If g(x) and f(u) Are differentiable functions of their arguments, respectively, at the 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 problems on derivatives is the automatic transfer of the rules of differentiation simple functions to complex functions. We will 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:

The right decision: again we define where is "apple" and where is "minced meat". Here the natural logarithm of the expression in parentheses is "apple", that is, a function by an intermediate argument u, and the expression in parentheses is "mince", that is, an intermediate argument u on the independent variable x.

Then (using formula 14 from the table of derivatives)

In many real tasks an expression with a logarithm can be somewhat more complicated, so there is a lesson

Example 3. Find the derivative of a function

Wrong solution:

The right decision. V Once again determine where is "apple" and where is "minced meat". Here, the cosine of the expression in brackets (formula 7 in the table of derivatives) is "apple", it is prepared in mode 1, affecting only it, and the expression in brackets (the derivative of the power is number 3 in the table of derivatives) is "minced meat", it prepares with mode 2, which affects only it. And, as always, we connect the two derivatives with a product sign. Result:

Derivative complex logarithmic function- a frequent assignment on test papers, so we strongly recommend visiting the lesson "Derivative of a logarithmic function".

The first examples were for complex functions in which the intermediate argument on the independent variable was a simple function. But in practical exercises it is often required to find the derivative 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 rules of differentiation. When the derivative of the intermediate argument is found, it is simply substituted in the right place in the formula. Below are two examples of how this is done.

It is also helpful 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:

To solve many of your homework assignments, you may need to open tutorials in new windows. Actions with powers and roots and Actions 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, the 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 a root, therefore

Thus, we got 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 raised to a power is an intermediate argument in the independent variable x.

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

We transform the degree of the first factor into a root, and 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 condition y:

Example 5. Find the derivative of a function

First, let's use the sum differentiation rule:

We got the sum of the derivatives of two complex functions. We find the first of them:

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

Now we find the second term from the generators of the derivative of the function y:

Here raising the cosine to a power is a complex function f, and the cosine itself is an intermediate argument with respect to the independent variable x... Let's use the rule of differentiation of a complex function again:

The result is the required derivative:

Derivative table of some complex functions

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

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

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 applied in reverse order, from the outermost function to the innermost. We decide:

It seems without 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) We take the derivative of the cosine.

6) And 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:

Let us bring the expression of the numerator to 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

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! ", Since all examples and solutions are taken from real control works 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 topics mathematical analysis- you will have to differentiate very often, and it is not always convenient (and not always necessary) to paint 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

Example 2

Find the derivative of a function

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.

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.

Example 4

Find the derivative of a function

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

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:

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 fractional degree, and then also from the 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 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. Sample design 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 :

In the "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 - a composite 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 be dependent on the order of “mixing”.

The chain rule naturally extends to a composition of three or more functions. In this case, there will be three or more "links" in the "chain" making up the derivative, respectively. There is also an analogy with multiplication: "we have" - a table of derivatives; “There” is the multiplication table; “We” is a chain rule and “there” is a rule of multiplication by “column”. 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, they "string" the corresponding links in the indicated order.

... Here, five operations are performed with the "x" to obtain the value of "igruka", that is, there is a composition of five functions: "external" (the last of them) - indicative - e ; further, in the reverse order, the exponential. (♦) 2; trigonometric sin (); sedate. () 3 and finally logarithmic ln. (). So

The following examples will "kill pairs of birds with one stone": we will practice differentiating complex functions and supplement the table of derivatives 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, applying the same technique, we will have

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

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

To obtain the derivatives of the inverse trigonometric functions, we use the relation which the derivatives of two reciprocal functions satisfy, that is, the functions φ (x) and f (x) connected by the relations:

This is the ratio

It is from this formula for mutually inverse functions

and
,

In the end, let us summarize these and some others, just as easily obtained derivatives, in the following table.

Functions complex kind do not always fit the definition of a complex function. If there is a function of the form y = sin x - (2 - 3) a r c t g x x 5 7 x 10 - 17 x 3 + x - 11, then it cannot be considered complex, unlike y = sin 2 x.

This article will show the concept of a complex function and its identification. Let's work with formulas for finding the derivative with examples of solutions in the conclusion. The use of the table of derivatives and the rule of differentiation significantly reduces the time to find the derivative.

Basic definitions

Definition 1

A complex function is a function whose argument is also a function.

It is denoted in this way: f (g (x)). We have that the function g (x) is considered an argument to f (g (x)).

Definition 2

If there is a function f and is a cotangent function, then g (x) = ln x is a natural logarithm function. We get that the complex function f (g (x)) will be written as arctan (lnx). Or a function f, which is a function raised to the 4th power, where g (x) = x 2 + 2 x - 3 is considered an entire rational function, we get that f (g (x)) = (x 2 + 2 x - 3) 4 ...

Obviously g (x) can be tricky. From the example y = sin 2 x + 1 x 3 - 5, you can see that the value of g has a cube root with a fraction. This expression is allowed to be denoted as y = f (f 1 (f 2 (x))). Whence we have that f is a sine function, and f 1 is a function located under square root, f 2 (x) = 2 x + 1 x 3 - 5 is a fractional rational function.

Definition 3

The nesting degree is determined by any natural number and is written as y = f (f 1 (f 2 (f 3 (... (F n (x)))))).

Definition 4

The concept of function composition refers to the number of nested functions by the condition of the problem. For the solution, the formula for finding the derivative of a complex function of the form

(f (g (x))) "= f" (g (x)) g "(x)

Examples of

Example 1

Find the derivative of a complex function of the form y = (2 x + 1) 2.

Solution

By the condition, you can see that f is a squaring function, and g (x) = 2 x + 1 is considered a linear function.

Let's apply the derivative formula for a complex function and write:

f "(g (x)) = ((g (x)) 2)" = 2 · (g (x)) 2 - 1 = 2 · g (x) = 2 · (2 x + 1); g "(x) = (2 x + 1)" = (2 x) "+ 1" = 2 x "+ 0 = 2 1 x 1 - 1 = 2 ⇒ (f (g (x))) "= f" (g (x)) g "(x) = 2 (2 x + 1) 2 = 8 x + 4

It is necessary to find a derivative with a simplified original form of the function. We get:

y = (2 x + 1) 2 = 4 x 2 + 4 x + 1

Hence we have that

y "= (4 x 2 + 4 x + 1)" = (4 x 2) "+ (4 x)" + 1 "= 4 · (x 2)" + 4 · (x) "+ 0 = = 4 2 x 2 - 1 + 4 1 x 1 - 1 = 8 x + 4

The results matched.

When solving problems of this kind, it is important to understand where the function of the form f and g (x) will be located.

Example 2

You should find the derivatives of complex functions of the form y = sin 2 x and y = sin x 2.

Solution

The first notation of the function says that f is a squaring function and g (x) is a sine function. Then we get that

y "= (sin 2 x)" = 2 sin 2 - 1 x (sin x) "= 2 sin x cos x

The second entry shows that f is a sine function, and g (x) = x 2 we denote a power function. It follows that the product of a complex function can be written as

y "= (sin x 2)" = cos (x 2) (x 2) "= cos (x 2) 2 x 2 - 1 = 2 x cos (x 2)

The formula for the derivative y = f (f 1 (f 2 (f 3 (... (Fn (x)))))) will be written as y "= f" (f 1 (f 2 (f 3 (... ( fn (x)))))) f 1 "(f 2 (f 3 (... (fn (x))))) f 2" (f 3 (.. (fn (x)) )) ·. ... ... · F n "(x)

Example 3

Find the derivative of the function y = sin (ln 3 a r c t g (2 x)).

Solution

This example shows the complexity of writing and locating functions. Then y = f (f 1 (f 2 (f 3 (f 4 (x))))) denote, where f, f 1, f 2, f 3, f 4 (x) is a sine function, a function of raising in 3 degree, function with logarithm and base e, arctangent function and linear.

From the formula for the definition of a complex function, we have that

y "= f" (f 1 (f 2 (f 3 (f 4 (x))))) f 1 "(f 2 (f 3 (f 4 (x)))) f 2" (f 3 (f 4 (x))) f 3 "(f 4 (x)) f 4" (x)

We get what to find

f "(f 1 (f 2 (f 3 (f 4 (x))))) as the sine derivative according to the table of derivatives, then f" (f 1 (f 2 (f 3 (f 4 (x)))) ) = cos (ln 3 arctan (2 x)).
f 1 "(f 2 (f 3 (f 4 (x)))) as the derivative of the power function, then f 1" (f 2 (f 3 (f 4 (x)))) = 3 ln 3 - 1 arctan (2 x) = 3 ln 2 arctan (2 x).
f 2 "(f 3 (f 4 (x))) as the derivative of the logarithmic, then f 2" (f 3 (f 4 (x))) = 1 a r c t g (2 x).
f 3 "(f 4 (x)) as the derivative of the arctangent, then f 3" (f 4 (x)) = 1 1 + (2 x) 2 = 1 1 + 4 x 2.
When finding the derivative f 4 (x) = 2 x, subtract 2 outside the sign of the derivative using the formula for the derivative of a power function with an exponent equal to 1, then f 4 "(x) = (2 x)" = 2 x "= 2 1 x 1 - 1 = 2.

We combine the intermediate results and get that

y "= f" (f 1 (f 2 (f 3 (f 4 (x))))) f 1 "(f 2 (f 3 (f 4 (x)))) f 2" (f 3 (f 4 (x))) f 3 "(f 4 (x)) f 4" (x) = = cos (ln 3 arctan (2 x)) 3 ln 2 arctan (2 x) 1 arctan (2 x) 1 1 + 4 x 2 2 = = 6 cos (ln 3 arctan (2 x)) ln 2 arctan (2 x) arctan (2 x) (1 + 4 x 2)

The analysis of such functions is reminiscent of nesting dolls. Differentiation rules cannot always be applied explicitly using a table of derivatives. It is often necessary to use a formula for finding derivatives of complex functions.

There are some differences between a complex view and a complex function. With an obvious ability to distinguish this, finding derivatives will be especially easy.

Example 4

It is necessary to consider giving a similar example. If there is a function of the form y = t g 2 x + 3 t g x + 1, then it can be considered as a complex form g (x) = t g x, f (g) = g 2 + 3 g + 1. Obviously, it is necessary to apply a formula for a complex derivative:

f "(g (x)) = (g 2 (x) + 3 g (x) + 1)" = (g 2 (x)) "+ (3 g (x))" + 1 "= = 2 · g 2 - 1 (x) + 3 · g "(x) + 0 = 2 g (x) + 3 · 1 · g 1 - 1 (x) = = 2 g (x) + 3 = 2 tgx + 3; g "(x) = (tgx)" = 1 cos 2 x ⇒ y "= (f (g (x)))" = f "(g (x)) g" (x) = (2 tgx + 3 ) 1 cos 2 x = 2 tgx + 3 cos 2 x

A function of the form y = t g x 2 + 3 t g x + 1 is not considered difficult, since it has the sum of t g x 2, 3 t g x and 1. However, t g x 2 is considered a complex function, then we obtain a power function of the form g (x) = x 2 and f, which is a function of the tangent. To do this, it is necessary to differentiate by the amount. We get that

y "= (tgx 2 + 3 tgx + 1)" = (tgx 2) "+ (3 tgx)" + 1 "= = (tgx 2)" + 3 · (tgx) "+ 0 = (tgx 2)" + 3 cos 2 x

We proceed to finding the derivative of a complex function (t g x 2) ":

f "(g (x)) = (tan (g (x)))" = 1 cos 2 g (x) = 1 cos 2 (x 2) g "(x) = (x 2)" = 2 x 2 - 1 = 2 x ⇒ (tgx 2) "= f" (g (x)) g "(x) = 2 x cos 2 (x 2)

We get that y "= (t g x 2 + 3 t g x + 1)" = (t g x 2) "+ 3 cos 2 x = 2 x cos 2 (x 2) + 3 cos 2 x

Complex functions can be included in complex functions, and complex functions themselves can be complex functions.

Example 5

For example, consider a complex function of the form y = log 3 x 2 + 3 cos 3 (2 x + 1) + 7 e x 2 + 3 3 + ln 2 x (x 2 + 1)

This function can be represented in the form y = f (g (x)), where the value of f is a function of the logarithm to base 3, and g (x) is considered the sum of two functions of the form h (x) = x 2 + 3 cos 3 (2 x + 1) + 7 ex 2 + 3 3 and k (x) = ln 2 x (x 2 + 1). Obviously, y = f (h (x) + k (x)).

Consider the function h (x). This is the ratio l (x) = x 2 + 3 cos 3 (2 x + 1) + 7 to m (x) = e x 2 + 3 3

We have that l (x) = x 2 + 3 cos 2 (2 x + 1) + 7 = n (x) + p (x) is the sum of two functions n (x) = x 2 + 7 and p (x) = 3 cos 3 (2 x + 1), where p (x) = 3 p 1 (p 2 (p 3 (x))) is a complex function with a numerical coefficient 3, and p 1 is a cubing function, p 2 as a cosine function, p 3 (x) = 2 x + 1 - a linear function.

We got that m (x) = ex 2 + 3 3 = q (x) + r (x) is the sum of two functions q (x) = ex 2 and r (x) = 3 3, where q (x) = q 1 (q 2 (x)) is a complex function, q 1 is a function with exponential function, q 2 (x) = x 2 is a power function.

This shows that h (x) = l (x) m (x) = n (x) + p (x) q (x) + r (x) = n (x) + 3 p 1 (p 2 ( p 3 (x))) q 1 (q 2 (x)) + r (x)

Passing to an expression of the form k (x) = ln 2 x s 2 (x)) with rational integer t (x) = x 2 + 1, where s 1 is the squaring function, and s 2 (x) = ln x is logarithmic with base e.

It follows that the expression takes the form k (x) = s (x) t (x) = s 1 (s 2 (x)) t (x).

Then we get that

y = log 3 x 2 + 3 cos 3 (2 x + 1) + 7 ex 2 + 3 3 + ln 2 x (x 2 + 1) = = fn (x) + 3 p 1 (p 2 (p 3 (x))) q 1 (q 2 (x)) = r (x) + s 1 (s 2 (x)) t (x)

By function structures, it became clear how and what formulas should be used to simplify an expression when differentiating it. To familiarize yourself with such problems and for the concept of their solution, it is necessary to turn to the point of differentiation of the function, that is, finding its derivative.

If you notice an error in the text, please select it and press Ctrl + Enter

Derivative table of some complex functions

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

Complex derivatives

Logarithmic derivative

The derivative of the exponential function

Basic definitions

Examples of

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