Probably, the concept of derivative is familiar to each of us since school. Usually students have difficulty understanding this, undoubtedly very important thing. It is actively used in various areas of people's lives, and many engineering developments were based precisely on mathematical calculations obtained using the derivative. But before we move on to an analysis of what derivatives of numbers are, how to calculate them and where they are useful to us, let’s dive a little into history.
Being the basis of mathematical analysis, it was discovered (it would be better to say “invented”, because it did not exist in nature as such) by Isaac Newton, whom we all know from the discovery of the law of universal gravitation. It was he who first used this concept in physics to connect the nature of speed and acceleration of bodies. And many scientists still praise Newton for this magnificent invention, because in fact he invented the basis of differential and integral calculus, in fact the basis of an entire field of mathematics called “mathematical analysis.” If there had been a Nobel Prize at that time, Newton would most likely have received it several times.
There were also other great minds. In addition to Newton, such eminent geniuses of mathematics as Leonard Euler, Louis Lagrange and Gottfried Leibniz worked on the development of the derivative and integral. It was thanks to them that we received the theory in the form in which it exists to this day. By the way, it was Leibniz who discovered the geometric meaning of the derivative, which turned out to be nothing more than the tangent of the angle of inclination of the tangent to the graph of the function.
What are derivatives of numbers? Let's repeat a little what we went through at school.
This concept can be defined in several different ways. The simplest explanation: the derivative is the rate of change of a function. Let's imagine a graph of some function y versus x. If it is not a straight line, then it has some bends in the graph, periods of increasing and decreasing. If we take any infinitesimal interval of this graph, it will represent a straight line segment. So, the ratio of the size of this infinitesimal segment along the y coordinate to the size along the x coordinate will be the derivative of this function at a given point. If we consider the function as a whole, and not at a specific point, then we get a derivative function, that is, a certain dependence of y on x.
In addition, in addition to the rate of change of a function, there is also a geometric meaning. We'll talk about it now.
Derivatives of numbers themselves represent a certain number, which without proper understanding does not carry any meaning. It turns out that the derivative not only shows the rate at which the function grows or decreases, but also the tangent of the tangent to the graph of the function at a given point. Not a very clear definition. Let's look at it in more detail. Let's say we have a graph of some function (for fun, let's take a curve). There are an infinite number of points on it, but there are areas where only one single point has a maximum or minimum. Through any such point you can draw a straight line that would be perpendicular to the graph of the function at this point. Such a line will be called a tangent. Let's say we draw it to the intersection with the OX axis. So, the angle obtained between the tangent and the OX axis will be determined by the derivative. More precisely, the tangent of this angle will be equal to it.
Let's talk a little about special cases and analyze the derivatives of numbers.
As we have already said, derivatives of numbers are the values of the derivative at a specific point. For example, let's take the function y=x 2 . The derivative of x is a number, and in the general case it is a function equal to 2*x. If we need to calculate the derivative, say, at the point x 0 = 1, then we get y"(1)=2*1=2. Everything is very simple. An interesting case is the derivative. We do not go into a detailed explanation of what a complex number is Let's just say that this is a number that contains the so-called imaginary unit - a number whose square is equal to -1. Calculation of such a derivative is possible only if the following conditions are met:
1) There must be first-order partial derivatives of the real and imaginary parts with respect to y and x.
2) The Cauchy-Riemann conditions associated with the equality of partial derivatives described in the first paragraph are satisfied.
Another interesting case, although not as difficult as the previous one, is the derivative of a negative number. In fact, any negative number can be represented as a positive number multiplied by -1. Well, the derivative of a constant and a function is equal to the constant multiplied by the derivative of the function.
It will be interesting to learn about the role of derivatives in everyday life, and this is what we will discuss now.
Probably, each of us at least once in our lives catches ourselves thinking that mathematics is unlikely to be useful to us. And such a complex thing as a derivative probably has no application at all. In fact, mathematics - and all its fruits - are developed mainly by physics, chemistry, astronomy and even economics. The derivative was the beginning that gave us the ability to draw conclusions from graphs of functions, and we learned to interpret the laws of nature and turn them to our advantage thanks to it.
Of course, not everyone may need a derivative in real life. But mathematics develops logic, which will certainly be needed. It’s not for nothing that mathematics is called the queen of sciences: it forms the basis for understanding other areas of knowledge.
Derivative calculation- one of the most important operations in differential calculus. Below is a table for finding derivatives of simple functions. For more complex differentiation rules, see other lessons: Explanation:
The derivative shows the rate at which the value of a function changes when its argument changes. Since the number does not change in any way under any conditions, the rate of its change is always zero.
2. Derivative of a variable equal to one
x´ = 1
Explanation:
With each increment of the argument (x) by one, the value of the function (the result of the calculation) increases by the same amount. Thus, the rate of change in the value of the function y = x is exactly equal to the rate of change in the value of the argument.
3. The derivative of a variable and a factor is equal to this factor
сx´ = с
Example:
(3x)´ = 3
(2x)´ = 2
Explanation:
In this case, every time the function argument changes ( X) its value (y) increases in With once. Thus, the rate of change of the function value in relation to the rate of change of the argument is exactly equal to the value With.
Whence it follows that
(cx + b)" = c
that is, the differential of the linear function y=kx+b is equal to the slope of the line (k).
5. Derivative of a variable to a power equal to the product of a number of this power and a variable to the power reduced by one
(x c)"= cx c-1, provided that x c and cx c-1 are defined and c ≠ 0
Example:
(x 2)" = 2x
(x 3)" = 3x 2
To remember the formula:
Move the degree of the variable down as a factor, and then reduce the degree itself by one. For example, for x 2 - the two was ahead of the x, and then the reduced power (2-1 = 1) simply gave us 2x. The same thing happened for x 3 - we “move down” the triple, reduce it by one and instead of a cube we have a square, that is, 3x 2. A little "unscientific" but very easy to remember.
6.Derivative of a fraction 1/x
(1/x)" = - 1 / x 2
Example:
Since a fraction can be represented as raising to a negative power
(1/x)" = (x -1)", then you can apply the formula from rule 5 of the table of derivatives
(x -1)" = -1x -2 = - 1 / x 2
7. Derivative of a fraction with a variable of arbitrary degree in the denominator
(1 / x c)" = - c / x c+1
Example:
(1 / x 2)" = - 2 / x 3
8. Derivative of the root(derivative of variable under square root)
(√x)" = 1 / (2√x) or 1/2 x -1/2
Example:
(√x)" = (x 1/2)" means you can apply the formula from rule 5
(x 1/2)" = 1/2 x -1/2 = 1 / (2√x)
9. Derivative of a variable under the root of an arbitrary degree
(n √x)" = 1 / (n n √x n-1)
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 calculate the value of the expression on a calculator (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 you to solve on your own (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 is a 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 a complex expression, which does not negate the validity of this formula. So, the result of applying the rule for differentiating a complex function 
next.
Very easy to remember.
Well, let’s not go far, let’s immediately consider the inverse function. Which function is the inverse of the exponential function? Logarithm:
In our case, the base is the number:
Such a logarithm (that is, a logarithm with a base) is called “natural”, and we use a special notation for it: we write instead.
What is it equal to? Of course.
The derivative of the natural logarithm is also very simple:
Examples:
Answers: The exponential and natural logarithm are uniquely simple functions from a derivative perspective. Exponential and logarithmic functions with any other base will have a different derivative, which we will analyze later, after we go through the rules of differentiation.
Rules of what? Again a new term, again?!...
Differentiation is the process of finding the derivative.
That's all. What else can you call this process in one word? Not derivative... Mathematicians call the differential the same increment of a function at. This term comes from the Latin differentia - difference. Here.
When deriving all these rules, we will use two functions, for example, and. We will also need formulas for their increments:
There are 5 rules in total.
If - some constant number (constant), then.
Obviously, this rule also works for the difference: .
Let's prove it. Let it be, or simpler.
Examples.
Find the derivatives of the functions:
Solutions:
Everything is similar here: let’s introduce a new function and find its increment:
Derivative:
Examples:
Solutions:
Now your knowledge is enough to learn how to find the derivative of any exponential function, and not just exponents (have you forgotten what that is yet?).
So, where is some number.
We already know the derivative of the function, so let's try to reduce our function to a new base:
To do this, we will use a simple rule: . Then:
Well, it worked. Now try to find the derivative, and don't forget that this function is complex.
Did it work?
Here, check yourself:
The formula turned out to be very similar to the derivative of an exponent: as it was, it remains the same, only a factor appeared, which is just a number, but not a variable.
Examples:
Find the derivatives of the functions:
Answers:
This is just a number that cannot be calculated without a calculator, that is, it cannot be written down in a simpler form. Therefore, we leave it in this form in the answer.
Note that here is a quotient of two functions, so we apply the corresponding differentiation rule:
In this example, the product of two functions:
It’s similar here: you already know the derivative of the natural logarithm:
Therefore, to find an arbitrary logarithm with a different base, for example:
We need to reduce this logarithm to the base. How do you change the base of a logarithm? I hope you remember this formula:
Only now we will write instead:
The denominator is simply a constant (a constant number, without a variable). The derivative is obtained very simply:
Derivatives of exponential and logarithmic functions are almost never found in the Unified State Examination, but it will not be superfluous to know them.
What is a "complex function"? No, this is not a logarithm, and not an arctangent. These functions can be difficult to understand (although if you find the logarithm difficult, read the topic “Logarithms” and you will be fine), but from a mathematical point of view, the word “complex” does not mean “difficult”.
Imagine a small conveyor belt: two people are sitting and doing some actions with some objects. For example, the first one wraps a chocolate bar in a wrapper, and the second one ties it with a ribbon. The result is a composite object: a chocolate bar wrapped and tied with a ribbon. To eat a chocolate bar, you need to do the reverse steps in reverse order.
Let's create a similar mathematical pipeline: first we will find the cosine of a number, and then square the resulting number. So, we are given a number (chocolate), I find its cosine (wrapper), and then you square what I got (tie it with a ribbon). What happened? Function. This is an example of a complex function: when, to find its value, we perform the first action directly with the variable, and then a second action with what resulted from the first.
In other words, a complex function is a function whose argument is another function: .
For our example, .
We can easily do the same steps in reverse order: first you square it, and I then look for the cosine of the resulting number: . It’s easy to guess that the result will almost always be different. An important feature of complex functions: when the order of actions changes, the function changes.
Second example: (same thing). .
The action we do last will be called "external" function, and the action performed first - accordingly "internal" function(these are informal names, I use them only to explain the material in simple language).
Try to determine for yourself which function is external and which internal:
Answers: Separating inner and outer functions is very similar to changing variables: for example, in a function
We change variables and get a function.
Well, now we will extract our chocolate bar - look for the derivative. The procedure is always reversed: first we look for the derivative of the outer function, then we multiply the result by the derivative of the inner function. In relation to the original example, it looks like this:
Another example:
So, let's finally formulate the official rule:
Algorithm for finding the derivative of a complex function:
It seems simple, right?
Let's check with examples:
Solutions:
1) Internal: ;
External: ;
2) Internal: ;
(just don’t try to cut it by now! Nothing comes out from under the cosine, remember?)
3) Internal: ;
External: ;
It is immediately clear that this is a three-level complex function: after all, this is already a complex function in itself, and we also extract the root from it, that is, we perform the third action (put the chocolate in a wrapper and with a ribbon in the briefcase). But there is no reason to be afraid: we will still “unpack” this function in the same order as usual: from the end.
That is, first we differentiate the root, then the cosine, and only then the expression in brackets. And then we multiply it all.
In such cases, it is convenient to number the actions. That is, let's imagine what we know. In what order will we perform actions to calculate the value of this expression? Let's look at an example:
The later the action is performed, the more “external” the corresponding function will be. The sequence of actions is the same as before:
Here the nesting is generally 4-level. Let's determine the course of action.
1. Radical expression. .
2. Root. .
3. Sine. .
4. Square. .
5. Putting it all together:
Derivative of a function- the ratio of the increment of the function to the increment of the argument for an infinitesimal increment of the argument:
Basic derivatives:
Rules of differentiation:
The constant is taken out of the derivative sign:
Derivative of the sum:
Derivative of the product:
Derivative of the quotient:
Derivative of a complex function:
Algorithm for finding the derivative of a complex function:
