Definition of derivative, its geometric meaning. Derivative of a function

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 Δx. 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 Δx 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 Δx=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 Δx=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 ( geometric meaning 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 algebraic sum derivatives of 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. Special case formulas 3.

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 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 knowledge of differential calculus or a wise brain for long years successfully got rid of this baggage, 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 teaching aids lead to the concept of derivative using some practical problems, and I also came up with 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 locate it topographic map. 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’s not a fact that the navigator and even 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 features does this graph have?

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 us also pay attention to singular 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 one more important feature: at intervals the function increases, but it increases With 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. It is obvious that this is quite 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 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 : numeric values The example under consideration corresponds to the proportions of the drawing only approximately.

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: how less value , the more accurately we will 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 allows you to direct the increment of the argument to zero: , that is, make it infinitesimal.

As a result, another one arises logical question: 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”.

Abstract open lesson teacher of GBPOU " Teachers College No. 4 St. Petersburg"

Martusevich Tatyana Olegovna

Date: 12/29/2014.

Topic: Geometric meaning of derivatives.

Lesson type: learning new material.

Teaching methods: visual, partly search.

The purpose of the lesson.

Introduce the concept of a tangent to the graph of a function at a point, find out what the geometric meaning of the derivative is, derive the equation of the tangent and teach how to find it.

Educational objectives:

Achieve an understanding of the geometric meaning of the derivative; deriving the tangent equation; learn to solve basic problems;

provide repetition of material on the topic “Definition of a derivative”;

create conditions for control (self-control) of knowledge and skills.

Developmental tasks:

promote the formation of skills to apply techniques of comparison, generalization, and highlighting the main thing;

continue the development of mathematical horizons, thinking and speech, attention and memory.

Educational tasks:

promote interest in mathematics;

education of activity, mobility, communication skills.

Lesson type – a combined lesson using ICT.

Equipment – multimedia installation, presentationMicrosoftPowerPoint.

Lesson stage

Time

Teacher's activities

Student activity

1. Organizational moment.

State the topic and purpose of the lesson.

Topic: Geometric meaning of derivatives.

The purpose of the lesson.

Introduce the concept of a tangent to the graph of a function at a point, find out what the geometric meaning of the derivative is, derive the equation of the tangent and teach how to find it.

Preparing students for work in class.

Preparation for work in class.

Understanding the topic and purpose of the lesson.

Note-taking.

2. Preparation for learning new material through repetition and updating of basic knowledge.

Organization of repetition and updating of basic knowledge: definition of derivative and formulation of its physical meaning.

Formulating the definition of a derivative and formulating its physical meaning. Repetition, updating and consolidation of basic knowledge.

Organization of repetition and development of the skill of finding a derivative power function and elementary functions.

Finding the derivative of these functions using formulas.

Repetition of properties linear function.

Repetition, perception of drawings and teacher’s statements

3. Working with new material: explanation.

Explanation of the meaning of the relationship between function increment and argument increment

Explanation of the geometric meaning of the derivative.

Introduction of new material through verbal explanations using images and visual aids: multimedia presentation with animation.

Perception of explanation, understanding, answering teacher's questions.

Formulating a question to the teacher in case of difficulty.

Perception of new information, its primary understanding and comprehension.

Formulation of questions to the teacher in case of difficulty.

Creating a note.

Formulation of the geometric meaning of the derivative.

Consideration of three cases.

Taking notes, making drawings.

4. Working with new material.

Primary comprehension and application of the studied material, its consolidation.

At what points is the derivative positive?

Negative?

Equal to zero?

Training in finding an algorithm for answering questions according to a schedule.

Understanding, making sense of, and applying new information to solve a problem.

5. Primary comprehension and application of the studied material, its consolidation.

Message of the task conditions.

Recording the conditions of the task.

Formulating a question to the teacher in case of difficulty

6. Application of knowledge: independent work of a teaching nature.

Solve the problem yourself:

Application of acquired knowledge.

Independent work on solving the problem of finding the derivative from a drawing. Discussion and verification of answers in pairs, formulation of a question to the teacher in case of difficulty.

7. Working with new material: explanation.

Deriving the equation of a tangent to the graph of a function at a point.

Detailed explanation deriving the equation of a tangent to the graph of a function at a point using a multimedia presentation for clarity, answering student questions.

Derivation of the tangent equation together with the teacher. Answers to the teacher's questions.

Taking notes, creating a drawing.

8. Working with new material: explanation.

In a dialogue with students, the derivation of an algorithm for finding the equation of a tangent to the graph of a given function at a given point.

In a dialogue with the teacher, derive an algorithm for finding the equation of the tangent to the graph of a given function at a given point.

Note-taking.

Message of the task conditions.

Training in the application of acquired knowledge.

Organizing the search for ways to solve a problem and their implementation. detailed analysis solutions with explanation.

Recording the conditions of the task.

Making assumptions about possible ways to solve the problem when implementing each item of the action plan. Solving the problem together with the teacher.

Recording the solution to the problem and the answer.

9. Application of knowledge: independent work of a teaching nature.

Individual control. Consulting and assistance to students as needed.

Check and explain the solution using a presentation.

Application of acquired knowledge.

Independent work on solving the problem of finding the derivative from a drawing. Discussion and verification of answers in pairs, formulation of a question to the teacher in case of difficulty

10. Homework.

§48, problems 1 and 3, understand the solution and write it down in a notebook, with drawings.

№ 860 (2,4,6,8),

Message homework with comments.

Recording homework.

11. Summing up.

We repeated the definition of the derivative; physical meaning derivative; properties of a linear function.

We learned what the geometric meaning of a derivative is.

We learned how to derive the equation of a tangent to the graph of a given function at a given point.

Correction and clarification of lesson results.

Listing the results of the lesson.

12. Reflection.

1. You found the lesson: a) easy; b) usually; c) difficult.

a) have mastered it completely, I can apply it;

b) have learned it, but find it difficult to apply;

c) didn’t understand.

3. Multimedia presentation in class:

a) helped to master the material; b) did not help master the material;

c) interfered with the assimilation of the material.

Conducting reflection.

Lecture: The concept of the derivative of a function, the geometric meaning of the derivative

Let us consider some function f(x), which will be continuous over the entire interval of consideration. On the interval under consideration, we select the point x 0, as well as the value of the function at this point.

So, let's look at the graph on which we mark our point x 0, as well as the point (x 0 + ∆x). Recall that ∆х is the distance (difference) between two selected points.

It is also worth understanding that each x corresponds to eigenvalue functions y.

The difference between the values ​​of the function at the point x 0 and (x 0 + ∆x) is called the increment of this function: ∆у = f(x 0 + ∆x) - f(x 0).

Let's pay attention to the additional information that is available on the graph - this is the secant, which is called KL, as well as the triangle that it forms with intervals KN and LN.

The angle at which the secant is located is called its angle of inclination and is denoted α. It can be easily determined that the degree measure of the angle LKN is also equal to α.

Now let's remember the ratios in right triangle tgα = LN / KN = ∆у / ∆х.

That is, the tangent of the secant angle is equal to the ratio of the increment of the function to the increment of the argument.

At one time, the derivative is the limit of the ratio of the increment of a function to the increment of the argument on infinitesimal intervals.

The derivative determines the rate at which a function changes over a certain area.

Geometric meaning of derivative

If you find the derivative of any function at a certain point, you can determine the angle at which the tangent to the graph in a given current will be located, relative to the OX axis. Pay attention to the graph - the tangential slope angle is denoted by the letter φ and is determined by the coefficient k in the equation of the straight line: y = kx + b.

That is, we can conclude that the geometric meaning of the derivative is the tangent of the tangent angle at some point of the function.

The derivative of a function is one of difficult topics V school curriculum. Not every graduate will answer the question of what a derivative is.

This article explains in a simple and clear way what a derivative is and why it is needed.. We will not now strive for mathematical rigor in the presentation. The most important thing is to understand the meaning.

Let's remember the definition:

The derivative is the rate of change of a function.

The figure shows graphs of three functions. Which one do you think is growing faster?

The answer is obvious - the third. It has the highest rate of change, that is, the largest derivative.

Here's another example.

Kostya, Grisha and Matvey got jobs at the same time. Let's see how their income changed during the year:

The graph shows everything at once, isn’t it? Kostya’s income more than doubled in six months. And Grisha’s income also increased, but just a little. And Matvey’s income decreased to zero. The starting conditions are the same, but the rate of change of the function, that is derivative, - different. As for Matvey, his income derivative is generally negative.

Intuitively, we easily estimate the rate of change of a function. But how do we do this?

What we're really looking at is how steeply the graph of a function goes up (or down). In other words, how quickly does y change as x changes? Obviously, the same function in different points may have different meaning derivative - that is, it can change faster or slower.

The derivative of a function is denoted .

We'll show you how to find it using a graph.

A graph of some function has been drawn. Let's take a point with an abscissa on it. Let us draw a tangent to the graph of the function at this point. We want to estimate how steeply the graph of a function goes up. A convenient value for this is tangent of the tangent angle.

The derivative of a function at a point is equal to the tangent of the tangent angle drawn to the graph of the function at this point.

Please note that as the angle of inclination of the tangent we take the angle between the tangent and the positive direction of the axis.

Sometimes students ask what a tangent to the graph of a function is. This is a straight line that has a single common point with the graph in this section, and as shown in our figure. It looks like a tangent to a circle.

Let's find it. We remember that the tangent of an acute angle in a right triangle is equal to the ratio of the opposite side to the adjacent side. From the triangle:

We found the derivative using a graph without even knowing the formula of the function. Such problems are often found in the Unified State Examination in mathematics under the number.

There is another important relationship. Recall that the straight line is given by the equation

The quantity in this equation is called slope of a straight line. It is equal to the tangent of the angle of inclination of the straight line to the axis.

.

We get that

Let's remember this formula. It expresses the geometric meaning of the derivative.

The derivative of a function at a point is equal to the slope of the tangent drawn to the graph of the function at that point.

In other words, the derivative is equal to the tangent of the tangent angle.

We have already said that the same function can have different derivatives at different points. Let's see how the derivative is related to the behavior of the function.

Let's draw a graph of some function. Let this function increase in some areas and decrease in others, and at different rates. And let this function have maximum and minimum points.

At a point the function increases. A tangent to the graph drawn at point forms an acute angle; with positive axis direction. This means that the derivative at the point is positive.

At the point our function decreases. The tangent at this point forms an obtuse angle; with positive axis direction. Since the tangent of an obtuse angle is negative, the derivative at the point is negative.

Here's what happens:

If a function is increasing, its derivative is positive.

If it decreases, its derivative is negative.

What will happen at the maximum and minimum points? We see that at the points (maximum point) and (minimum point) the tangent is horizontal. Therefore, the tangent of the tangent at these points is zero, and the derivative is also zero.

Point - maximum point. At this point, the increase in the function is replaced by a decrease. Consequently, the sign of the derivative changes at the point from “plus” to “minus”.

At the point - the minimum point - the derivative is also zero, but its sign changes from “minus” to “plus”.

Conclusion: using the derivative we can find out everything that interests us about the behavior of a function.

If the derivative is positive, then the function increases.

If the derivative is negative, then the function decreases.

At the maximum point, the derivative is zero and changes sign from “plus” to “minus”.

At the minimum point, the derivative is also zero and changes sign from “minus” to “plus”.

Let's write these conclusions in the form of a table:

Let's make two small clarifications. You will need one of them when solving the problem. Another - in the first year, with a more serious study of functions and derivatives.

It is possible that the derivative of a function at some point is equal to zero, but the function has neither a maximum nor a minimum at this point. This is the so-called :

At a point, the tangent to the graph is horizontal and the derivative is zero. However, before the point the function increased - and after the point it continues to increase. The sign of the derivative does not change - it remains positive as it was.

It also happens that at the point of maximum or minimum the derivative does not exist. On the graph, this corresponds to a sharp break, when it is impossible to draw a tangent at a given point.

How to find the derivative if the function is given not by a graph, but by a formula? In this case it applies