Very easy to remember.

Well, let’s not go far, let’s immediately consider the inverse function. Which function is the inverse of the exponential function? Logarithm:

In our case, the base is the number:

Such a logarithm (that is, a logarithm with a base) is called “natural”, and we use a special notation for it: we write instead.

What is it equal to? Of course, .

The derivative of the natural logarithm is also very simple:

Examples:

1. Find the derivative of the function.
2. What is the derivative of the function?

Answers: The exponential and natural logarithm are uniquely simple functions from a derivative perspective. Exponential and logarithmic functions with any other base will have a different derivative, which we will analyze later, after we go through the rules of differentiation.

Rules of differentiation

Rules of what? Again a new term, again?!...

Differentiation is the process of finding the derivative.

That's all. What else can you call this process in one word? Not derivative... Mathematicians call the differential the same increment of a function at. This term comes from the Latin differentia - difference. Here.

When deriving all these rules, we will use two functions, for example, and. We will also need formulas for their increments:

There are 5 rules in total.

The constant is taken out of the derivative sign.

If - some constant number (constant), then.

Obviously, this rule also works for the difference: .

Let's prove it. Let it be, or simpler.

Examples.

Find the derivatives of the functions:

1. at a point;
2. at a point;
3. at a point;
4. at the point.

Solutions:

1. (the derivative is the same at all points, since it is a linear function, remember?);

Derivative of the product

Everything is similar here: let’s introduce a new function and find its increment:

Derivative:

Examples:

1. Find the derivatives of the functions and;
2. Find the derivative of the function at a point.

Solutions:

Derivative of an exponential function

Now your knowledge is enough to learn how to find the derivative of any exponential function, and not just exponents (have you forgotten what that is yet?).

So, where is some number.

We already know the derivative of the function, so let's try to reduce our function to a new base:

To do this, we will use a simple rule: . Then:

Well, it worked. Now try to find the derivative, and don't forget that this function is complex.

Happened?

Here, check yourself:

The formula turned out to be very similar to the derivative of an exponent: as it was, it remains the same, only a factor appeared, which is just a number, but not a variable.

Examples:
Find the derivatives of the functions:

Answers:

This is just a number that cannot be calculated without a calculator, that is, it cannot be written down in a simpler form. Therefore, we leave it in this form in the answer.

Note that here is the quotient of two functions, so we apply the corresponding differentiation rule:

In this example, the product of two functions:

Derivative of a logarithmic function

It’s similar here: you already know the derivative of the natural logarithm:

Therefore, to find an arbitrary logarithm with a different base, for example:

We need to reduce this logarithm to the base. How do you change the base of a logarithm? I hope you remember this formula:

Only now we will write instead:

The denominator is simply a constant (a constant number, without a variable). The derivative is obtained very simply:

Derivatives of exponential and logarithmic functions are almost never found in the Unified State Examination, but it will not be superfluous to know them.

Derivative of a complex function.

What is a "complex function"? No, this is not a logarithm, and not an arctangent. These functions can be difficult to understand (although if you find the logarithm difficult, read the topic “Logarithms” and you will be fine), but from a mathematical point of view, the word “complex” does not mean “difficult”.

Imagine a small conveyor belt: two people are sitting and doing some actions with some objects. For example, the first one wraps a chocolate bar in a wrapper, and the second one ties it with a ribbon. The result is a composite object: a chocolate bar wrapped and tied with a ribbon. To eat a chocolate bar, you need to do the reverse steps in reverse order.

Let's create a similar mathematical pipeline: first we will find the cosine of a number, and then square the resulting number. So, we are given a number (chocolate), I find its cosine (wrapper), and then you square what I got (tie it with a ribbon). What happened? Function. This is an example of a complex function: when, to find its value, we perform the first action directly with the variable, and then a second action with what resulted from the first.

In other words, a complex function is a function whose argument is another function: .

For our example, .

We can easily do the same steps in reverse order: first you square it, and I then look for the cosine of the resulting number: . It’s easy to guess that the result will almost always be different. An important feature of complex functions: when the order of actions changes, the function changes.

Second example: (same thing). .

The action we do last will be called "external" function, and the action performed first - accordingly "internal" function(these are informal names, I use them only to explain the material in simple language).

Try to determine for yourself which function is external and which internal:

Answers: Separating inner and outer functions is very similar to changing variables: for example, in a function

1. What action will we perform first? First, let's calculate the sine, and only then cube it. This means that it is an internal function, but an external one.
   And the original function is their composition: .
2. Internal: ; external: .
   Examination: .
3. Internal: ; external: .
   Examination: .
4. Internal: ; external: .
   Examination: .
5. Internal: ; external: .
   Examination: .

We change variables and get a function.

Well, now we will extract our chocolate bar and look for the derivative. The procedure is always reversed: first we look for the derivative of the outer function, then we multiply the result by the derivative of the inner function. In relation to the original example, it looks like this:

Another example:

So, let's finally formulate the official rule:

Algorithm for finding the derivative of a complex function:

It seems simple, right?

Let's check with examples:

Solutions:

1) Internal: ;

External: ;

2) Internal: ;

(Just don’t try to cut it by now! Nothing comes out from under the cosine, remember?)

3) Internal: ;

External: ;

It is immediately clear that this is a three-level complex function: after all, this is already a complex function in itself, and we also extract the root from it, that is, we perform the third action (put the chocolate in a wrapper and with a ribbon in the briefcase). But there is no reason to be afraid: we will still “unpack” this function in the same order as usual: from the end.

That is, first we differentiate the root, then the cosine, and only then the expression in brackets. And then we multiply it all.

In such cases, it is convenient to number the actions. That is, let's imagine what we know. In what order will we perform actions to calculate the value of this expression? Let's look at an example:

The later the action is performed, the more “external” the corresponding function will be. The sequence of actions is the same as before:

Here the nesting is generally 4-level. Let's determine the course of action.

1. Radical expression. .

2. Root. .

3. Sine. .

4. Square. .

5. Putting it all together:

DERIVATIVE. BRIEFLY ABOUT THE MAIN THINGS

Derivative of a function- the ratio of the increment of the function to the increment of the argument for an infinitesimal increment of the argument:

Basic derivatives:

Rules of differentiation:

The constant is taken out of the derivative sign:

Derivative of the sum:

Derivative of the product:

Derivative of the quotient:

Derivative of a complex function:

Algorithm for finding the derivative of a complex function:

1. We define the “internal” function and find its derivative.
2. We define the “external” function and find its derivative.
3. We multiply the results of the first and second points.

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 resolve 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? 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. 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 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 instantaneous and polite response: .

The first example will be immediately intended for independent solution.

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 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:

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 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 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 an 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

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 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 :

Proof and derivation of formulas for the derivative of the natural logarithm and the logarithm to base a. Examples of calculating derivatives of ln 2x, ln 3x and ln nx. Proof of the formula for the derivative of the nth order logarithm using the method of mathematical induction.

Content

See also: Logarithm - properties, formulas, graph
Natural logarithm - properties, formulas, graph

Derivation of formulas for the derivatives of the natural logarithm and the logarithm to base a

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's 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 in terms of 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 . The inverse function to the natural logarithm is the exponential:
.
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.

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:
.

; ; .

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 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:
.

See also:

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 many difficult tasks in the Unified State Exam that 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 pass the Unified State Exam with the 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 is a school curriculum, and if you have problems with these concepts individually, it is worth repeating the topic.

Derivative of logarithm

In the Unified State Exam assignments on this topic, you can give several tasks as an example. 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 of the decimal logarithm, and the derivative of ln x is the derivative of the natural logarithm (based on 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 (even more so in higher educational institutions).

Derivative of the natural logarithm

At its core, it is the derivative of the logarithm to the base e (which is an irrational number that 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.

Let's imagine it as a complex function consisting of two simple ones.

It is enough to convert

We are looking for the derivative of u with respect to x

Let's continue with the second

We use the method of solving the derivative of a complex function by substituting u=nx.

What happened in the end?

Now let's remember what n meant in this example? This is any number that can appear in front of x in the natural logarithm. It is important for you to understand that the answer does not depend on her. Substitute whatever you want, the answer will still be 1/x.

As you can see, there is nothing complicated here; you just need to understand the principle to quickly and effectively solve problems on this topic. Now you know the theory, all you have to do is put it into practice. Practice solving problems in order to remember the principle of their solution for a long time. You may not need this knowledge after graduating from school, but in the exam it will be more relevant than ever. Good luck to you!