Calculate derivatives using the rule for differentiating a complex function. Rules for calculating derivatives

Since you came here, you probably already saw this formula in the textbook

and make a face like this:

Friend, don't worry! In fact, everything is simply outrageous. You will definitely understand everything. Just one request - read the article slowly, try to understand every step. I wrote as simply and clearly as possible, but you still need to understand the idea. And be sure to solve the tasks from the article.

What is a complex function?

Imagine that you are moving to another apartment and therefore packing things into large boxes. Suppose you need to collect some small items, for example, school writing materials. If you just throw them into a huge box, they will get lost among other things. To avoid this, you first put them, for example, in a bag, which you then put in a large box, after which you seal it. This “complex” process is presented in the diagram below:

It would seem, what does mathematics have to do with it? Yes, despite the fact that a complex function is formed in EXACTLY THE SAME way! Only we “pack” not notebooks and pens, but \(x\), while the “packages” and “boxes” are different.

For example, let's take x and “pack” it into a function:

As a result, we get, of course, \(\cos⁡x\). This is our “bag of things”. Now let’s put it in a “box” - pack it, for example, into a cubic function.

What will happen in the end? Yes, that’s right, there will be a “bag of things in a box,” that is, “cosine of X cubed.”

The resulting design is a complex function. It differs from simple one in that SEVERAL “influences” (packages) are applied to one X in a row and it turns out as if “function from function” - “packaging within packaging”.

In the school course there are very few types of these “packages”, only four:

Let's now “pack” X first into an exponential function with base 7, and then into a trigonometric function. We get:

\(x → 7^x → tg⁡(7^x)\)

Now let’s “pack” x twice into trigonometric functions, first in and then in:

\(x → sin⁡x → cotg⁡ (sin⁡x)\)

Simple, right?

Now write the functions yourself, where x:
- first it is “packed” into a cosine, and then into an exponential function with base \(3\);
- first to the fifth power, and then to the tangent;
- first to the logarithm to the base \(4\) , then to the power \(-2\).

Find the answers to this task at the end of the article.

Can we “pack” X not two, but three times? No problem! And four, and five, and twenty-five times. Here, for example, is a function in which x is “packed” \(4\) times:

\(y=5^(\log_2⁡(\sin⁡(x^4)))\)

But such formulas will not be found in school practice (students are luckier - theirs may be more complicated☺).

"Unpacking" a complex function

Look at the previous function again. Can you figure out the “packing” sequence? What X was stuffed into first, what then, and so on until the very end. That is, which function is nested within which? Take a piece of paper and write down what you think. You can do this with a chain with arrows as we wrote above or in any other way.

Now the correct answer is: first, x was “packed” into the \(4\)th power, then the result was packed into a sine, it, in turn, was placed into the logarithm to the base \(2\), and in the end this whole construction was stuffed into a power fives.

That is, you need to unwind the sequence IN REVERSE ORDER. And here’s a hint on how to do it easier: immediately look at the X – you should dance from it. Let's look at a few examples.

For example, here is the following function: \(y=tg⁡(\log_2⁡x)\). We look at X - what happens to it first? Taken from him. And then? The tangent of the result is taken. The sequence will be the same:

\(x → \log_2⁡x → tg⁡(\log_2⁡x)\)

Another example: \(y=\cos⁡((x^3))\). Let's analyze - first we cubed X, and then took the cosine of the result. This means the sequence will be: \(x → x^3 → \cos⁡((x^3))\). Pay attention, the function seems to be similar to the very first one (where it has pictures). But this is a completely different function: here in the cube is x (that is, \(\cos⁡((x·x·x)))\), and there in the cube is the cosine \(x\) (that is, \(\cos⁡ x·\cos⁡x·\cos⁡x\)). This difference arises from different "packing" sequences.

The last example (with important information in it): \(y=\sin⁡((2x+5))\). It is clear that here they first did arithmetic operations with x, then took the sine of the result: \(x → 2x+5 → \sin⁡((2x+5))\). And this is an important point: despite the fact that arithmetic operations are not functions in themselves, here they also act as a way of “packing”. Let's delve a little deeper into this subtlety.

As I said above, in simple functions x is “packed” once, and in complex functions - two or more. Moreover, any combination of simple functions (that is, their sum, difference, multiplication or division) is also a simple function. For example, \(x^7\) is a simple function and so is \(ctg x\). This means that all their combinations are simple functions:

\(x^7+ ctg x\) - simple,
\(x^7· cot x\) – simple,
\(\frac(x^7)(ctg x)\) – simple, etc.

However, if one more function is applied to such a combination, it will become a complex function, since there will be two “packages”. See diagram:

Okay, go ahead now. Write the sequence of “wrapping” functions:
\(y=cos(⁡(sin⁡x))\)
\(y=5^(x^7)\)
\(y=arctg⁡(11^x)\)
\(y=log_2⁡(1+x)\)
The answers are again at the end of the article.

Internal and external functions

Why do we need to understand function nesting? What does this give us? The fact is that without such an analysis we will not be able to reliably find derivatives of the functions discussed above.

And in order to move on, we will need two more concepts: internal and external functions. This is a very simple thing, moreover, in fact, we have already analyzed them above: if we remember our analogy at the very beginning, then the internal function is a “package”, and the external function is a “box”. Those. what X is “wrapped” in first is an internal function, and what the internal function is “wrapped” in is already external. Well, it’s clear why - she’s outside, that means external.

In this example: \(y=tg⁡(log_2⁡x)\), the function \(\log_2⁡x\) is internal, and
- external.

And in this: \(y=\cos⁡((x^3+2x+1))\), \(x^3+2x+1\) is internal, and
- external.

Complete the last practice of analyzing complex functions, and let's finally move on to what we were all started for - we will find derivatives of complex functions:

Fill in the blanks in the table:

Derivative of a complex function

Bravo to us, we finally got to the “boss” of this topic - in fact, the derivative of a complex function, and specifically, to that very terrible formula from the beginning of the article.☺

\((f(g(x)))"=f"(g(x))\cdot g"(x)\)

This formula reads like this:

The derivative of a complex function is equal to the product of the derivative of the external function with respect to a constant internal function and the derivative of the internal function.

And immediately look at the “word by word” parsing diagram to understand what is what:

I hope the terms “derivative” and “product” do not cause any difficulties. “Complex function” - we have already sorted it out. The catch is in the “derivative of an external function with respect to a constant internal function.” What it is?

Answer: This is the usual derivative of an external function, in which only the external function changes, and the internal one remains the same. Still not clear? Okay, let's use an example.

Let us have a function \(y=\sin⁡(x^3)\). It is clear that the internal function here is \(x^3\), and the external
. Let us now find the derivative of the exterior with respect to the constant interior.

In “old” textbooks it is also called the “chain” rule. So if y = f (u), and u = φ (x), that is

y = f (φ (x))

complex - compound function (composition of functions) then

Where , after calculation is considered at u = φ (x).

Note that here we took “different” compositions from the same functions, and the result of differentiation naturally turned out to depend on the order of “mixing”.

The chain rule naturally extends to compositions of three or more functions. In this case, there will be three or more “links” in the “chain” that makes up the derivative. Here is an analogy with multiplication: “we have” a table of derivatives; “there” - multiplication table; “with us” is the chain rule and “there” is the “column” multiplication rule. When calculating such “complex” derivatives, no auxiliary arguments (u¸v, etc.), of course, are introduced, but, having noted for themselves the number and sequence of functions involved in the composition, the corresponding links are “strung” in the indicated order.

. Here, with the “x” to obtain the value of the “y”, five operations are performed, that is, there is a composition of five functions: “external” (the last of them) - exponential - e  ; then in reverse order, power. (♦) 2 ; trigonometric sin(); sedate. () 3 and finally logarithmic ln.(). That's why

With the following examples we will “kill a couple of birds with one stone”: we will practice differentiating complex functions and add to the table of derivatives of elementary functions. So:

4. For a power function - y = x α - rewriting it using the well-known “basic logarithmic identity” - b=e ln b - in the form x α = x α ln x we ​​obtain

5. For an arbitrary exponential function, using the same technique we will have

6. For an arbitrary logarithmic function, using the well-known formula for transition to a new base, we consistently obtain

.

7. To differentiate the tangent (cotangent), we use the rule for differentiating quotients:

To obtain the derivatives of inverse trigonometric functions, we use the relation that is satisfied by the derivatives of two mutually inverse functions, that is, the functions φ (x) and f (x) related by the relations:

This is the ratio

It is from this formula for mutually inverse functions

And
,

Finally, let us summarize these and some other derivatives that are also easily obtained in the following table.

If g(x) And f(u) – differentiable functions of their arguments, respectively, at points x And u= g(x), then the complex function is also differentiable at the point x and is found by the formula

A typical mistake when solving derivative problems is mechanically transferring the rules for differentiating simple functions to complex functions. Let's learn to avoid this mistake.

Example 2. Find the derivative of a function

Wrong solution: calculate the natural logarithm of each term in parentheses and look for the sum of the derivatives:

Correct solution: again we determine where the “apple” is and where the “minced meat” is. Here the natural logarithm of the expression in parentheses is an “apple”, that is, a function over the intermediate argument u, and the expression in brackets is “minced meat”, that is, an intermediate argument u by independent variable x.

Then (using formula 14 from the derivatives table)

In many real-life problems, the expression with a logarithm can be somewhat more complicated, which is why there is a lesson

Example 3. Find the derivative of a function

Wrong solution:

Correct solution. Once again we determine where the “apple” is and where the “mincemeat” is. Here, the cosine of the expression in brackets (formula 7 in the table of derivatives) is an “apple”, it is prepared in mode 1, which affects only it, and the expression in brackets (the derivative of the degree is number 3 in the table of derivatives) is “minced meat”, it is prepared under mode 2, which affects only it. And as always, we connect two derivatives with the product sign. Result:

The derivative of a complex logarithmic function is a frequent task in tests, so we strongly recommend that you attend the lesson “Derivative of a logarithmic function.”

The first examples were on complex functions, in which the intermediate argument on the independent variable was a simple function. But in practical tasks it is often necessary to find the derivative of a complex function, where the intermediate argument is either itself a complex function or contains such a function. What to do in such cases? Find derivatives of such functions using tables and differentiation rules. When the derivative of the intermediate argument is found, it is simply substituted into the right place in the formula. Below are two examples of how this is done.

In addition, it is useful to know the following. If a complex function can be represented as a chain of three functions

then its derivative should be found as the product of the derivatives of each of these functions:

Many of your homework assignments may require you to open your guides in new windows. Actions with powers and roots And Operations with fractions .

Example 4. Find the derivative of a function

We apply the rule of differentiation of a complex function, not forgetting that in the resulting product of derivatives there is an intermediate argument with respect to the independent variable x does not change:

We prepare the second factor of the product and apply the rule for differentiating the sum:

The second term is the root, so

Thus, we found that the intermediate argument, which is a sum, contains a complex function as one of the terms: raising to a power is a complex function, and what is being raised to a power is an intermediate argument with respect to the independent variable x.

Therefore, we again apply the rule for differentiating a complex function:

We transform the degree of the first factor into a root, and when differentiating the second factor, do not forget that the derivative of the constant is equal to zero:

Now we can find the derivative of the intermediate argument needed to calculate the derivative of a complex function required in the problem statement y:

Example 5. Find the derivative of a function

First, we use the rule for differentiating the sum:

We obtained the sum of the derivatives of two complex functions. Let's find the first one:

Here, raising the sine to a power is a complex function, and the sine itself is an intermediate argument for the independent variable x. Therefore, we will use the rule of differentiation of a complex function, along the way taking the factor out of brackets :

Now we find the second term of the derivatives of the function y:

Here raising the cosine to a power is a complex function f, and the cosine itself is an intermediate argument in the independent variable x. Let us again use the rule for differentiating a complex function:

The result is the required derivative:

Table of derivatives of some complex functions

For complex functions, based on the rule of differentiation of a complex function, the formula for the derivative of a simple function takes a different form.

1. Derivative of a complex power function, where u x
2. Derivative of the root of the expression
3. Derivative of an exponential function
4. Special case of exponential function
5. Derivative of a logarithmic function with an arbitrary positive base A
6. Derivative of a complex logarithmic function, where u– differentiable function of the argument x
7. Derivative of sine
8. Derivative of cosine
9. Derivative of tangent
10. Derivative of cotangent
11. Derivative of arcsine
12. Derivative of arccosine
13. Derivative of arctangent
14. Derivative of arc cotangent

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? 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 instant 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 :

Solving physical problems or examples in mathematics is completely impossible without knowledge of the derivative and methods for calculating it. The derivative is one of the most important concepts in mathematical analysis. We decided to devote today’s article to this fundamental topic. What is a derivative, what is its physical and geometric meaning, how to calculate the derivative of a function? All these questions can be combined into one: how to understand the derivative?

Geometric and physical meaning of derivative

Let there be a function f(x) , specified in a certain interval (a, b) . Points x and x0 belong to this interval. When x changes, the function itself changes. Changing the argument - the difference in its values x-x0 . This difference is written as delta x and is called argument increment. A change or increment of a function is the difference between the values ​​of a function at two points. Definition of derivative:

The derivative of a function at a point is the limit of the ratio of the increment of the function at a given point to the increment of the argument when the latter tends to zero.

Otherwise it can be written like this:

What's the point of finding such a limit? And here's what it is:

the derivative of a function at a point is equal to the tangent of the angle between the OX axis and the tangent to the graph of the function at a given point.

Physical meaning of the derivative: the derivative of the path with respect to time is equal to the speed of rectilinear motion.

Indeed, since school days everyone knows that speed is a particular path x=f(t) and time t . Average speed over a certain period of time:

To find out the speed of movement at a moment in time t0 you need to calculate the limit:

Rule one: set a constant

The constant can be taken out of the derivative sign. Moreover, this must be done. When solving examples in mathematics, take it as a rule - If you can simplify an expression, be sure to simplify it .

Example. Let's calculate the derivative:

Rule two: derivative of the sum of functions

The derivative of the sum of two functions is equal to the sum of the derivatives of these functions. The same is true for the derivative of the difference of functions.

We will not give a proof of this theorem, but rather consider a practical example.

Find the derivative of the function:

Rule three: derivative of the product of functions

The derivative of the product of two differentiable functions is calculated by the formula:

Example: find the derivative of a function:

Solution:

It is important to talk about calculating derivatives of complex functions here. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument and the derivative of the intermediate argument with respect to the independent variable.

In the above example we come across the expression:

In this case, the intermediate argument is 8x to the fifth power. In order to calculate the derivative of such an expression, we first calculate the derivative of the external function with respect to the intermediate argument, and then multiply by the derivative of the intermediate argument itself with respect to the independent variable.

Rule four: derivative of the quotient of two functions

Formula for determining the derivative of the quotient of two functions:

We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it seems, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.

With any questions on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult test and understand the tasks, even if you have never done derivative calculations before.