmic derivative is not very justified.

The derivative of the exponential function

We have not considered this function yet. An exponential function is a function in which and the degree and base depend on "x"... A classic example that will be given to you in any textbook or in any lecture:

How to find the derivative of an exponential function?

It is necessary to use the technique just considered - the logarithmic derivative. We hang logarithms on both sides:

As a rule, the degree is taken out from under the logarithm on the right side:

As a result, on the right-hand side, we got a product of two functions, which will be differentiated according to the standard formula .

We find the derivative, for this we enclose both parts under the strokes:

Further actions are simple:

Finally:

If any transformation is not entirely clear, please carefully re-read the explanations in Example 11.

In practical tasks, the exponential function will always be more complicated than the considered lecture example.

Example 13

Find the derivative of a function

We use the logarithmic derivative.

On the right side we have a constant and a product of two factors - "x" and "logarithm of the logarithm of x" (another logarithm is embedded under the logarithm). When differentiating the constant, as we remember, it is better to immediately take out the sign of the derivative so that it does not get in the way under your feet; and of course we apply the familiar rule :

When differentiating an exponential 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 implies the ability to use the table of derivatives, the rules of differentiation and knowledge of the formula for the derivative of a complex function.

Derivation of the formula for the logarithmic derivative.

First, we make the logarithm to the base e, simplify the form of the function using the properties of the logarithm, and then find the derivative of the implicitly given function:

As an example, let us find the derivative of an exponential function x to the power of x.

Taking the logarithm gives. By 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 go from differentiating an exponential function to finding the derivative of a complex function:

Example.

Find the derivative of a function .

Solution.

In this example, the function is a fraction and its derivative can be searched for using the rules of differentiation. But due to the cumbersomeness of the expression, this will require many transformations. In such cases, it is wiser to use the formula for the logarithmic derivative ... Why? You will understand now.

Let's find it first. In the transformations, we will use the properties of the logarithm (the logarithm of the fraction is equal to the difference between the logarithms, and the logarithm of the product is equal to the sum of the logarithms, and 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 obtained result into the formula for the logarithmic derivative and we get the answer:

To consolidate the material, we will give a couple more examples without detailed explanations.

Example.

Find the derivative of the exponential function

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

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

Proof

Let there be some positive number that is not equal to one. Consider a function that depends on the variable x, which is the logarithm to the base:
.
This function is defined at. Let us 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 the well-known mathematical properties and rules. To do this, we need to know the following facts:
A) Logarithm properties. We need the following formulas:
(4) ;
(5) ;
(6) ;
B) Continuity of the logarithm and the property of limits for a continuous function:
(7) .
Here is some function that has a limit and this limit is positive.
V) The meaning of the second remarkable limit:
(8) .

We apply these facts to our limit. First, we transform the algebraic expression
.
For 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 base e called natural logarithm... It is designated as follows:
.
Then ;
.

Thus, we have 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 with respect to the 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 in terms of the natural logarithm using property (6):
.

The base derivative of the logarithm can be found from formula (1), if the constant is taken out of the differentiation sign:
.

Other ways to prove the derivative of the logarithm

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

Let us prove the formula for the derivative of the natural logarithm, by applying the formula for the derivative of the inverse function:
.
In our case . The function inverse to the natural logarithm is the exponent:
.
Its derivative is determined by formula (9). Variables can be designated with 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 complex function differentiation rules... Since the functions and are inverse to each other, then
.
We differentiate this equation with respect to the variable x:
(10) .
The x-derivative is equal to one:
.
We apply the rule of differentiating a complex function:
.
Here . Substitute in (10):
.
From here
.

Example

Find derivatives of ln 2x, ln 3x and ln nx.

Solution

The original functions are similar. Therefore, we will find the derivative of the function y = ln nx... Then plug in 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 = ln nx .
Let's imagine this function as a complex function, consisting of two functions:
1) Variable-dependent functions:;
2) Variable-dependent functions:.
Then the original function is composed of functions and:
.

Let us 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 up.

So we found:
(11) .
We see that the derivative is independent of n. This result is quite natural if we transform the original function using the formula for the logarithm of the product:
.
is constant. Its derivative is zero. Then, according to the rule for differentiating the sum, we have:
.

Answer

; ; .

Derivative of the logarithm of the modulus x

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

Let's consider a case. Then the function has the form:
.
Its derivative is determined by the formula (1):
.

Now consider the case. Then the function has the form:
,
where .
But we also found the derivative of this function in the above example. It does not depend on n and is equal to
.
Then
.

We combine these two cases into one formula:
.

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

Higher order derivatives of the natural logarithm

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

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

It can be seen that the nth order derivative has the form:
(14) .
Let us prove this by the method of mathematical induction.

Proof

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

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

Indeed, for n = k we have:
.
We 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 derivative of the nth order, is valid for any n.

Higher-order derivatives of the logarithm with the base a

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

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

At first glance, logarithms seem incredibly complex, but a detailed analysis makes the situation much simpler. If you want to pass the exam for the highest score, you should understand the concept in question, which we propose to do in this article.

Let's start by separating these definitions. What is a logarithm (log)? This is an indicator of the degree to which the base must be raised in order to obtain the indicated number. If it's not clear, let's look at an elementary example.

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

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

Derivative of the logarithm

In the tasks of the exam on this topic, several tasks can be cited as an example. For starters, the simplest logarithmic derivative. It is necessary to find the derivative of the following function.

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

Now just plug these values into the formula. Try it yourself, then check the answer.

What could be the problem here for some? We have introduced the concept of the natural logarithm. We will tell you about it, and at the same time we will figure out how to solve problems with it. You will not see anything complicated, especially when you understand how it works. You should get used to it, as it is often used in mathematics (especially in higher education).

Derivative of the natural logarithm

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