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 of 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 contents, lead to the same mathematical operation that must be performed on a function. This operation has 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
. Notice, 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 point x, then their sum is differentiable at this point, 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)

Create a ratio and calculate the limit.

Where did it come from? table of derivatives and differentiation rules? Thanks to the only limit. It seems like magic, but in reality it is sleight of hand and no fraud. At the lesson What is a derivative? I began to look at specific examples where, using the definition, I found the derivatives of a linear and quadratic function. For the purpose of cognitive warm-up, we will continue to disturb table of derivatives, honing the algorithm and technical solutions:

Example 1

Essentially, you need to prove a special case of the derivative of a power function, which usually appears in the table: .

Solution technically formalized in two ways. Let's start with the first, already familiar approach: the ladder starts with a plank, and the derivative function starts with the derivative at a point.

Let's consider some(specific) point belonging to domain of definition function in which there is a derivative. Let us set the increment at this point (of course, within the scopeo/o -I) and compose the corresponding increment of the function:

Let's calculate the limit:

The uncertainty 0:0 is eliminated by a standard technique, considered back in the first century BC. Multiply the numerator and denominator by the conjugate expression :

The technique for solving such a limit is discussed in detail in the introductory lesson. about the limits of functions.

Since you can choose ANY point of the interval as quality, then, having made the replacement, we get:

Answer

Once again let's rejoice at logarithms:

Example 2

Find the derivative of a function using the definition of derivative

Solution: Let's consider a different approach to promoting the same task. It is exactly the same, but more rational in terms of design. The idea is to get rid of the subscript at the beginning of the solution and use the letter instead of the letter.

Let's consider arbitrary point belonging to domain of definition function (interval) and set the increment in it. But here, by the way, as in most cases, you can do without any reservations, since the logarithmic function is differentiable at any point in the domain of definition.

Then the corresponding increment of the function is:

Let's find the derivative:

The simplicity of the design is balanced by the confusion that may arise for beginners (and not only). After all, we are used to the fact that the letter “X” changes in the limit! But here everything is different: - an antique statue, and - a living visitor, briskly walking along the corridor of the museum. That is, “x” is “like a constant.”

I will comment on the elimination of uncertainty step by step:

(1) We use the property of the logarithm .

(2) In parentheses, divide the numerator by the denominator term by term.

(3) In the denominator, we artificially multiply and divide by “x” to take advantage of remarkable limit , while as infinitesimal stands out.

Answer: by definition of derivative:

Or in short:

I propose to construct two more table formulas yourself:

Example 3

In this case, it is convenient to immediately reduce the compiled increment to a common denominator. An approximate sample of the assignment at the end of the lesson (first method).

Example 3:Solution : consider some point , belonging to the domain of definition of the function . Let us set the increment at this point and compose the corresponding increment of the function:

Let's find the derivative at the point :


Since as a you can select any point function domain , That And
Answer : by definition of derivative

Example 4

Find derivative by definition

And here everything needs to be reduced to wonderful limit. The solution is formalized in the second way.

A number of other tabular derivatives. The complete list can be found in the school textbook, or, for example, the 1st volume of Fichtenholtz. I don’t see much point in copying proofs of differentiation rules from books - they are also generated by the formula.

Example 4:Solution , belonging to , and set the increment in it

Let's find the derivative:

Using a wonderful limit

Answer : a-priory

Example 5

Find the derivative of a function , using the definition of derivative

Solution: we use the first design style. Let's consider some point belonging to , and specify the increment of the argument at it. Then the corresponding increment of the function is:

Perhaps some readers have not yet fully understood the principle by which increments need to be made. Take a point (number) and find the value of the function in it: , that is, into the function instead of"X" should be substituted. Now we also take a very specific number and also substitute it into the function instead of"iksa": . We write down the difference, and it is necessary put in brackets completely.

Compiled function increment It can be beneficial to immediately simplify. For what? Facilitate and shorten the solution to a further limit.

We use formulas, open the brackets and reduce everything that can be reduced:

The turkey is gutted, no problem with the roast:

Eventually:

Since we can choose any real number as a value, we make the replacement and get .

Answer: a-priory.

For verification purposes, let’s find the derivative using differentiation rules and tables:

It is always useful and pleasant to know the correct answer in advance, so it is better to differentiate the proposed function in a “quick” way, either mentally or in a draft, at the very beginning of the solution.

Example 6

Find the derivative of a function by definition of derivative

This is an example for you to solve on your own. The result is obvious:

Example 6:Solution : consider some point , belonging to , and set the increment of the argument in it . Then the corresponding increment of the function is:


Let's calculate the derivative:


Thus:
Because as you can choose any real number, then And
Answer : a-priory.

Let's go back to style #2:

Example 7

Let's find out immediately what should happen. By rule of differentiation of complex functions:

Solution: consider an arbitrary point belonging to , set the increment of the argument at it and compose the increment of the function:

Let's find the derivative:


(1) Use trigonometric formula .

(2) Under the sine we open the brackets, under the cosine we present similar terms.

(3) Under the sine we reduce the terms, under the cosine we divide the numerator by the denominator term by term.

(4) Due to the oddness of the sine, we take out the “minus”. Under the cosine we indicate that the term .

(5) We carry out artificial multiplication in the denominator in order to use first wonderful limit. Thus, the uncertainty is eliminated, let’s tidy up the result.

Answer: a-priory

As you can see, the main difficulty of the problem under consideration rests on the complexity of the limit itself + a slight uniqueness of the packaging. In practice, both methods of design occur, so I describe both approaches in as much detail as possible. They are equivalent, but still, in my subjective impression, it is more advisable for dummies to stick to option 1 with “X-zero”.

Example 8

Using the definition, find the derivative of the function

Example 8:Solution : consider an arbitrary point , belonging to , let us set the increment in it and compose the increment of the function:

Let's find the derivative:

We use the trigonometric formula and the first remarkable limit:

Answer : a-priory

Let's look at a rarer version of the problem:

Example 9

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

Firstly, what should be the bottom line? Number

Let's calculate the answer in the standard way:

Solution: from the point of view of clarity, this task is much simpler, since the formula instead considers a specific value.

Let's set the increment at the point and compose the corresponding increment of the function:

Let's calculate the derivative at the point:

We use a very rare tangent difference formula and once again we reduce the solution to the first wonderful limit:

Answer: by definition of derivative at a point.

The problem is not so difficult to solve “in general” - it is enough to replace with or simply depending on the design method. In this case, it is clear that the result will not be a number, but a derived function.

Example 10

Using the definition, find the derivative of the function at a point (one of which may turn out to be infinite), which I have already described in general terms on theoretical lesson about derivative.

Some piecewise defined functions are also differentiable at the “junction” points of the graph, for example, catdog has a common derivative and a common tangent (x-axis) at the point. Curve, but differentiable by ! Those interested can verify this for themselves using the example just solved.

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Ministry of Education of the Russian Federation

“MATI” - RUSSIAN STATE

TECHNOLOGICAL UNIVERSITY named after. K. E. TSIOLKOVSKY

Department of “Higher Mathematics”

Course assignment options

Guidelines for the course assignment

“Limits of functions. Derivatives"

Kulakova R. D.

Titarenko V. I.

Moscow 1999

annotation

The proposed guidelines are aimed at helping first-year students master theoretical and practical material on the topic “Mathematical Analysis”.

In each section, after the theoretical part, typical problems are analyzed.

The guidelines cover the following topics: limits of functions, differentiation of functions given in various forms, derivatives and differentials of higher orders, L'Hopital's rule, application of the derivative to problems of geometry and mechanics.

To consolidate the material, students are asked to complete coursework on the topics listed above.

These guidelines can be used in all faculties and specialties.

1. Function limits

Some well-known techniques are used to determine the limits of sequences and functions:

If you need to find a limit

can be preliminarily reduced to a common denominator

Dividing by the term that has the maximum degree, we get a constant value in the numerator, and all terms tending to 0 in the denominator, that is

.

Then substituting x=a, we get:
;

4.
, when substituting x=0, we get
.

5. However, if it is necessary to find the limit of a rational function

, then when dividing by the term with the minimum degree, we get

; and, directing x to 0, we get:

If the limits contain irrational expressions, then new variables have to be introduced to obtain a rational expression, or irrationalities must be transferred from the denominator to the numerator and vice versa.

6.
; Let's make a variable change. We will replace
, at
, we get
.

7.
. If the numerator and denominator are multiplied by the same number, the limit does not change. Multiply the numerator by
and divide by the same expression so that the limit does not change, and multiply the denominator by
and divide by the same expression. Then we get:

The following remarkable limits are often used to define limits:

; (1)

. (2)

8.
.

To calculate such a limit, we reduce it to the 1st remarkable limit (1). To do this, multiply and divide the numerator by
, and the denominator is
, Then.

9.
To calculate this limit, we reduce it to the second remarkable limit. For this purpose, we select the whole part from the rational expression in brackets and present it in the form of a proper fraction. This is done in cases where
, Where
, A
, Where
;

, A
, then finally
. Here the continuity of the composition of continuous functions was used.

2. Derivative

Derivative of a function
is called the final limit of the ratio of the increment of a function to the increment of the argument when the latter tends to zero:

, or
.

Geometrically, the derivative is the slope of the tangent to the graph of the function
at point x, that is
.

The derivative is the rate of change of a function at point x.

Finding the derivative is called differentiating the function.

Formulas for differentiating basic functions:

3. Basic rules of differentiation

Let then:

7) If , that is
, Where
And
have derivatives, then
(rule for differentiating a complex function).

4. Logarithmic differentiation

If you need to find from Eq.
, then you can:

a) logarithm both sides of the equation

b) differentiate both sides of the resulting equality, where
there is a complex function of x,

.

c) replace its expression in terms of x

.

Example:

5. Differentiation of implicit functions

Let the equation
defines as an implicit function of x.

a) differentiate both sides of the equation with respect to x
, we obtain an equation of the first degree with respect to ;

b) from the resulting equation we express .

Example:
.

6. Differentiation of functions given

parametrically

Let the function be given by parametric equations
,

Then
, or

Example:

7. Application of the derivative to problems

geometry and mechanics

Let
And
, Where - the angle formed with the positive direction of the OX axis by the tangent to the curve at the point with the abscissa .

Equation of a tangent to a curve
at the point
has the form:

, Where -derivative at
.

The normal to a curve is a line perpendicular to the tangent and passing through the point of tangency.

The normal equation has the form

.

Angle between two curves
And
at the point of their intersection
is the angle between the tangents to these curves at a point
. This angle is found by the formula

.

8. Higher order derivatives

If is the derivative of the function
, then the derivative of is called the second derivative, or derivative of the second order and is denoted , or
, or .

Derivatives of any order are defined similarly: third order derivative
; nth order derivative:

.

For the product of two functions, you can obtain a derivative of any nth order using the Leibniz formula:

9. Second derivative of an implicit function

-the equation determines , as an implicit function of x.

a) define
;

b) differentiate with respect to x the left and right sides of the equality
,

Moreover, differentiating the function
by variable x, remember that there is a function of x:

;

c) replacing through
, we get:
etc.

10. Derivatives of functions specified parametrically

Find
If
.

11. Differentials of the first and higher orders

First order differential of the function
is called the main part, linear with respect to the argument. The differential of an argument is the increment of an argument:
.

The differential of a function is equal to the product of its derivative and the differential of the argument:

.

Basic properties of the differential:

Where
.

If the increment
argument is small in absolute value, then
And.

Thus, the differential of a function can be used for approximate calculations.

Second order differential of the function
is called the differential of the first order differential:
.

Likewise:
.

.

If
And is an independent variable, then higher order differentials are calculated using the formulas

Find the first and second order differentials of the function

12. Calculation of limits using L'Hopital's rule

All of the above limits did not use the apparatus of differential calculus. However, if you need to find

and at
both of these functions are infinitesimal or both are infinitely large, then their ratio is not defined at the point
and therefore represents an uncertainty type or respectively. Since this is a relationship at a point
may have a limit, finite or infinite, then finding this limit is called the disclosure of uncertainty (L'Hopital Bernoulli's rule),

and the following equality holds:

, If
And
.

=
.

A similar rule holds if
And
, i.e.
.

=

=
.

L'Hopital's rule also makes it possible to resolve uncertainties of the type
And
. To calculate
, Where
- infinitesimal, and
- infinitely large at
(type uncertainty disclosure
) the product should be converted to the form

(uncertainty of type) or to species (type uncertainty ) and then use Lapital's rule.

To calculate
, Where
And
- infinitely large at
(type uncertainty disclosure
) the difference should be converted to the form
, then reveal the uncertainty type . If
, That
.

If
, then we get an uncertainty of the type (
), which is revealed similarly to example 12).

Because
, then we end up with an uncertainty of the type
and then we have

.

L'Hopital's rule can also be used to resolve uncertainties of the type
. In these cases, we mean calculating the limit of the expression
, Where
when
is infinitesimal, in the case
- infinitely large, and in the case
- a function whose limit is equal to unity.

Function
in the first two cases it is an infinitely small function, and in the last case it is an infinitely large function.

Before looking for the limit of such expressions, they are taken logarithmically, i.e. If
, That
, then find the limit
, and then find the limit . In all of the above cases
is a type uncertainty
, which is opened similarly to example 12).

5.

(use L'Hopital's rule)=

=
.

In this product of limits, the first factor is equal to 1, the second factor is the first remarkable limit and it is also equal to 1, and the last factor tends to 0, therefore:

and then
.

=
;

.

7.
;

=
;

.

8.
;

=
;

.

THE COURSE WORK INCLUDES 21 TASKS.

No. 1-4 – Calculation of function limits;

No. 5-10 – Find derivatives of functions;

No. 11 – Find the first derivative;

#12 – Calculate function specified in parametric form;

#13 – Find d 2 y;

#14 – Find y ( n ) ;

No. 15 – Create an equation for the normal and tangent to the curve at a point x 0 ;

No. 16 – Calculate the value of the function approximately using a differential;

#17 – Find
;

#18 – Find ;

#19 – Find ;

No. 20-21 – Calculate the limit using L'Hopital's rule.

Option 1

№1.
.

№2.
.

№3.
.

№4.
.

Calculate Derivative

№5.
.

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 fact 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 flat 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 down up(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! Designations 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 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 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 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 simply 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”.

In the coordinate plane xOy consider the graph of the function y=f(x). Let's fix the point M(x 0 ; f (x 0)). Let's add an abscissa x 0 increment Δх. We will get a new abscissa x 0 +Δx. This is the abscissa of the point N, and the ordinate will be equal f (x 0 +Δx). The change in the abscissa entailed a change in the ordinate. This change is called the function increment and is denoted Δy.

Δy=f (x 0 +Δx) - f (x 0). Through dots M And N let's draw a secant MN, which forms an angle φ with positive axis direction Oh. Let's determine the tangent of the angle φ from a right triangle MPN.

Let Δх tends to zero. Then the secant MN will tend to take a tangent position MT, and the angle φ will become an angle α . So, the tangent of the angle α is the limiting value of the tangent of the angle φ :

The limit of the ratio of the increment of a function to the increment of the argument, when the latter tends to zero, is called the derivative of the function at a given point:

Geometric meaning of derivative lies in the fact that the numerical derivative of the function at a given point is equal to the tangent of the angle formed by the tangent drawn through this point to the given curve and the positive direction of the axis Oh:

Examples.

1. Find the increment of the argument and the increment of the function y= x 2, if the initial value of the argument was equal to 4 , and new - 4,01 .

Solution.

New argument value x=x 0 +Δx. Let's substitute the data: 4.01=4+Δх, hence the increment of the argument Δх=4.01-4=0.01. The increment of a function, by definition, is equal to the difference between the new and previous values ​​of the function, i.e. Δy=f (x 0 +Δx) - f (x 0). Since we have a function y=x2, That Δу=(x 0 +Δx) 2 - (x 0) 2 =(x 0) 2 +2x 0 · Δx+(Δx) 2 - (x 0) 2 =2x 0 · Δx+(Δx) 2 =

2 · 4 · 0,01+(0,01) 2 =0,08+0,0001=0,0801.

Answer: argument increment Δх=0.01; function increment Δу=0,0801.

The function increment could be found differently: Δy=y (x 0 +Δx) -y (x 0)=y(4.01) -y(4)=4.01 2 -4 2 =16.0801-16=0.0801.

2. Find the angle of inclination of the tangent to the graph of the function y=f(x) at the point x 0, If f "(x 0) = 1.

Solution.

The value of the derivative at the point of tangency x 0 and is the value of the tangent of the tangent angle (the geometric meaning of the derivative). We have: f "(x 0) = tanα = 1 → α = 45°, because tg45°=1.

Answer: the tangent to the graph of this function forms an angle with the positive direction of the Ox axis equal to 45°.

3. Derive the formula for the derivative of the function y=x n.

Differentiation is the action of finding the derivative of a function.

When finding derivatives, use formulas that were derived based on the definition of a derivative, in the same way as we derived the formula for the derivative degree: (x n)" = nx n-1.

These are the formulas.

Table of derivatives It will be easier to memorize by pronouncing verbal formulations:

1. The derivative of a constant quantity is zero.

2. X prime is equal to one.

3. The constant factor can be taken out of the sign of the derivative.

4. The derivative of a degree is equal to the product of the exponent of this degree by a degree with the same base, but the exponent is one less.

5. The derivative of a root is equal to one divided by two equal roots.

6. The derivative of one divided by x is equal to minus one divided by x squared.

7. The derivative of the sine is equal to the cosine.

8. The derivative of the cosine is equal to minus sine.

9. The derivative of the tangent is equal to one divided by the square of the cosine.

10. The derivative of the cotangent is equal to minus one divided by the square of the sine.

We teach differentiation rules.

1. The derivative of an algebraic sum is equal to the algebraic sum of the derivatives of the terms.

2. The derivative of a product is equal to the product of the derivative of the first factor and the second plus the product of the first factor and the derivative of the second.

3. The derivative of “y” divided by “ve” is equal to a fraction in which the numerator is “y prime multiplied by “ve” minus “y multiplied by ve prime”, and the denominator is “ve squared”.

4. A special case of the formula 3.

Let's learn together!

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