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

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

But from my point of view, the following approach is more pragmatic: first of all, it is advisable to UNDERSTAND WELL function limit, and especially infinitesimals. The fact is that the definition of the derivative is based on the concept of a limit, which is poorly considered in the school course. That is why a significant part of young consumers of granite knowledge poorly penetrate into the very essence of the derivative. Thus, if you are not well versed in differential calculus, or the wise brain has successfully rid itself of this baggage over the years, please start with function limits. At the same time master / remember their decision.

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

I recommend starting the materials on this page after reading the article. The simplest problems with a derivative, where, in particular, the problem of the tangent to the graph of a function is considered. But it can be delayed. 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 intervals of increase/decrease and extremums functions. Moreover, he was in the subject for quite a long time " Functions and Graphs”, until I decided to put it in 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 tutorials lead to the concept of a derivative with the help of some practical problems, and I also came up with an interesting example. Imagine that we have to travel to a city that can be reached in different ways. We immediately discard the curved winding paths, and we will consider only straight lines. However, straight-line directions are also different: you can get to the city along a flat autobahn. Or on a hilly highway - up and down, up and down. Another road goes only uphill, and another one goes downhill all the time. Thrill-seekers will choose a route through the gorge with a steep cliff and a steep ascent.

But whatever your preferences, it is desirable to know the area, or at least have a topographical map of it. What if there is no such information? After all, you can choose, for example, a flat path, but as a result, stumble upon a ski slope with funny Finns. Not the fact that the navigator and even a satellite image will give reliable data. Therefore, it would be nice to formalize the relief of the path by means of mathematics.

Consider some road (side view):

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

What are the features of this chart?

At intervals function increases, that is, each of its next value more the previous one. Roughly speaking, the schedule goes upwards(we climb the hill). And on the interval the function decreasing- each next value less the previous one, and our schedule goes top down(going 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 on which the value will be the largest (highest). At the same point, minimum, and exists such its neighborhood, in which the value is the smallest (lowest).

More rigorous terminology and definitions will be considered in the lesson. about the extrema of the function, but for now let's study one more important feature: on the intervals the function is increasing, but it is increasing at different speeds. And the first thing that catches your eye is that the chart soars up on the interval much more cool than on the interval. Is it possible to measure the steepness of the road using mathematical tools?

Function change rate

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

1) Let's look at the leftmost point: bypassing the distance , we climb the slope to a height (green line). The value is called function increment, and in this case this increment is positive (the difference of values ​​along the axis is greater than zero). Let's make the ratio , which will be the measure of the steepness of our road. Obviously, 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 applies to the function's increment symbol.

Let's explore the nature of the resulting fraction more meaningful. Suppose initially we are at a height of 20 meters (in the left black dot). Having overcome the distance of meters (left red line), we will be at a height of 60 meters. Then the increment of the function will be meters (green line) and: . In this way, on every meter this section of the road height increases average by 4 meters…did you forget your climbing equipment? =) In other words, the constructed ratio characterizes the AVERAGE RATE OF CHANGE (in this case, growth) of the function.

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

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

3) A little adventure on the mountainside. Let's look at the top black dot located on the y-axis. Let's assume that this is a mark of 50 meters. Again we overcome the distance, as a result of which we find ourselves lower - at the level of 30 meters. Since the movement has been made top down(in the "opposite" direction of the axis), then the final the increment of the function (height) will be negative: meters (brown line in the drawing). And in this case we are talking about decay rate features: , that is, for each meter of the path of this section, the height decreases average by 2 meters. Take care of clothes on the fifth point.

Now let's ask the question: what is the best value of "measuring standard" to use? It is clear that 10 meters is very rough. A good dozen bumps can easily fit on them. Why are there bumps, there may be a deep gorge below, and after a few meters - its other side with a further steep ascent. Thus, with a ten-meter one, we will not get an intelligible characteristic of such sections of the path through the ratio.

From the above discussion, the following conclusion follows: the smaller the value, the more accurately we will describe the relief of the road. Moreover, the following facts are true:

– For any lifting points you can choose a value (albeit a very small one) that fits within the boundaries of one or another rise. And 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. Therefore, the corresponding increase in height is unequivocally negative, and the inequality will correctly show the decrease in the function at each point of the given interval.

– Of particular interest is the case when the rate of change of the function is zero: . First, a zero height increment () is a sign of an even path. And secondly, there are other curious situations, examples of which you see in the figure. Imagine that fate has taken 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, then the change in height will be negligible, and we can say that the rate of change of the function is actually zero. The same pattern is observed at points.

Thus, we have approached an amazing opportunity to perfectly accurately characterize the rate of change of a function. After all, mathematical analysis allows us 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 tell us about all flats, uphills, downhills, peaks, lowlands, as well as the rate of increase / decrease at each point of the path?

What is a derivative? Definition of a derivative.
The geometric meaning of the derivative and differential

Please read thoughtfully and not too quickly - the material is simple and accessible to everyone! It's okay if in some places something seems not 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 qualitatively 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 will replace it with:

What have we come to? And we came to the conclusion that for a function according to the law is aligned other function, which is called derivative function(or simply derivative).

The derivative characterizes rate of change functions . How? The thought goes like a red thread from the very beginning of the article. Consider some point domains 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 if very small) containing the 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 “from top to bottom”).

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

Some semantics. What does the verb "differentiate" mean in a broad sense? To differentiate means to single out a feature. Differentiating the function , we "select" the rate of its change in the form of a derivative of the function . And what, by the way, is meant by the word "derivative"? Function happened from the function.

The terms very successfully interpret the mechanical meaning of the derivative :
Let's consider the law of change of the body's coordinates, which depends on time, and the function of the speed of motion of the given body. The function characterizes the rate of change of the body coordinate, therefore it is the first derivative of the function with respect to time: . If the concept of “body motion” did not exist in nature, then there would not exist derivative concept of "velocity".

The acceleration of a body is the rate of change of speed, therefore: . If the original concepts of “body movement” and “body movement speed” did not exist in nature, then there would be no derivative the concept of acceleration of a body.

Synopsis of an open lesson by a teacher at Pedagogical College No. 4 of St. Petersburg

Martusevich Tatyana Olegovna

Date: 12/29/2014.

Topic: The geometric meaning of the derivative.

Lesson type: learning new material.

Teaching methods: visual, partly exploratory.

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 tangent equation and teach how to find it.

Educational tasks:

To achieve an understanding of the geometric meaning of the derivative; derivation of the tangent equation; learn how to solve basic problems;

to provide a repetition of the material on the topic "Definition of a derivative";

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

Development tasks:

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

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

Educational tasks:

to promote the education of interest in mathematics;

education of activity, mobility, ability to communicate.

Lesson type - a combined lesson using ICT.

Equipment – multimedia installation, presentationMicrosoftpowerpoint.

Lesson stage

Time

Teacher activity

Student activity

1. Organizational moment.

Message about the topic and purpose of the lesson.

Topic: The geometric meaning of the derivative.

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 tangent equation and teach how to find it.

Preparing students for work in the classroom.

Preparation for work in class.

Awareness of the topic and purpose of the lesson.

Note-taking.

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

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

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

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

Finding the derivative of these functions by formulas.

Repetition of the properties of a linear function.

Repetition, perception of drawings and teacher's statements

3. Working with new material: explanation.

Explanation of the meaning of the ratio of function increment to 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, answers to teacher's questions.

Formulation of 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.

Create an outline.

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 point is the derivative positive?

Negative?

Equal to zero?

Learning to search for an algorithm for answers to the questions posed by the schedule.

Understanding and comprehending and applying new information to solve a problem.

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

Task condition message.

Recording a task condition.

Formulation of 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 of the figure. Discussion and verification of answers in pairs, formulating a question to the teacher in case of difficulty.

7. Working with new material: explanation.

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

A detailed explanation of the derivation of the equation of the tangent to the function graph at a point, using as a visual aid in the form of a multimedia presentation, answers to students' questions.

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

Sketching, drawing.

8. Working with new material: explanation.

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

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

Note-taking.

Task condition message.

Training in the application of acquired knowledge.

Organization of the search for ways to solve the problem and their implementation. detailed analysis of the solution with an explanation.

Recording a task condition.

Making assumptions about possible ways to solve the problem in the implementation of each item of the action plan. Problem solving together with the teacher.

Recording the solution of the problem and the answer.

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

Individual control. Advice and assistance to students as needed.

Verification and explanation of the solution using the presentation.

Application of acquired knowledge.

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

10. Homework.

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

№ 860 (2,4,6,8),

Homework message with comments.

Recording homework.

11. Summing up.

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

We learned what the geometric meaning of the derivative is.

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

Correction and clarification of the results of the lesson.

Enumeration of the results of the lesson.

12. Reflection.

1. Did you have a lesson: a) easy; b) usually; c) difficult.

a) learned (a) completely, I can apply;

b) learned (a), but find it difficult to apply;

c) didn't get it.

3. Multimedia presentation in the lesson:

a) helped the assimilation of the material; b) did not help the assimilation of the material;

c) interfered with the assimilation of the material.

Conducting reflection.

Job type: 7

Condition

The line y=3x+2 is tangent to the graph of the function y=-12x^2+bx-10. Find b , given that the abscissa of the touch point is less than zero.

Show Solution

Solution

Let x_0 be the abscissa of the point on the graph of the function y=-12x^2+bx-10 through which the tangent to this graph passes.

The value of the derivative at the point x_0 is equal to the slope of the tangent, i.e. y"(x_0)=-24x_0+b=3. On the other hand, the tangent point belongs to both the graph of the function and the tangent, i.e. -12x_0^2+bx_0-10= 3x_0 + 2. We get a system of equations \begin(cases) -24x_0+b=3,\\-12x_0^2+bx_0-10=3x_0+2. \end(cases)

Solving this system, we get x_0^2=1, which means either x_0=-1 or x_0=1. According to the condition of the abscissa, the touch points are less than zero, therefore x_0=-1, then b=3+24x_0=-21.

Answer

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

The line y=-3x+4 is parallel to the tangent to the graph of the function y=-x^2+5x-7. Find the abscissa of the point of contact.

Show Solution

Solution

The slope of the line to the graph of the function y=-x^2+5x-7 at an arbitrary point x_0 is y"(x_0). But y"=-2x+5, so y"(x_0)=-2x_0+5. Angular the coefficient of the line y=-3x+4 specified in the condition is -3.Parallel lines have the same slope coefficients.Therefore, we find such a value x_0 that =-2x_0 +5=-3.

We get: x_0 = 4.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

Show Solution

Solution

From the figure, we determine that the tangent passes through the points A(-6; 2) and B(-1; 1). Denote by C(-6; 1) the point of intersection of the lines x=-6 and y=1, and by \alpha the angle ABC (it can be seen in the figure that it is sharp). Then the line AB forms an obtuse angle \pi -\alpha with the positive direction of the Ox axis.

As you know, tg(\pi -\alpha) will be the value of the derivative of the function f(x) at the point x_0. notice, that tg \alpha =\frac(AC)(CB)=\frac(2-1)(-1-(-6))=\frac15. From here, by the reduction formulas, we obtain: tg(\pi -\alpha) =-tg \alpha =-\frac15=-0.2.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

The line y=-2x-4 is tangent to the graph of the function y=16x^2+bx+12. Find b , given that the abscissa of the touch point is greater than zero.

Show Solution

Solution

Let x_0 be the abscissa of the point on the graph of the function y=16x^2+bx+12 through which

is tangent to this graph.

The value of the derivative at the point x_0 is equal to the slope of the tangent, i.e. y "(x_0)=32x_0+b=-2. On the other hand, the tangent point belongs to both the graph of the function and the tangent, i.e. 16x_0^2+bx_0+12=- 2x_0-4 We get a system of equations \begin(cases) 32x_0+b=-2,\\16x_0^2+bx_0+12=-2x_0-4. \end(cases)

Solving the system, we get x_0^2=1, which means either x_0=-1 or x_0=1. According to the condition of the abscissa, the touch points are greater than zero, therefore x_0=1, then b=-2-32x_0=-34.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

The figure shows a graph of the function y=f(x) defined on the interval (-2; 8). Determine the number of points where the tangent to the graph of the function is parallel to the straight line y=6.

Show Solution

Solution

The line y=6 is parallel to the Ox axis. Therefore, we find such points at which the tangent to the function graph is parallel to the Ox axis. On this chart, such points are extremum points (maximum or minimum points). As you can see, there are 4 extremum points.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

The line y=4x-6 is parallel to the tangent to the graph of the function y=x^2-4x+9. Find the abscissa of the point of contact.

Show Solution

Solution

The slope of the tangent to the graph of the function y \u003d x ^ 2-4x + 9 at an arbitrary point x_0 is y "(x_0). But y" \u003d 2x-4, which means y "(x_0) \u003d 2x_0-4. The slope of the tangent y \u003d 4x-7 specified in the condition is equal to 4. Parallel lines have the same slopes. Therefore, we find such a value x_0 that 2x_0-4 \u003d 4. We get: x_0 \u003d 4.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

The figure shows the graph of the function y=f(x) and the tangent to it at the point with the abscissa x_0. Find the value of the derivative of the function f(x) at the point x_0.

Show Solution

Solution

From the figure, we determine that the tangent passes through the points A(1; 1) and B(5; 4). Denote by C(5; 1) the point of intersection of the lines x=5 and y=1, and by \alpha the angle BAC (it can be seen in the figure that it is acute). Then the line AB forms an angle \alpha with the positive direction of the Ox axis.

To find out the geometric value of the derivative, consider the graph of the function y = f(x). Take an arbitrary point M with coordinates (x, y) and a point N close to it (x + $\Delta $x, y + $\Delta $y). Let us draw the ordinates $\overline(M_(1) M)$ and $\overline(N_(1) N)$, and draw a line parallel to the OX axis from the point M.

The ratio $\frac(\Delta y)(\Delta x) $ is the tangent of the angle $\alpha $1 formed by the secant MN with the positive direction of the OX axis. As $\Delta $x tends to zero, point N will approach M, and the tangent MT to the curve at point M will become the limiting position of the secant MN. Thus, the derivative f`(x) is equal to the tangent of the angle $\alpha $ formed by the tangent to curve at the point M (x, y) with a positive direction to the OX axis - the slope of the tangent (Fig. 1).

Figure 1. Graph of a function

When calculating the values ​​using formulas (1), it is important not to make a mistake in the signs, because increment can be negative.

The point N lying on the curve can approach M from any side. So, if in Figure 1, the tangent is given the opposite direction, the angle $\alpha $ will change by $\pi $, which will significantly affect the tangent of the angle and, accordingly, the slope.

Conclusion

It follows that the existence of the derivative is connected with the existence of a tangent to the curve y = f(x), and the slope -- tg $\alpha $ = f`(x) is finite. Therefore, the tangent must not be parallel to the OY axis, otherwise $\alpha $ = $\pi $/2, and the tangent of the angle will be infinite.

At some points, a continuous curve may not have a tangent or have a tangent parallel to the OY axis (Fig. 2). Then the function cannot have a derivative in these values. There can be any number of such points on the function curve.

Figure 2. Exceptional points of the curve

Consider Figure 2. Let $\Delta $x tend to zero from negative or positive values:

\[\Delta x\to -0\begin(array)(cc) () & (\Delta x\to +0) \end(array)\]

If in this case relations (1) have a finite aisle, it is denoted as:

In the first case, the derivative on the left, in the second, the derivative on the right.

The existence of a limit speaks of the equivalence and equality of the left and right derivatives:

If the left and right derivatives are not equal, then at this point there are tangents that are not parallel to OY (point M1, Fig. 2). At points M2, M3, relations (1) tend to infinity.

For N points to the left of M2, $\Delta $x $

To the right of $M_2$, $\Delta $x $>$ 0, but the expression is also f(x + $\Delta $x) -- f(x) $

For point $M_3$ on the left $\Delta $x $$ 0 and f(x + $\Delta $x) -- f(x) $>$ 0, i.e. expressions (1) are both positive on the left and right and tend to +$\infty $ both when $\Delta $x approaches -0 and +0.

The case of the absence of a derivative at specific points of the line (x = c) is shown in Figure 3.

Figure 3. Absence of derivatives

Example 1

Figure 4 shows the graph of the function and the tangent to the graph at the point with the abscissa $x_0$. Find the value of the derivative of the function in the abscissa.

Solution. The derivative at a point is equal to the ratio of the increment of the function to the increment of the argument. Let's choose two points with integer coordinates on the tangent. Let, for example, these be points F (-3.2) and C (-2.4).