Functions complex type 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 derivative table and differentiation rules significantly reduces the time for finding the derivative.
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Basic definitions
Definition 1A complex function is one whose argument is also a function.
It is denoted this way: f (g (x)). We have that the function g (x) is considered an argument f (g (x)).
Definition 2
If there is a function f and is a cotangent function, then g(x) = ln x is the function natural logarithm. We find that the complex function f (g (x)) will be written as arctg(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 obtain that f (g (x)) = (x 2 + 2 x - 3) 4 .
Obviously g(x) can be complex. From the example y = sin 2 x + 1 x 3 - 5 it is clear that the value of g has the cube root of the fraction. This expression can be denoted as y = f (f 1 (f 2 (x))). From where 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 - fractional rational function.
Definition 3
The degree of nesting 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 according to the conditions of the problem. To solve, use the formula for finding the derivative of a complex function of the form
(f (g (x))) " = f " (g (x)) g " (x)
Examples
Example 1Find the derivative of a complex function of the form y = (2 x + 1) 2.
Solution
The condition shows 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 the derivative with a simplified original form of the function. We get:
y = (2 x + 1) 2 = 4 x 2 + 4 x + 1
From here 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 were the same.
When solving problems of this type, 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 function notation says that f is the squaring function and g(x) is the 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 denotes a power function. It follows that we write the product of a complex function 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 (. . . (f n (x))))) will be written as y " = f " (f 1 (f 2 (f 3 (. . . ( f n (x))))) · f 1 " (f 2 (f 3 (. . . (f n (x)))) · · f 2 " (f 3 (. . . (f n (x))) )) · . . . fn "(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 difficulty of writing and determining the location of 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 the sine function, the function of raising to 3 degree, function with logarithm and base e, arctangent and linear function.
From the formula for defining 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 we need to find
- f " (f 1 (f 2 (f 3 (f 4 (x))))) as the derivative of the sine according to the table of derivatives, then f " (f 1 (f 2 (f 3 (f 4 (x)))) ) = cos (ln 3 a r c t g (2 x)) .
- f 1 "(f 2 (f 3 (f 4 (x)))) as derivative power function, then f 1 "(f 2 (f 3 (f 4 (x)))) = 3 · ln 3 - 1 a r c t g (2 x) = 3 · ln 2 a r c t g (2 x) .
- f 2 " (f 3 (f 4 (x))) as a logarithmic derivative, 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, remove 2 from 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 a r c t g (2 x)) 3 ln 2 a r c t g (2 x) 1 a r c t g (2 x) 1 1 + 4 x 2 2 = = 6 cos (ln 3 a r c t g (2 x)) ln 2 a r c t g (2 x) a r c t g (2 x) (1 + 4 x 2)
Analysis of such functions is reminiscent of nesting dolls. The rules of differentiation cannot always be applied explicitly using a derivative table. Often you need to use a formula for finding derivatives of complex functions.
There are some differences between complex appearance and complex functions. With a clear ability to distinguish this, finding derivatives will be especially easy.
Example 4
It is necessary to consider giving such an 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 function of the form g (x) = t g x, f (g) = g 2 + 3 g + 1. Obviously, it is necessary to use the 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 t g x + 3 ; g " (x) = (t g x) " = 1 cos 2 x ⇒ y " = (f (g (x))) " = f " (g (x)) g " (x) = (2 t g x + 3 ) · 1 cos 2 x = 2 t g x + 3 cos 2 x
A function of the form y = t g x 2 + 3 t g x + 1 is not considered complex, 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 tangent function. To do this, differentiate by amount. We get that
y " = (t g x 2 + 3 t g x + 1) " = (t g x 2) " + (3 t g x) " + 1 " = = (t g x 2) " + 3 (t g x) " + 0 = (t g x 2) " + 3 cos 2 x
Let's move on to finding the derivative of a complex function (t g x 2) ":
f " (g (x)) = (t g (g (x))) " = 1 cos 2 g (x) = 1 cos 2 (x 2) g " (x) = (x 2) " = 2 x 2 - 1 = 2 x ⇒ (t g x 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
Functions of a complex type can be included in complex functions, and complex functions themselves can be components of functions of a complex type.
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 as y = f (g (x)), where the value of f is a function of the base 3 logarithm, and g (x) is considered the sum of two functions of the form h (x) = x 2 + 3 cos 3 (2 x + 1) + 7 e x 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 numerical coefficient 3, and p 1 - by the cube function, p 2 by the cosine function, p 3 (x) = 2 x + 1 - by the linear function.
We found that m (x) = e x 2 + 3 3 = q (x) + r (x) is the sum of two functions q (x) = e x 2 and r (x) = 3 3, where q (x) = q 1 (q 2 (x)) is a complex function, q 1 is a function with an exponential, 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)
When moving to an expression of the form k (x) = ln 2 x (x 2 + 1) = s (x) t (x), it is clear that the function is presented in the form of a complex s (x) = ln 2 x = s 1 ( s 2 (x)) with a rational integer t (x) = x 2 + 1, where s 1 is a squaring function, and s 2 (x) = ln x is logarithmic with base e.
It follows that the expression will take 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 e x 2 + 3 3 + ln 2 x (x 2 + 1) = = f n (x) + 3 p 1 (p 2 (p 3 (x))) q 1 (q 2 (x)) = r (x) + s 1 (s 2 (x)) t (x)
Based on the structures of the function, it became clear how and what formulas need to be used to simplify the expression when differentiating it. To become familiar with such problems and for the concept of their solution, it is necessary to turn to the point of differentiating a function, that is, finding its derivative.
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On which we examined the simplest derivatives, and also became acquainted with the rules of differentiation and some technical techniques for finding derivatives. Thus, if you are not very good with derivatives of functions or some points in this article are not entirely clear, then first read the above lesson. Please get in a serious mood - the material is not simple, but I will still try to present it simply and clearly.
In practice, you have to deal with the derivative of a complex function very often, I would even say, almost always, when you are given tasks to find derivatives.
We look at the table at the rule (No. 5) for differentiating a complex function:
Let's figure it out. First of all, let's pay attention to the entry. Here we have two functions – and , and the function, figuratively speaking, is nested within the function . A function of this type (when one function is nested within another) is called a complex function.
I will call the function external function, and the function – internal (or nested) function.
! These definitions are not theoretical and should not appear in the final design of assignments. I use informal expressions “external function”, “internal” function only to make it easier for you to understand the material.
To clarify the situation, consider:
Example 1
Find the derivative of a function
Under the sine we have not just the letter “X”, but an entire expression, so finding the derivative right away from the table will not work. We also notice that it is impossible to apply the first four rules here, there seems to be a difference, but the fact is that the sine cannot be “torn into pieces”:
In this example, it is already intuitively clear from my explanations that a function is a complex function, and the polynomial is an internal function (embedding), and an external function.
First step what you need to do when finding the derivative of a complex function is to understand which function is internal and which is external.
In the case of simple examples, it seems clear that a polynomial is embedded under the sine. But what if everything is not obvious? How to accurately determine which function is external and which is internal? To do this, I suggest using the following technique, which can be done mentally or in a draft.
Let's imagine that we need to use a calculator to calculate the value of the expression at (instead of one there can be any number).
What will we calculate first? First of all you will need to perform the following action: , therefore the polynomial will be an internal function: 
Secondly will need to be found, so sine – will be an external function: 
After we SOLD OUT with internal and external functions, it’s time to apply the rule of differentiation of complex functions 
.
Let's start deciding. From the lesson How to find the derivative? we remember that the design of a solution to any derivative always begins like this - we enclose the expression in brackets and put a stroke at the top right:
At first we find the derivative of the external function (sine), look at the table of derivatives of elementary functions and notice that . All table formulas are also applicable if “x” is replaced with a complex expression, in this case:
Please note that the inner function hasn't changed, we don't touch it.
Well, it's quite obvious that
The result of applying the formula 
in its final form it looks like this:
The constant factor is usually placed at the beginning of the expression:
If there is any misunderstanding, write the solution down on paper and read the explanations again.
Example 2
Find the derivative of a function
Example 3
Find the derivative of a function
As always, we write down: 
Let's figure out where we have an external function and where we have an internal one. To do this, we try (mentally or in a draft) to calculate the value of the expression at . What should you do first? First of all, you need to calculate what the base is equal to: therefore, the polynomial is the internal function: 
And only then is the exponentiation performed, therefore, the power function is an external function: 
According to the formula 
, first you need to find the derivative of the external function, in this case, the degree. We look for the required formula in the table: . We repeat again: any tabular formula is valid not only for “X”, but also for a complex expression. Thus, the result of applying the rule for differentiating a complex function 
next:
I emphasize again that when we take the derivative of the external function, our internal function does not change: 
Now all that remains is to find a very simple derivative of the internal function and tweak the result a little:
Example 4
Find the derivative of a function
This is an example for independent decision(answer at the end of the lesson).
To consolidate your understanding of the derivative of a complex function, I will give an example without comments, try to figure it out on your own, reason where the external and where the internal function is, why the tasks are solved this way?
Example 5
a) Find the derivative of the function
b) Find the derivative of the function
Example 6
Find the derivative of a function 
Here we have a root, and in order to differentiate the root, it must be represented as a power. Thus, first we bring the function into the form appropriate for differentiation:
Analyzing the function, we come to the conclusion that the sum of the three terms is an internal function, and raising to a power is an external function. We apply the rule of differentiation of complex functions 
:
We again represent the degree as a radical (root), and for the derivative of the internal function we apply a simple rule for differentiating the sum:
Ready. You can also reduce the expression to a common denominator in brackets and write everything down as one fraction. It’s beautiful, of course, but when you get cumbersome long derivatives, it’s better not to do this (it’s easy to get confused, make an unnecessary mistake, and it will be inconvenient for the teacher to check).
Example 7
Find the derivative of a function
This is an example for you to solve on your own (answer at the end of the lesson).
It is interesting to note that sometimes instead of the rule for differentiating a complex function, you can use the rule for differentiating a quotient 
, but such a solution will look like an unusual perversion. Here typical example:
Example 8
Find the derivative of a function
Here you can use the rule of differentiation of the quotient 
, but it is much more profitable to find the derivative through the rule of differentiation of a complex function:
We prepare the function for differentiation - we move the minus out of the derivative sign, and raise the cosine into the numerator:
Cosine is an internal function, exponentiation is an external function.
Let's use our rule 
:
We find the derivative of the internal function and reset the cosine back down:
Ready. In the example considered, it is important not to get confused in the signs. By the way, try to solve it using the rule 
, the answers must match.
Example 9
Find the derivative of a function
This is an example for you to solve on your own (answer at the end of the lesson).
So far we have looked at cases where we had only one nesting in a complex function. In practical tasks, you can often find derivatives, where, like nesting dolls, one inside the other, 3 or even 4-5 functions are nested at once.
Example 10
Find the derivative of a function
Let's understand the attachments of this function. Let's try to calculate the expression using the experimental value. How would we count on a calculator?
First you need to find , which means the arcsine is the deepest embedding:
This arcsine of one should then be squared:
And finally, we raise seven to a power: 
That is, in this example we have three different functions and two embeddings, while the innermost function is the arcsine, and the outermost function is the exponential function.
Let's start deciding
According to the rule 
First you need to take the derivative of the outer function. We look at the table of derivatives and find the derivative of the exponential function: The only difference is that instead of “x” we have complex expression, which does not negate the validity of this formula. So, the result of applying the rule for differentiating a complex function 
next.
The problem of finding the derivative of a given function is one of the main ones in high school mathematics courses and in higher education. educational institutions. It is impossible to fully explore a function and construct its graph without taking its derivative. The derivative of a function can be easily found if you know the basic rules of differentiation, as well as the table of derivatives of basic functions. Let's figure out how to find the derivative of a function.
The derivative of a function is the limit of the ratio of the increment of the function to the increment of the argument when the increment of the argument tends to zero.
Understanding this definition is quite difficult, since the concept of a limit is not fully studied in school. But in order to find derivatives of various functions, it is not necessary to understand the definition; let’s leave it to mathematicians and move straight to finding the derivative.
The process of finding the derivative is called differentiation. When we differentiate a function, we will obtain a new function.
To denote them we will use latin letters f, g, etc.
There are many different notations for derivatives. We will use a stroke. For example, writing g" means that we will find the derivative of the function g.
Derivatives table
In order to answer the question of how to find the derivative, it is necessary to provide a table of derivatives of the main functions. To calculate the derivatives of elementary functions, it is not necessary to perform complex calculations. It is enough just to look at its value in the table of derivatives.
- (sin x)"=cos x
- (cos x)"= –sin x
- (x n)"=n x n-1
- (e x)"=e x
- (ln x)"=1/x
- (a x)"=a x ln a
- (log a x)"=1/x ln a
- (tg x)"=1/cos 2 x
- (ctg x)"= – 1/sin 2 x
- (arcsin x)"= 1/√(1-x 2)
- (arccos x)"= - 1/√(1-x 2)
- (arctg x)"= 1/(1+x 2)
- (arcctg x)"= - 1/(1+x 2)
Example 1. Find the derivative of the function y=500.
We see that this is a constant. From the table of derivatives it is known that the derivative of a constant is equal to zero (formula 1).
Example 2. Find the derivative of the function y=x 100.
This is a power function whose exponent is 100, and to find its derivative you need to multiply the function by the exponent and reduce it by 1 (formula 3).
(x 100)"=100 x 99
Example 3. Find the derivative of the function y=5 x
This is an exponential function, let's calculate its derivative using formula 4.
Example 4. Find the derivative of the function y= log 4 x
We find the derivative of the logarithm using formula 7.
(log 4 x)"=1/x ln 4
Rules of differentiation
Let's now figure out how to find the derivative of a function if it is not in the table. Most of the functions studied are not elementary, but are combinations of elementary functions using simple operations (addition, subtraction, multiplication, division, and multiplication by a number). To find their derivatives, you need to know the rules of differentiation. Below, the letters f and g denote functions, and C is a constant.
1. The constant coefficient can be taken out of the sign of the derivative
Example 5. Find the derivative of the function y= 6*x 8
We take out a constant factor of 6 and differentiate only x 4. This is a power function, the derivative of which is found using formula 3 of the table of derivatives.
(6*x 8)" = 6*(x 8)"=6*8*x 7 =48* x 7
2. The derivative of a sum is equal to the sum of the derivatives
(f + g)"=f" + g"
Example 6. Find the derivative of the function y= x 100 +sin x
A function is the sum of two functions, the derivatives of which we can find from the table. Since (x 100)"=100 x 99 and (sin x)"=cos x. The derivative of the sum will be equal to the sum of these derivatives:
(x 100 +sin x)"= 100 x 99 +cos x
3. The derivative of the difference is equal to the difference of the derivatives
(f – g)"=f" – g"
Example 7. Find the derivative of the function y= x 100 – cos x
This function is the difference of two functions, the derivatives of which we can also find from the table. Then the derivative of the difference is equal to the difference of the derivatives and don’t forget to change the sign, since (cos x)"= – sin x.
(x 100 – cos x)"= 100 x 99 + sin x
Example 8. Find the derivative of the function y=e x +tg x– x 2.
This function has both a sum and a difference; let’s find the derivatives of each term:
(e x)"=e x, (tg x)"=1/cos 2 x, (x 2)"=2 x. Then the derivative original function is equal to:
(e x +tg x– x 2)"= e x +1/cos 2 x –2 x
4. Derivative of the product
(f * g)"=f" * g + f * g"
Example 9. Find the derivative of the function y= cos x *e x
To do this, we first find the derivative of each factor (cos x)"=–sin x and (e x)"=e x. Now let's substitute everything into the product formula. We multiply the derivative of the first function by the second and add the product of the first function by the derivative of the second.
(cos x* e x)"= e x cos x – e x *sin x
5. Derivative of the quotient
(f / g)"= f" * g – f * g"/ g 2
Example 10. Find the derivative of the function y= x 50 /sin x
To find the derivative of a quotient, we first find the derivative of the numerator and denominator separately: (x 50)"=50 x 49 and (sin x)"= cos x. Substituting the derivative of the quotient into the formula, we get:
(x 50 /sin x)"= 50x 49 *sin x – x 50 *cos x/sin 2 x
Derivative of a complex function
A complex function is a function represented by a composition of several functions. There is also a rule for finding the derivative of a complex function:
(u (v))"=u"(v)*v"
Let's figure out how to find the derivative of such a function. Let y= u(v(x)) be a complex function. Let's call the function u external, and v - internal.
For example:
y=sin (x 3) is a complex function.
Then y=sin(t) is an external function
t=x 3 - internal.
Let's try to calculate the derivative of this function. According to the formula, you need to multiply the derivatives of the internal and external functions.
(sin t)"=cos (t) - derivative of the external function (where t=x 3)
(x 3)"=3x 2 - derivative of the internal function
Then (sin (x 3))"= cos (x 3)* 3x 2 is the derivative of a complex function.
How to find the derivative, how to take the derivative? In this lesson we will learn how to find derivatives of functions. But before studying this page, I strongly recommend that you familiarize yourself with the methodological material Hot formulas for school mathematics course. The reference manual can be opened or downloaded on the page Mathematical formulas and tables. Also from there we will need Derivatives table, it is better to print it out; you will often have to refer to it, not only now, but also offline.
Eat? Let's get started. I have two news for you: good and very good. The good news is this: to learn how to find derivatives, you don’t have to know and understand what a derivative is. Moreover, the definition of the derivative of a function, mathematical, physical, geometric meaning It is more appropriate to digest the derivative later, since a high-quality elaboration of the theory, in my opinion, requires the study of a number of other topics, as well as some practical experience.
And now our task is to master these same derivatives technically. The very good news is that learning to take derivatives is not so difficult; there is a fairly clear algorithm for solving (and explaining) this task; integrals or limits, for example, are more difficult to master.
I recommend the following order of studying the topic:: First, this article. Then you need to read the most important lesson Derivative of a complex function. These two basic lessons will improve your skills from complete zero. Next you can get acquainted with more complex derivatives in the article Complex derivatives. Logarithmic derivative. If the bar is too high, read the thing first The simplest typical problems with derivatives. In addition to the new material, the lesson covers other, simpler types of derivatives, and is a great opportunity to improve your differentiation technique. In addition, test papers almost always contain tasks on finding derivatives of functions that are specified implicitly or parametrically. There is also such a lesson: Derivatives of implicit and parametrically defined functions.
I will try in an accessible form, step by step, to teach you how to find derivatives of functions. All information is presented in detail, in simple words.
Actually, let’s immediately look at an example:
Example 1
Find the derivative of a function
Solution: 
This simplest example, please find it in the table of derivatives of elementary functions. Now let's look at the solution and analyze what happened? And the following thing happened: we had a function, which, as a result of the solution, turned into a function.
To put it quite simply, in order to find the derivative of a function, you need certain rules turn it into another function. Look again at the table of derivatives - there functions turn into other functions. The only exception is the exponential function, which turns into itself. The operation of finding the derivative is called differentiation .
Designations: The derivative is denoted by or .
ATTENTION, IMPORTANT! Forgetting to put a stroke (where it is necessary), or to draw an extra stroke (where it is not necessary) - BIG MISTAKE! A function and its derivative are two different functions!
Let's return to our table of derivatives. From this table it is desirable memorize: rules of differentiation and derivatives of some elementary functions, especially:
derivative of the constant:
, where is a constant number;
derivative of a power function:
, in particular: , , .
Why remember? This knowledge is basic knowledge about derivatives. And if you cannot answer the teacher’s question “What is the derivative of a number?”, then your studies at the university may end for you (I personally know two real cases from life). In addition, these are the most common formulas that we have to use almost every time we come across derivatives.
In reality, simple tabular examples are rare; usually, when finding derivatives, differentiation rules are first used, and then a table of derivatives of elementary functions.
In this regard, we move on to consider differentiation rules:
1) A constant number can (and should) be taken out of the derivative sign
Where is a constant number (constant)
Example 2
Find the derivative of a function
Let's look at the table of derivatives. The derivative of the cosine is there, but we have .
It's time to use the rule, we take the constant factor out of the sign of the derivative:
Now we convert our cosine according to the table:
Well, it’s advisable to “comb” the result a little - put the minus sign in first place, at the same time getting rid of the brackets:
2) The derivative of the sum is equal to the sum of the derivatives
Example 3
Find the derivative of a function
Let's decide. As you probably already noticed, the first step that is always performed when finding a derivative is that we enclose the entire expression in parentheses and put a prime at the top right:
Let's apply the second rule:
Please note that for differentiation, all roots and degrees must be represented in the form, and if they are in the denominator, then move them up. How to do this is discussed in my teaching materials.
Now let’s remember the first rule of differentiation - we take the constant factors (numbers) outside the sign of the derivative:
Usually, during the solution, these two rules are applied simultaneously (so as not to rewrite a long expression again).
All functions located under the strokes are elementary table functions; using the table we carry out the transformation:
You can leave everything as is, since there are no more strokes, and the derivative has been found. However, expressions like this usually simplify:
It is advisable to represent all powers of the type again in the form of roots; powers with negative exponents should be reset to the denominator. Although you don't have to do this, it won't be a mistake.
Example 4
Find the derivative of a function 
Try to solve this example yourself (answer at the end of the lesson). Those interested can also use intensive course in pdf format, which is especially relevant if you have very little time at your disposal.
3) Derivative of the product of functions
It seems that the analogy suggests the formula ...., but the surprise is that:
This is an unusual rule (as, in fact, others) follows from derivative definitions. But we’ll hold off on the theory for now - now it’s more important to learn how to solve:
Example 5
Find the derivative of a function
Here we have the product of two functions depending on .
First we apply our strange rule, and then we transform the functions using the derivative table:
Difficult? Not at all, quite accessible even for a teapot.
Example 6
Find the derivative of a function 
This function contains the sum and product of two functions - quadratic trinomial and logarithm. From school we remember that multiplication and division take precedence over addition and subtraction.
It's the same here. AT FIRST we use the product differentiation rule:
Now for the bracket we use the first two rules:
As a result of applying the rules of differentiation under the strokes, we are left with only elementary functions; using the table of derivatives, we turn them into other functions:
Ready.
At certain experience finding derivatives; simple derivatives do not need to be described in such detail. In general, they are usually decided orally, and it is immediately written down that 
.
Example 7
Find the derivative of a function 
This is an example for you to solve on your own (answer at the end of the lesson)
4) Derivative of quotient functions
A hatch opened in the ceiling, don't be alarmed, it's a glitch.
But this is the harsh reality: 
Example 8
Find the derivative of a function
What’s missing here – sum, difference, product, fraction…. Where to start?! There are doubts, there are no doubts, but, ANYWAY First, draw brackets and put a stroke at the top right:
Now we look at the expression in brackets, how can we simplify it? In this case, we notice a factor, which, according to the first rule, it is advisable to place outside the sign of the derivative.
After preliminary artillery preparation, examples with 3-4-5 nestings of functions will be less scary. Perhaps the following two examples will seem complicated to some, but if you understand them (someone will suffer), then almost everything else in differential calculus It 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 useful trick: we take the experimental meaning of “x”, for example, and try (mentally or in a draft) to substitute this meaning 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:
It seems without errors:
1) Take the derivative of the square root.
2) 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.
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, let’s see if it’s possible to turn the product of three functions into the product of two functions? For example, if we had two polynomials in the product, 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's reduce the expression of the numerator to a common denominator and get rid of the three-story structure of the fraction:
The disadvantage of additional simplifications is that there is a risk of making a mistake not when finding the derivative, but during banal school transformations. On the other hand, teachers often reject the assignment and ask to “bring it to mind” the derivative.
A simpler example to solve on your own:
Example 7
Find the derivative of a function
We continue to master the methods of finding the derivative, and now we will consider a typical case when a “terrible” logarithm is proposed for differentiation
