What is equal to the derived function. What is derivative
Solve physical tasks Or the examples of mathematics are completely impossible without knowledge about the derivative and methods of its calculation. The derivative is one of the most important concepts of mathematical analysis. We decided to devote this fundamental topic to the current article. What is a derivative, what is its physical and geometric meaning, how to calculate the derived function? All these questions can be combined into one: how to understand the derivative?
Geometric and physical meaning derivative
Let there be a function f (X) set in a certain interval (A, B) . Points x and x0 belong to this interval. When changing X changes the function itself. Change of argument - the difference of its values x-x0. . This difference is written as delta X. and is called the increment of the argument. Changing or incrementing the function is the difference of function values \u200b\u200bat two points. Definition of the derivative:
The derivative function at the point is the limit of the function of the function 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 is the point in finding such a limit? But what:
The derivative of the function at the point is equal to the angle tangent between the OX axis and tangent to the graph of the function at this point.
Physical meaning derivative: The time derivative is equal to the rate of straight movement.
Indeed, since school times everyone knows that speed is a private path x \u003d f (t) and time t. . average speed For some time:
To find out the speed of movement at time t0. You need to calculate the limit:
Rule first: we take a constant
The constant can be taken out of the sign of the derivative. Moreover, it needs to be done. When solving examples in mathematics, take a rule - if you can simplify the expression, be sure to simplify .
Example. Calculate the derivative:
Rule Second: Derivative Functions
The derivative of the 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 lead the proof of this theorem, and it is better to consider a practical example.
Find a derivative function:
Rule Third: derived works of functions
The derivative of the work of two differentiable functions is calculated by the formula:
Example: Find a derivative function:
Decision: 
It is important to say about the calculation of derivatives of complex functions. The derivative of the complex function is equal to the product of the derivative of this function by an intermediate argument on the derivative of the intermediate argument on an independent variable.
In the above example, we encounter an expression:
In this case, the intermediate argument is 8x to the fifth degree. In order to calculate the derivative of such an expression, we first consider the derivative of the external function by an intermediate argument, and then multiply on the derivative directly the very intermediate argument on an independent variable.
Rule fourth: Derivative of private two functions
Formula for determining the derivative of private two functions:
We tried to talk about derivatives for teapots from scratch. This topic is not so simple, as it seems, so I warn it: in the examples there are often traps, so be careful when calculating derivatives.
With any question on this and other topics you can contact the student service. In a short time, we will help solve the most difficult control and deal with the tasks, even if you have never been involved in the calculation of derivatives.
When the first formula itself is derived, we will proceed from the definition of derivatives in the point. Take where x. - any valid number, that is, x. - Any number from the function of determining the function. We write the limit of the relationship of the increment function to the increment of the argument at: 
It should be noted that under the sign of the limit it turns out an expression that is not a single zero to divide on zero, since in the numerator there is not an infinitely small value, namely zero. In other words, the increment of a constant function is always zero.
In this way, derivative permanent functionequal to zero throughout the field of definition.
The derivative of the power function.
The formula derivative power function Has appearance 
where the indicator of the degree p. - Any valid number.
We first prove the formula for the natural indicator, that is, for p \u003d 1, 2, 3, ...
We will use the definition of the derivative. We write the limit of the ratio of the increment of the power function to the increment of the argument: 
To simplify the expression in the numerator, we turn to the Newton Binoma formula: 
Hence, 
This proved the formula for the derivative of the power function for the natural indicator.
Derivative indicative function.
The derivative of the derivative formula is based on the definition:
Came to uncertainty. For its disclosure, we introduce a new variable, and with. Then. In the last transition, we used the transition formula to the new base of the logarithm.
Perform a substitution to the initial limit: 
If we recall the second wonderful limit, then we will come to the formula of the derivative of the indicative function: 
Derivative logarithmic function.
We prove the formula of the derivative logarithmic function for all x. from the definition area and all allowable base values a. Logarithm. By definition, we have: 
As you noticed, when proofing the transformation was carried out using the properties of the logarithm. Equality 
fairly due to the second remarkable limit.
Derived trigonometric functions.
To display the formulas of derivative trigonometric functions, we will have to recall some formulas trigonometry, as well as the first wonderful limit.
By definition of the derivative for the sinus function we have 
.
We use the sinus difference formula: 
It remains to contact the first wonderful limit: 
Thus, derivative function sIN X. there is cOS X..
Absolutely similarly proved the formula of the cosine derivative. 
Consequently, derived function cOS X. there is -Sin X..
The output of the formulas of the tables of derivatives for Tangent and Kotangens will carry out using proven differentiation rules (derivative of the fraction). 
Derivatives of hyperbolic functions.
The rules of differentiation and the formula of the derivative indicative function from the derivative table allow us to derive the formula of derivatives of hyperbolic sinus, cosine, tangent and catangent. 
Derived reverse function.
In order not to be confused when exposing, let's refer to the lower index the argument of the function to which differentiation is performed, that is, it is derived f (X) by x..
Now formulate the rule of finding a derivative of the feedback.
Let functions y \u003d f (x) and x \u003d g (y) Mutually reverse, determined at intervals and, accordingly. If at the point there is a finite different derivative function from zero f (X), then at the point there is a finite derivative of the feedback g (Y), and 
. In another record 
.
You can reformulate this rule for any x. From the gap, then we get 
.
Let's check the validity of these formulas.
Find a reverse function for a natural logarithm 
(here y. - function, and x.- argument). Resolving this equation relative x., get here (here x. - function, and y. - its argument). I.e, 
and mutually reverse functions.
From the table derivatives we see that 
and 
.
Correct that formulas for finding derivative feedback leads us to the same results: 
Definition. Let the function \\ (y \u003d f (x) \\) define in a certain interval containing within itself the point \\ (x_0 \\). We give the argument the increment \\ (\\ Delta X \\) is so as not to get out of this interval. Find the appropriate increment of the function \\ (\\ Delta Y \\) (when moving from point \\ (x_0 \\) to the point \\ (x_0 + \\ deelta x \\)) and amounted to the ratio \\ (\\ FRAC (\\ Delta Y) (\\ Delta X) \\). If there is a limit of this relationship with \\ (\\ delta x \\ rightarrow 0 \\), then the specified limit is called derived function \\ (y \u003d f (x) \\) at point \\ (x_0 \\) and denote \\ (F "(x_0) \\).
$$ \\ LIM _ (\\ Delta X \\ To 0) \\ FRAC (\\ Delta Y) (\\ Delta X) \u003d F "(x_0) $$
To designate the derivative, the Y symbol often use. Note that Y "\u003d F (x) is a new function, but naturally associated with the function y \u003d f (x) defined in all points x in which the above limit exists . This feature is called this: derivative function y \u003d f (x).
Geometric meaning of the derivative It consists next. If the function of the function y \u003d F (x) at the abscissa point x \u003d a can be carried out by a tangent, non-parallel axis y, then f (a) expresses the angular coefficient of tangent:
\\ (k \u003d f "(a) \\)
Since \\ (k \u003d tg (a) \\), then the equality \\ (F "(a) \u003d Tg (A) \\) is true.
And now we interpret the definition of the derivative from the point of view of approximate equalities. Let the function \\ (y \u003d f (x) \\) has a derivative at a specific point \\ (x \\):
$$ \\ LIM _ (\\ Delta X \\ To 0) \\ FRAC (\\ Delta Y) (\\ Delta X) \u003d F "(x) $$
This means that approximate equality \\ (\\ FRAC (\\ Delta Y) (\\ Delta X) \\ APPROX F "(X) \\), i.e. \\ (\\ Delta Y \\ Approx F" (x) \\ Cdot \\ Delta X \\). The meaningful meaning of the obtained approximate equality is as follows: the increment of the function "almost proportionately" the increment of the argument, and the proportionality coefficient is the value of the derivative in set point x. For example, for the function \\ (y \u003d x ^ 2 \\), the approximate equality \\ (\\ Delta Y \\ Approx 2x \\ Cdot \\ Delta X \\) is true. If you carefully analyze the definition of the derivative, then we will find that it is put on it algorithm.
Word it.
How to find the derivative function y \u003d f (x)?
1. Fix the value \\ (x \\), to find \\ (f (x) \\)
2. Give the argument \\ (x \\) increment \\ (\\ Delta X \\), go to new point \\ (X + \\ Delta X \\), Find \\ (F (x + \\ Delta X) \\)
3. Find the increment of the function: \\ (\\ Delta Y \u003d F (X + \\ Delta X) - F (x) \\)
4. Make a relation \\ (\\ FRAC (\\ Delta Y) (\\ Delta X) \\)
5. Calculate $$ \\ LIM _ (\\ Delta X \\ To 0) \\ FRAC (\\ Delta Y) (\\ Delta X) $$
This limit is derived from the point x.
If the function y \u003d f (x) has a derivative at point x, it is called differentiable at point x. The procedure for finding the derivative function y \u003d f (x) is called differentiation Functions y \u003d f (x).
Let us discuss such a question: how are the continuity of the continuity and differentiability of the function at the point.
Let the function y \u003d f (x) differentiate at the point x. Then to the graph of the function at the point M (x; f (x)), it is possible to carry out a tangent, and, we recall, the angular coefficient of tangent is f "(x). Such a chart cannot" break "at the point M, i.e. the function is obliged be continuous at point x.
These were reasoning "on the fingers." We give a more stringent reasoning. If the function y \u003d f (x) is differentiable at the point x, then approximate equality is performed \\ (\\ Delta Y \\ Approx F "(x) \\ Cdot \\ Delta X \\). If in this equality \\ (\\ Delta X \\) rushed to zero, then \\ (\\ delta y \\) will strive for zero, and this is the condition of the continuity of the function at the point.
So, if the function is differentiable at the point x, it is continuous at this point.
The opposite statement is incorrect. For example: function y \u003d | x | Continuous everywhere, in particular at point x \u003d 0, but tangent to the graphics of the function in the "point of the joint" (0; 0) does not exist. If at some point to the graphics of the function can not be tanged, then at this point there is no derivative.
One more example. The function \\ (y \u003d \\ sqrt (x) \\) is continuous on the entire numeric line, including at the point x \u003d 0. And the function to the graphic function exists at any point, including at the point x \u003d 0. But at this point The tangent coincides with the axis y, i.e. perpendicular to the abscissa axis, its equation has the form x \u003d 0. Corner coefficient There is no other direct, it means that there is no and \\ (F "(0) \\)
So, we got acquainted with the new feature of the function - differentiability. And how can the function of the function be concluded about its differentiability?
The answer is actually obtained above. If at some point to the graph of the function you can spend a tangential, non-perpendicular abscissa axis, then at this point the function is differentiable. If at some point tangent to the graphics function does not exist or it is perpendicular to the abscissa axis, then at this point the function is not differentiated.
Differentiation rules
Operation finding a derivative called differentiation. When performing this operation, it often has to work with private, sums, works of functions, as well as with "functions functions", that is, complex functions. Based on the definition of the derivative, you can withdraw the differentiation rules that facilitate this work. If C is a constant number and f \u003d f (x), G \u003d G (x) - some differentiable functions, then the following are valid differentiation rules:
$$ F "_X (G (x)) \u003d f" _g \\ cdot g "_x $$
Table of derivatives of some functions
$$ \\ left (\\ FRAC (1) (X) \\ RIGHT) "\u003d - \\ FRAC (1) (x ^ 2) $$$$ (\\ SQRT (X))" \u003d \\ FRAC (1) (2 \\ $$$$ \\ left (E ^ X \\ RIGHT) "\u003d E ^ x $$$$ (\\ ln x)" \u003d \\ FRAC (1) (X) $$$$ (\\ log_a x) "\u003d \\ FRAC (1) (x \\ ln a) $$$$ (\\ sin x) "\u003d \\ cos x $$$$ (\\ cos x)" \u003d - \\ sin x $$$$ (\\ Text (TG) X) "\u003d \\ FRAC (1) (\\ cos ^ 2 x) $$$$ (\\ Text (CTG) X)" \u003d - \\ FRAC (1) (\\ sin ^ 2 x) $$$$ (\\ arcsin x) "\u003d \\ FRAC (1) (\\ sqrt (1-x ^ 2)) $$$$ (\\ arccos x)" \u003d \\ FRAC (-1) (\\ SQRT (1-x ^ 2)) $$$$ (\\ Text (arctg) x) "\u003d \\ FRAC (1) (1 + x ^ 2) $$$$ (\\ Text (ArcCTG) X)" \u003d \\ FRAC (-1) (1 + x ^ 2) $ $Proof and output of the formulas derivative derivative (E to degree x) and an indicative function (A to degree x). Examples of calculating derivatives from E ^ 2x, E ^ 3x and E ^ nx. Formulas derivatives of higher orders.
The derivative of the exhibit is equal to the exhibitor itself (the derivative E to the degree x is equal to e to the degree x):
(1)
(E X) '\u003d E X.
The derivative of the indicative function with the base of the degree A is equal to the function itself multiplied by natural logarithm From a:
(2)
.
Output of the formula of the derivative of the exponent, E to the degree x
The exhibitor is an indicative function in which the degree base is equal to the number e, which is the following limit:
.
Here can be both natural and actual number. Next, we derive the formula (1) of the derivative of the exhibit.
Output of the formula derivative exhibit
Consider the exhibitor, E to the degree x:
y \u003d E X.
This feature is defined for all. Find its derivative in the variable x. By definition, the derivative is the following limit:
(3)
.
We transform this expression to reduce it to well-known mathematical properties and rules. For this, we will need the following facts:
BUT) Property Exhibitors:
(4)
;
B) Logarithm property:
(5)
;
IN) Continuity of logarithm and property of limits for continuous function:
(6)
.
Here is some function that has a limit and this limit is positive.
D) The value of the second remarkable limit:
(7)
.
We use these facts to our limit (3). We use property (4):
;
.
Make a substitution. Then; .
Due to continuity of exhibitors,
.
Therefore, when,. As a result, we get:
.
Make a substitution. Then. With ,. And we have:
.
Applicate Logarithm property (5):
. Then
.
Apply Property (6). Since there is a positive limit and logarithm continuously, then:
.
Here we also took advantage of the second wonderful limit (7). Then
.
Thus, we obtained formula (1) of the derivative of the exhibit.
Output of the formula of the derivative of the indicative function
Now we will derive the formula (2) of the derivative of the indicative function with the basis of degree a. We believe that. Then the indicative function
(8)
Defined for all.
We transform formula (8). To do this, we use properties of an indicative function and logarithm.
;
.
So, we transformed formula (8) to the following form:
.
Derivatives of higher orders from E to degree x
Now we find derivatives of higher orders. First consider the exhibitor:
(14)
.
(1)
.
We see that the derivative of the function (14) is equal to the function itself (14). Differentiating (1), we obtain derivatives of the second and third order:
;
.
It can be seen that the derivative of the N-th order is also equal to the source function:
.
Derivatives of higher orders of indicative function
Now consider an indicative function with the basis of degree A:
.
We found her first order derivative:
(15)
.
Differentiating (15), we obtain derivatives of the second and third order:
;
.
We see that each differentiation leads to multiplication of the original function on. Therefore, the derivative of the N-th order has the following form:
.
Date: 11/20/2014
What is a derivative?
Table derivatives.
Derivative - one of the main concepts higher Mathematics. In this lesson, we will get acquainted with this concept. It is to get acquainted, without strict mathematical formulations and evidence.
This acquaintance will allow:
Understand the essence of simple tasks with the derivative;
Successfully solve these very simple tasks;
Prepare for more serious derivative lessons.
First, a pleasant surprise.)
Strict determination of the derivative is based on the theory of limits and the thing is quite complicated. It grieves. But the practical application of the derivative, as a rule, does not require such extensive and deep knowledge!
For the successful implementation of most tasks at school and university, it is enough to know just a few terms - To understand the task, and just a few rules - To solve it. And that's it. This makes me happy.
Let's start acquaintance?)
Terms and designations.
In elementary mathematics, many mathematical operations. Addition, subtraction of multiplication, exercise into a degree, logarithm, etc. If you add more to these operations, elementary mathematics becomes the highest. This new operation is called differentiation. The definition and meaning of this operation will be discussed in separate lessons.
It is important here it is important to understand that differentiation is just a mathematical operation of the function. We take any function and, according to a certain rules, we convert it. The result will be a new feature. This is this new feature and is called: derivative.
Differentiation - action on function.
Derivative - The result of this action.
Just like, for example, sum - Result of addition. Or private - The result of the division.
Knowing the terms, you can, at a minimum, understand the tasks.) The wording is such: find a derived function; take a derivative; Differentiate function; Calculate the derivative etc. It's all same. Of course, there are more complex tasks where the finding of the derivative (differentiation) will be only one of the steps to solve the task.
The derivative is denoted by the stroke at the top of the right above the function. Like this: y " or f "(x) or S "(t) etc.
Reading cheregor Barcode, Ef Barcter from X, Es Strike from TE, Well, you understood ...)
The barcode may also denote a derivative of a particular function, for example: (2x + 3) ", (X. 3 )" , (SINX) " etc. Often the derivative is denoted by differentials, but we will not consider such a designation in this lesson.
Suppose we learned to understand the tasks. There is nothing left to learn to decide.) Let's remind once again: finding a derivative - this convert function according to a specific rules. These rules, surprisingly, quite a bit.
To find a derivative function, you need to know only three things. Three whales on which all differentiation costs. Here they are these three whales:
1. Table of derivatives (differentiation formulas).
3. Derivative complex function.
Let's start in order. In this lesson, consider the table of derivatives.
Table derivatives.
In the world - infinite many functions. Among this set there are features that are most important for practical application. These functions are sitting in all laws of nature. From these functions, like from bricks, you can construct all the others. This class of functions is called elementary functions. It is these functions that are studied in school - linear, quadratic, hyperbole, etc.
Differentiation of functions "from scratch", i.e. Based on the determination of the derivative and the theory of limits - the thing is quite time consuming. And mathematicians are also people, yes - yes!) So they simply simplified life (and us). They calculated derivative elementary functions to us. It turned out a table of derivatives, where everything is already ready.)
Here it is, this plate for the most popular features. Left - elementary functionOn the right - its derivative.
| Function y. |
Derivative function y. y " |
|
| 1 | C (permanent value) | C "\u003d 0 |
| 2 | x. | x "\u003d 1 |
| 3 | x n (n - any number) | (x n) "\u003d NX N-1 |
| x 2 (n \u003d 2) | (x 2) "\u003d 2x | |
 |
||
 |
 |
|
| 4 | sIN X. | (SIN X) "\u003d COSX |
| cOS X. | (COS X) "\u003d - SIN X | |
| tG X. |  |
|
| cTG X. |  |
|
| 5 | arcsin X. |  |
| arccos X. |  |
|
| arctg X. |  |
|
| arcCTG X. |  |
|
| 4 | a. X. |  |
| e. X. | ||
| 5 | log. A.x. |  |
| ln x ( a \u003d E.) |
I recommend paying attention to the third group of functions in this table of derivatives. The derivative of the power function is one of the most common formulas, unless the most consuming! A hint is understandable?) Yes, the table of derivatives is desirable to know by heart. By the way, it is not as hard as it may seem. Try to solve more examples, the table itself will also be remembered!)
To find table value The derivative, as you understand, the task is not the most difficult. Therefore, very often there are additional chips in such tasks. Either in the wording of the task, or in the original function, which in the table seems to be no ...
Consider a few examples:
1. Find the derivative function y \u003d x 3
There is no such function in the table. But there is a derivative of the power function in general (Third Group). In our case, n \u003d 3. So we substitute the top three instead of n and carefully write the result:
(X. 3) "\u003d 3 · X 3-1 = 3X. 2
That's all things.
Answer: y "\u003d 3x 2
2. Find the value of the derivative function y \u003d sinx at point x \u003d 0.
This task means that you must first find a derivative of sinus, and then substitute the value x \u003d 0. To this very derivative. It is in this order! And then, it happens, immediately substitute the zero in the original function ... we are also asked to find the meaning of the original function, but the value its derivative. The derivative, remind - this is already a new feature.
On the table, we find sinus and the corresponding derivative:
y "\u003d (sin x)" \u003d COSX
We substitute zero to the derivative:
y "(0) \u003d cos 0 \u003d 1
This will answer.
3. Differentiate function:
What inspires?) Such a function in the table of derivatives and is not close.
Let me remind you that the function is to directly find the derivative of this function. If you forget the elementary trigonometry, search for the derivative of our function is quite troublesome. The table does not help ...
But if you see that our function is kosinus double cornerthen everything immediately gets up!
Yes Yes! Remember that the conversion of the original function before differentiation It is fully allowed! And, it happens, great facilitates life. By the cosine formula of a double angle:
Those. Our tricky function is nothing but y \u003d COSX.. And this is a tabular function. Immediately get:
Answer: y "\u003d - sin x.
Example for advanced graduates and students:
4. Find a derivative function:
There is no such function in the table of derivatives, of course. But if you recall the elementary mathematics, action with degrees ... then it is quite possible to simplify this function. Like this:
And the degree is one tenth to the degree - this is a table function! Third group, n \u003d 1/10. Directly by the formula and write down:
That's all. It will be the answer.
I hope that with the first cyt of differentiation - the table of derivatives - everything is clear. It remains to deal with the two remaining whales. In the next lesson, I will master the differentiation rules.
