What is a derivative?

Table of derivatives.

Derivative is one of the main concepts of higher mathematics. In this lesson we will introduce this concept. Let's get to know each other, without strict mathematical formulations and proofs.

This acquaintance will allow you to:

Understand the essence of simple tasks with derivatives;

Successfully solve these simplest tasks;

Prepare for more serious lessons on derivatives.

First - a pleasant surprise.)

The strict definition of the derivative is based on the theory of limits and the thing is quite complicated. This is upsetting. But the practical application of derivatives, as a rule, does not require such extensive and deep knowledge!

To successfully complete 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. That's all. This makes me happy.

Let's start getting acquainted?)

Terms and designations.

There are many different mathematical operations in elementary mathematics. Addition, subtraction, multiplication, exponentiation, logarithm, etc. If you add one more operation to these operations, elementary mathematics becomes higher. This new operation is called differentiation. The definition and meaning of this operation will be discussed in separate lessons.

It is important to understand here that differentiation is simply a mathematical operation on a function. We take any function and, according to certain rules, transform it. The result will be a new function. This new function is called: derivative.

Differentiation- action on a function.

Derivative- the result of this action.

Just like, for example, sum- the result of addition. Or private- the result of division.

Knowing the terms, you can at least understand the tasks.) The formulations are as follows: find the derivative of a function; take the derivative; differentiate the function; calculate derivative etc. This is all the same thing. Of course, there are also more complex tasks, where finding the derivative (differentiation) will be just one of the steps in solving the problem.

The derivative is indicated by a dash at the top right of the function. Like this: y" or f"(x) or S"(t) and so on.

Reading igrek stroke, ef stroke from x, es stroke from te, well, you understand...)

A prime can also indicate the derivative of a particular function, for example: (2x+3)", (x 3 )" , (sinx)" etc. Often derivatives are denoted using differentials, but we will not consider such notation in this lesson.

Let's assume that we have learned to understand the tasks. All that’s left is to learn how to solve them.) Let me remind you once again: finding the derivative is transformation of a function according to certain rules. Surprisingly, there are very few of these rules.

To find the derivative of a function, you need to know only three things. Three pillars on which all differentiation stands. Here they are these three pillars:

1. Table of derivatives (differentiation formulas).

3. Derivative of a complex function.

Let's start in order. In this lesson we will look at the table of derivatives.

Table of derivatives.

There are an infinite number of functions in the world. Among this set there are functions that are most important for practical use. These functions are found 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 at school - linear, quadratic, hyperbola, etc.

Differentiation of functions "from scratch", i.e. Based on the definition of derivative and the theory of limits, this is a rather labor-intensive thing. And mathematicians are people too, yes, yes!) So they simplified their (and us) life. They calculated the derivatives of elementary functions before us. The result is a table of derivatives, where everything is ready.)

Here it is, this plate for the most popular functions. On the left is an elementary function, on the right is its derivative.

	Function y	Derivative of function y y"
1	C (constant value)	C" = 0
2	x	x" = 1
3	x n (n - any number)	(x n)" = nx n-1
	x 2 (n = 2)	(x 2)" = 2x
4	sin x	(sin x)" = cosx
	cos x	(cos x)" = - sin x
	tg x
	ctg x
5	arcsin x
	arccos x
	arctan x
	arcctg x
4	a x
	e x
5	log a x
	ln x ( a = e)

I recommend paying attention to the third group of functions in this table of derivatives. The derivative of a power function is one of the most common formulas, if not the most common! Do you get the hint?) Yes, it is advisable to know the table of derivatives by heart. By the way, this is not as difficult as it might seem. Try to solve more examples, the table itself will be remembered!)

Finding the table value of the derivative, as you understand, is not the most difficult task. Therefore, very often in such tasks there are additional chips. Either in the wording of the task, or in the original function, which doesn’t seem to be in the table...

Let's look at a few examples:

1. Find the derivative of the function y = x 3

There is no such function in the table. But there is a derivative of a power function in general form (third group). In our case n=3. So we substitute three instead of n and carefully write down the result:

(x 3) " = 3 x 3-1 = 3x 2

That's it.

Answer: y" = 3x 2

2. Find the value of the derivative of the function y = sinx at the point x = 0.

This task means that you must first find the derivative of the sine, and then substitute the value x = 0 into this very derivative. Exactly in that order! Otherwise, it happens that they immediately substitute zero into the original function... We are asked to find not the value of the original function, but the value its derivative. The derivative, let me remind you, is a new function.

Using the tablet we find the sine and the corresponding derivative:

y" = (sin x)" = cosx

We substitute zero into the derivative:

y"(0) = cos 0 = 1

This will be the answer.

3. Differentiate the function:

What, does it inspire?) There is no such function in the table of derivatives.

Let me remind you that to differentiate a function is simply to find the derivative of this function. If you forget elementary trigonometry, looking for the derivative of our function is quite troublesome. The table doesn't help...

But if we see that our function is double angle cosine, then everything gets better right away!

Yes, yes! Remember that transforming the original function before differentiation quite acceptable! And it happens to make life a lot easier. Using the double angle cosine formula:

Those. our tricky function is nothing more than y = cosx. And this is a table function. We immediately get:

Answer: y" = - sin x.

Example for advanced graduates and students:

4. Find the derivative of the function:

There is no such function in the derivatives table, of course. But if you remember elementary mathematics, operations with powers... Then it is quite possible to simplify this function. Like this:

And x to the power of one tenth is already a tabular function! Third group, n=1/10. We write directly according to the formula:

That's it. This will be the answer.

I hope that everything is clear with the first pillar of differentiation - the table of derivatives. It remains to deal with the two remaining whales. In the next lesson we will learn the rules of differentiation.

Derivative of a function of one variable.

Introduction.

These methodological developments are intended for students of the Faculty of Industrial and Civil Engineering. They were compiled in relation to the mathematics course program in the section “Differential calculus of functions of one variable.”

The developments represent a single methodological guide, including: brief theoretical information; “standard” problems and exercises with detailed solutions and explanations for these solutions; test options.

There are additional exercises at the end of each paragraph. This structure of developments makes them suitable for independent mastery of the section with minimal assistance from the teacher.

§1. Definition of derivative.

Mechanical and geometric meaning

derivative.

The concept of derivative is one of the most important concepts in mathematical analysis. It arose back in the 17th century. The formation of the concept of derivative is historically associated with two problems: the problem of the speed of alternating motion and the problem of the tangent to a curve.

These problems, despite their different content, lead to the same mathematical operation that must be performed on the function. This operation received a special name in mathematics. It is called the operation of differentiation of a function. The result of the differentiation operation is called the derivative.

So, the derivative of the function y=f(x) at the point x0 is the limit (if it exists) of the ratio of the increment of the function to the increment of the argument
at
.

The derivative is usually denoted as follows:
.

Thus, by definition

The symbols are also used to denote derivatives
.

Mechanical meaning of derivative.

If s=s(t) is the law of rectilinear motion of a material point, then
is the speed of this point at time t.

Geometric meaning of derivative.

If the function y=f(x) has a derivative at the point , then the angular coefficient of the tangent to the graph of the function at the point
equals
.

Example.

Find the derivative of the function
at the point =2:

1) Let's give it a point =2 increment
. Note that.

2) Find the increment of the function at the point =2:

3) Let’s create the ratio of the increment of the function to the increment of the argument:

Let us find the limit of the ratio at
:

.

Thus,
.

§ 2. Derivatives of some

simplest functions.

The student needs to learn how to calculate derivatives of specific functions: y=x,y= and in generaly= .

Let's find the derivative of the function y=x.

those. (x)′=1.

Let's find the derivative of the function

Derivative

Let
Then

It is easy to notice a pattern in the expressions for the derivatives of the power function
with n=1,2,3.

Hence,

. (1)

This formula is valid for any real n.

In particular, using formula (1), we have:

;

.

Example.

Find the derivative of the function

.

.

This function is a special case of a function of the form

at
.

Using formula (1), we have

.

Derivatives of the functions y=sin x and y=cos x.

Let y=sinx.

Divide by ∆x, we get

Passing to the limit at ∆x→0, we have

Let y=cosx.

Passing to the limit at ∆x→0, we obtain

;
. (2)

§3. Basic rules of differentiation.

Let's consider the rules of differentiation.

Theorem1 . If the functions u=u(x) and v=v(x) are differentiable at a given pointx, then at this point their sum is also differentiable, and the derivative of the sum is equal to the sum of the derivatives of the terms: (u+v)"=u"+v".(3 )

Proof: consider the function y=f(x)=u(x)+v(x).

The increment ∆x of the argument x corresponds to the increments ∆u=u(x+∆x)-u(x), ∆v=v(x+∆x)-v(x) of the functions u and v. Then the function y will increase

∆y=f(x+∆x)-f(x)=

=--=∆u+∆v.

Hence,

So, (u+v)"=u"+v".

Theorem2. If the functions u=u(x) and v=v(x) are differentiable at a given pointx, then their product is differentiable at the same point. In this case, the derivative of the product is found by the following formula: (uv)"=u"v+uv". ( 4)

Proof: Let y=uv, where u and v are some differentiable functions of x. Let's give x an increment of ∆x; then u will receive an increment of ∆u, v will receive an increment of ∆v, and y will receive an increment of ∆y.

We have y+∆y=(u+∆u)(v+∆v), or

y+∆y=uv+u∆v+v∆u+∆u∆v.

Therefore, ∆y=u∆v+v∆u+∆u∆v.

From here

Passing to the limit at ∆x→0 and taking into account that u and v do not depend on ∆x, we will have

Theorem 3. The derivative of the quotient of two functions is equal to a fraction, the denominator of which is equal to the square of the divisor, and the numerator is the difference between the product of the derivative of the dividend and the divisor and the product of the dividend and the derivative of the divisor, i.e.

If
That
(5)

Theorem 4. The derivative of a constant is zero, i.e. if y=C, where C=const, then y"=0.

Theorem 5. The constant factor can be taken out of the sign of the derivative, i.e. if y=Cu(x), where С=const, then y"=Cu"(x).

Example 1.

Find the derivative of the function

.

This function has the form
, whereu=x,v=cosx. Applying the differentiation rule (4), we find

.

Example 2.

Find the derivative of the function

.

Let's apply formula (5).

Here
;
.

Tasks.

Find the derivatives of the following functions:

;

11)

2)
; 12)
;

3)
13)

4)
14)

5)
15)

6)
16)

7 )
17)

8)
18)

9)
19)

10)
20)

In this article we will give the basic concepts on which all further theory on the topic of the derivative of a function of one variable will be based.

Path x is the argument of the function f(x) and is a small number different from zero.

(read “delta x”) is called incrementing a function argument. In the figure, the red line shows the change in the argument from value x to value (hence the essence of the name “increment” of the argument).

When moving from the value of the argument to the values ​​of the function change accordingly from to, provided that the function is monotonic on the interval. The difference is called increment of function f(x), corresponding to this argument increment. In the figure, the function increment is shown with a blue line.

Let's look at these concepts using a specific example.

Let's take, for example, the function . Let us fix the point and the increment of the argument. In this case, the increment of the function when moving from to will be equal to

A negative increment indicates a decrease in the function on the segment.

Graphic illustration

Determining the derivative of a function at a point.

Let the function f(x) be defined on the interval (a; b) and and be the points of this interval. Derivative of the function f(x) at the point is called the limit of the ratio of the increment of a function to the increment of the argument at . Designated .

When the last limit takes on a specific final value, we speak of the existence finite derivative at the point. If the limit is infinite, then they say that derivative is infinite at a given point. If the limit does not exist, then the derivative of the function at this point does not exist.

The function f(x) is called differentiable at the point, when it has a finite derivative in it.

If a function f(x) is differentiable at each point of a certain interval (a; b), then the function is called differentiable on this interval. Thus, any point x from the interval (a; b) can be associated with the value of the derivative of the function at this point, that is, we have the opportunity to define a new function, which is called derivative of the function f(x) on the interval (a; b).

The operation of finding the derivative is called differentiation.

Let us make a distinction in the nature of the concepts of the derivative of a function at a point and on an interval: the derivative of a function at a point is a number, and the derivative of a function on an interval is a function.

Let's look at this with examples to make the picture clearer. When differentiating, we will use the definition of derivative, that is, we will proceed to finding limits. If difficulties arise, we recommend that you refer to the theory section.

Example.

Find the derivative of the function at the point using the definition.

Solution.

Since we are looking for the derivative of a function at a point, the answer must contain a number. Let's write down the limit of the ratio of the increment of a function to the increment of the argument and use the trigonometry formulas:

What is a derivative?
Definition and meaning of a derivative function

Many will be surprised by the unexpected placement of this article in my author’s course on the derivative of a function of one variable and its applications. After all, as it has been since school: the standard textbook first of all gives the definition of a derivative, its geometric, mechanical meaning. Next, students find derivatives of functions by definition, and, in fact, only then they perfect the technique of differentiation using derivative tables.

But from my point of view, the following approach is more pragmatic: first of all, it is advisable to UNDERSTAND WELL limit of a function, and, in particular, infinitesimal quantities. The point is that the definition of derivative is based on the concept of limit, which is poorly considered in the school course. That is why a significant part of young consumers of the granite of knowledge do not understand the very essence of the derivative. Thus, if you have little understanding of differential calculus or a wise brain has successfully gotten rid of this baggage over many years, please start with function limits. At the same time, master/remember their solution.

The same practical sense dictates that it is advantageous first learn to find derivatives, including derivatives of complex functions. Theory is theory, but, as they say, you always want to differentiate. In this regard, it is better to work through the listed basic lessons, and maybe master of differentiation without even realizing the essence of their actions.

I recommend starting with the materials on this page after reading the article. The simplest problems with derivatives, where, in particular, the problem of the tangent to the graph of a function is considered. But you can wait. The fact is that many applications of the derivative do not require understanding it, and it is not surprising that the theoretical lesson appeared quite late - when I needed to explain finding increasing/decreasing intervals and extrema functions. Moreover, he was on the topic for quite a long time. Functions and graphs”, until I finally decided to put it earlier.

Therefore, dear teapots, do not rush to absorb the essence of the derivative like hungry animals, because the saturation will be tasteless and incomplete.

The concept of increasing, decreasing, maximum, minimum of a function

Many textbooks introduce the concept of derivatives with the help of some practical problems, and I also came up with an interesting example. Imagine that we are about to travel to a city that can be reached in different ways. Let’s immediately discard the curved winding paths and consider only straight highways. However, straight-line directions are also different: you can get to the city along a smooth highway. Or along a hilly highway - up and down, up and down. Another road goes only uphill, and another one goes downhill all the time. Extreme enthusiasts will choose a route through a gorge with a steep cliff and a steep climb.

But whatever your preferences, it is advisable to know the area or at least have a topographic map of it. What if such information is missing? After all, you can choose, for example, a smooth path, but as a result stumble upon a ski slope with cheerful Finns. It is not a fact that a navigator or even a satellite image will provide reliable data. Therefore, it would be nice to formalize the relief of the path using mathematics.

Let's look at some road (side view):

Just in case, I remind you of an elementary fact: travel happens from left to right. For simplicity, we assume that the function continuous in the area under consideration.

What are the features of this graph?

At intervals function increases, that is, each next value of it more previous one. Roughly speaking, the schedule is on from bottom to top(we climb the hill). And on the interval the function decreases– each next value less previous, and our schedule is on top down(we go down the slope).

Let's also pay attention to special points. At the point we reach maximum, that is exists such a section of the path where the value will be the largest (highest). At the same point it is achieved minimum, And exists its neighborhood in which the value is the smallest (lowest).

We will look at more strict terminology and definitions in class. about the extrema of the function, but for now let’s study another important feature: on intervals the function increases, but it increases at different speeds. And the first thing that catches your eye is that the graph soars up during the interval much more cool, than on the interval . Is it possible to measure the steepness of a road using mathematical tools?

Rate of change of function

The idea is this: let's take some value (read "delta x"), which we'll call argument increment, and let’s start “trying it on” to various points on our path:

1) Let's look at the leftmost point: passing the distance, we climb the slope to a height (green line). The quantity is called function increment, and in this case this increment is positive (the difference in values ​​along the axis is greater than zero). Let's create a ratio that will be a measure of the steepness of our road. Obviously, this is a very specific number, and since both increments are positive, then .

Attention! Designation are ONE symbol, that is, you cannot “tear off” the “delta” from the “X” and consider these letters separately. Of course, the comment also concerns the function increment symbol.

Let's explore the nature of the resulting fraction more meaningfully. Let us initially be at a height of 20 meters (at the left black point). Having covered the distance of meters (left red line), we will find ourselves at an altitude of 60 meters. Then the increment of the function will be meters (green line) and: . Thus, on every meter this section of the road height increases on average by 4 meters...forgot your climbing equipment? =) In other words, the constructed relationship characterizes the AVERAGE RATE OF CHANGE (in this case, growth) of the function.

Note : The numerical values ​​of the example in question correspond only approximately to the proportions of the drawing.

2) Now let's go the same distance from the rightmost black point. Here the rise is more gradual, so the increment (crimson line) is relatively small, and the ratio compared to the previous case will be very modest. Relatively speaking, meters and function growth rate is . That is, here for every meter of the path there are on average half a meter of rise.

3) A little adventure on the mountainside. Let's look at the top black dot located on the ordinate axis. Let's assume that this is the 50 meter mark. We overcome the distance again, as a result of which we find ourselves lower - at the level of 30 meters. Since the movement is carried out top down(in the “counter” direction of the axis), then the final the increment of the function (height) will be negative: meters (brown segment in the drawing). And in this case we are already talking about rate of decrease Features: , that is, for every meter of path of this section, the height decreases on average by 2 meters. Take care of your clothes at the fifth point.

Now let's ask ourselves the question: what value of the “measuring standard” is best to use? It’s completely understandable, 10 meters is very rough. A good dozen hummocks can easily fit on them. No matter the bumps, there may be a deep gorge below, and after a few meters there is its other side with a further steep rise. Thus, with a ten-meter we will not get an intelligible description of such sections of the path through the ratio .

From the above discussion the following conclusion follows: the lower the value, the more accurately we describe the road topography. Moreover, the following facts are true:

– For anyone lifting points you can select a value (even if very small) that fits within the boundaries of a particular rise. This means that the corresponding height increment will be guaranteed to be positive, and the inequality will correctly indicate the growth of the function at each point of these intervals.

- Likewise, for any slope point there is a value that will fit completely on this slope. Consequently, the corresponding increase in height is clearly negative, and the inequality will correctly show the decrease in the function at each point of the given interval.

– A particularly interesting case is when the rate of change of the function is zero: . Firstly, zero height increment () is a sign of a smooth path. And secondly, there are other interesting situations, examples of which you see in the figure. Imagine that fate has brought us to the very top of a hill with soaring eagles or the bottom of a ravine with croaking frogs. If you take a small step in any direction, the change in height will be negligible, and we can say that the rate of change of the function is actually zero. This is exactly the picture observed at the points.

Thus, we have come to an amazing opportunity to perfectly accurately characterize the rate of change of a function. After all, mathematical analysis makes it possible to direct the increment of the argument to zero: , that is, to make it infinitesimal.

As a result, another logical question arises: is it possible to find for the road and its schedule another function, which would let us know about all the flat sections, ascents, descents, peaks, valleys, as well as the rate of growth/decrease at each point along the way?

What is a derivative? Definition of derivative.
Geometric meaning of derivative and differential

Please read carefully and not too quickly - the material is simple and accessible to everyone! It’s okay if in some places something doesn’t seem very clear, you can always return to the article later. I will say more, it is useful to study the theory several times in order to thoroughly understand all the points (the advice is especially relevant for “technical” students, for whom higher mathematics plays a significant role in the educational process).

Naturally, in the very definition of the derivative at a point we replace it with:

What have we come to? And we came to the conclusion that for the function according to the law is put in accordance other function which is called derivative function(or just derivative).

The derivative characterizes rate of change functions How? The idea runs like a red thread from the very beginning of the article. Let's consider some point domain of definition functions Let the function be differentiable at a given point. Then:

1) If , then the function increases at the point . And obviously there is interval(even a very small one), containing a point at which the function grows, and its graph goes “from bottom to top”.

2) If , then the function decreases at the point . And there is an interval containing a point at which the function decreases (the graph goes “top to bottom”).

3) If , then infinitely close near a point the function maintains its speed constant. This happens, as noted, with a constant function and at critical points of the function, in particular at minimum and maximum points.

A bit of semantics. What does the verb “differentiate” mean in a broad sense? To differentiate means to highlight a feature. By differentiating a function, we “isolate” the rate of its change in the form of a derivative of the function. What, by the way, is meant by the word “derivative”? Function happened from function.

The terms are very successfully interpreted by the mechanical meaning of the derivative :
Let us consider the law of change in the coordinates of a body, depending on time, and the function of the speed of movement of a given body. The function characterizes the rate of change of body coordinates, therefore it is the first derivative of the function with respect to time: . If the concept of “body movement” did not exist in nature, then there would be no derivative concept of "body speed".

The acceleration of a body is the rate of change of speed, therefore: . If the initial concepts of “body motion” and “body speed” did not exist in nature, then there would not exist derivative concept of “body acceleration”.