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 simply and clearly explains what a derivative is and why it is needed.. We will not now strive for mathematical rigor of presentation. The most important thing is to understand the meaning.

Let's remember the definition:

The derivative is the rate of change of the function.

The figure shows graphs of three functions. Which one do you think grows the fastest?

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

Here is another example.

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

You can see everything on the chart right away, right? Kostya's income has more than doubled in six months. And Grisha's income also increased, but just a little bit. And Matthew's income decreased to zero. The starting conditions are the same, but the rate of change of the function, i.e. derivative, - different. As for Matvey, the derivative of his income is generally negative.

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

What we are really looking at is how steeply the graph of the function goes up (or down). In other words, how fast y changes with x. Obviously, the same function at different points can have different meaning derivative - that is, it can change faster or slower.

The derivative of a function is denoted by .

Let's show how to find using the graph.

A graph of some function is drawn. Take a point on it with an abscissa. Draw a tangent to the graph of the function at this point. We want to evaluate how steeply the graph of the function goes up. A handy value for this is tangent of the slope of the tangent.

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

Please note - 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 is the tangent to the graph of a function. This is a straight line that has the only common point with the graph in this section, moreover, as shown in our figure. It looks like a tangent to a circle.

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

We found the derivative using the graph without even knowing the formula of the function. Such tasks are often found in the exam in mathematics under the number.

There is another important correlation. 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 slope of the tangent.

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 is increasing. The tangent to the graph, drawn at the point, forms an acute angle with the positive direction of the axis. So the derivative is positive at the point.

At the point, our function is decreasing. The tangent at this point forms an obtuse angle with the positive direction of the axis. 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.

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

The point is the maximum point. At this point, the increase of 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 equal to zero, but its sign changes from "minus" to "plus".

Conclusion: with the help of the derivative, you can find out everything that interests us about the behavior of the function.

If the derivative is positive, then the function is increasing.

If the derivative is negative, then the function is decreasing.

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.

We write these findings in the form of a table:

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

A case is possible when 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 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 has remained 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.

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

Dear friends! The group of tasks related to the derivative includes tasks - in the condition, the graph of the function is given, several points on this graph and the question is:

At what point is the value of the derivative the largest (smallest)?

Let's briefly repeat:

The derivative at the point is equal to the slope of the tangent passing throughthis point on the graph.

Atthe global coefficient of the tangent, in turn, is equal to the tangent of the slope of this tangent.

*This refers to the angle between the tangent and the x-axis.

1. On intervals of increasing function, the derivative has positive value.

2. On the intervals of its decrease, the derivative has a negative value.

Consider the following sketch:

At points 1,2,4, the derivative of the function has a negative value, since these points belong to the decreasing intervals.

At points 3,5,6, the derivative of the function has a positive value, since these points belong to the intervals of increase.

As you can see, everything is clear with the value of the derivative, that is, it is not difficult to determine what sign it has (positive or negative) at a certain point on the graph.

Moreover, if we mentally construct tangents at these points, we will see that the lines passing through points 3, 5 and 6 form angles with the oX axis lying in the range from 0 to 90 °, and the lines passing through points 1, 2 and 4 form with the oX axis, angles ranging from 90 o to 180 o.

* The relationship is clear: tangents passing through points belonging to intervals of increasing functions form acute angles with the oX axis, tangents passing through points belonging to intervals of decreasing functions form obtuse angles with the oX axis.

Now the important question!

How does the value of the derivative change? After all, the tangent at different points of the graph of a continuous function forms different angles, depending on which point of the graph it passes through.

*Or, speaking plain language, the tangent is located, as it were, “more horizontally” or “more vertically”. Look:

Straight lines form angles with the oX axis ranging from 0 to 90 o

Straight lines form angles with the oX axis ranging from 90 o to 180 o

So if there are any questions:

- at which of the given points on the graph does the value of the derivative have the smallest value?

- at which of the given points on the graph does the value of the derivative have the greatest value?

then for the answer it is necessary to understand how the value of the tangent of the angle of the tangent changes in the range from 0 to 180 o.

*As already mentioned, the value of the derivative of the function at a point is equal to the tangent of the slope of the tangent to the x-axis.

The tangent value changes as follows:

When the slope of the straight line changes from 0 o to 90 o, the value of the tangent, and hence the derivative, changes from 0 to +∞, respectively;

When the slope of the straight line changes from 90 o to 180 o, the value of the tangent, and hence the derivative, changes accordingly –∞ to 0.

This can be clearly seen from the graph of the tangent function:

In simple terms:

When the angle of inclination of the tangent is from 0 o to 90 o

The closer it is to 0 o, the greater the value of the derivative will be close to zero (on the positive side).

The closer the angle is to 90°, the more the value of the derivative will increase towards +∞.

When the angle of inclination of the tangent is from 90 o to 180 o

The closer it is to 90 o, the more the value of the derivative will decrease towards –∞.

The closer the angle is to 180 o, the greater the value of the derivative will be close to zero (on the negative side).

317543. The figure shows a graph of the function y = f(x) and marked points–2, –1, 1, 2. At which of these points is the value of the derivative greatest? Please indicate this point in your answer.

We have four points: two of them belong to the intervals on which the function decreases (these are points –1 and 1) and two to the intervals on which the function increases (these are points –2 and 2).

We can immediately conclude that at points -1 and 1 the derivative has a negative value, at points -2 and 2 it has a positive value. Therefore, in this case, it is necessary to analyze points -2 and 2 and determine which of them will have the largest value. Let's construct tangents passing through the indicated points:

The value of the tangent of the angle between line a and the abscissa axis will be greater than the value of the tangent of the angle between line b and this axis. This means that the value of the derivative at the point -2 will be the largest.

Let's answer the following question: at which of the points -2, -1, 1 or 2 is the value of the derivative the largest negative? Please indicate this point in your answer.

The derivative will have a negative value at the points belonging to the decreasing intervals, so consider the points -2 and 1. Let's construct the tangents passing through them:

We see that the obtuse angle between the straight line b and the oX axis is "closer" to 180 O , so its tangent will be greater than the tangent of the angle formed by the straight line a and the x-axis.

Thus, at the point x = 1, the value of the derivative will be the largest negative.

317544. The figure shows a graph of the function y = f(x) and marked points–2, –1, 1, 4. At which of these points is the value of the derivative the smallest? Please indicate this point in your answer.

We have four points: two of them belong to the intervals on which the function decreases (these are points –1 and 4) and two to the intervals on which the function increases (these are points –2 and 1).

We can immediately conclude that at points -1 and 4 the derivative has a negative value, at points -2 and 1 it has a positive value. Therefore, in this case, it is necessary to analyze points –1 and 4 and determine which of them will have the smallest value. Let's construct tangents passing through the indicated points:

The value of the tangent of the angle between line a and the abscissa axis will be greater than the value of the tangent of the angle between line b and this axis. This means that the value of the derivative at the point x = 4 will be the smallest.

Answer: 4

I hope I didn't "overload" you with the amount of writing. In fact, everything is very simple, one has only to understand the properties of the derivative, its geometric meaning and how the value of the tangent of the angle changes from 0 to 180 o.

1. First, determine the signs of the derivative at these points (+ or -) and select the necessary points (depending on the question posed).

2. Construct tangents at these points.

3. Using the tangesoid plot, schematically mark the corners and displayAlexander.

P.S: I would be grateful if you tell about the site in social networks.

In the interim ( A,b), A X- is a randomly chosen point of the given interval. Let's give an argument X incrementΔx (positive or negative).

The function y \u003d f (x) will receive an increment Δy equal to:

Δy = f(x + Δx)-f(x).

For infinitely small Δх incrementΔy is also infinitely small.

For example:

Consider the solution of a derivative function using an example free fall body.

Since t 2 \u003d t l + Δt, then

.

Calculating the limit, we find:

The notation t 1 is introduced to emphasize the constancy of t when calculating the limit of a function. Since t 1 is an arbitrary value of time, index 1 can be dropped; then we get:

It can be seen that the speed v, like the way s, There is function time. Function type v depends entirely on the type of function s, so the function s sort of "produces" a function v. Hence the name " derivative function».

Consider another example.

Find the value of the derivative of a function:

y = x 2 at x = 7.

Solution. At x = 7 we have y=7 2=49. Let's give an argument X increment Δ X. The argument becomes 7 + Δ X, and the function will get the value (7 + Δ x) 2.

Sergei Nikiforov

If the derivative of a function is of constant sign on an interval, and the function itself is continuous on its boundaries, then the boundary points are attached to both increasing and decreasing intervals, which fully corresponds to the definition of increasing and decreasing functions.

Farit Yamaev 26.10.2016 18:50

Hello. How (on what basis) can it be argued that at the point where the derivative is equal to zero, the function increases. Give reasons. Otherwise, it's just someone's whim. By what theorem? And also proof. Thank you.

Support

The value of the derivative at a point is not directly related to the increase of the function on the interval. Consider, for example, functions - they all increase on the segment

Vladlen Pisarev 02.11.2016 22:21

If a function is increasing on the interval (a;b) and is defined and continuous at the points a and b, then it is increasing on the segment . Those. the point x=2 is included in the given interval.

Although, as a rule, increase and decrease is considered not on a segment, but on an interval.

But at the very point x=2, the function has a local minimum. And how to explain to children that when they are looking for points of increase (decrease), then we do not count the points of local extremum, but they enter into the intervals of increase (decrease).

Considering that the first part of the exam For " middle group kindergarten", then perhaps such nuances are too much.

Separately, Thanks a lot for "I will decide the exam" for all employees - an excellent benefit.

Sergei Nikiforov

A simple explanation can be obtained if we start from the definition of an increasing / decreasing function. Let me remind you that it sounds like this: a function is called increasing/decreasing on the interval if the larger argument of the function corresponds to a larger/smaller value of the function. Such a definition does not use the concept of a derivative in any way, so questions about the points where the derivative vanishes cannot arise.

Irina Ishmakova 20.11.2017 11:46

Good afternoon. Here in the comments I see beliefs that borders should be included. Let's say I agree with this. But look, please, at your solution to problem 7089. When specifying intervals of increase, the boundaries are not included there. And that affects the response. Those. the solutions of tasks 6429 and 7089 contradict each other. Please clarify this situation.

Alexander Ivanov

Tasks 6429 and 7089 have completely different questions.

In one, there are intervals of increase, and in the other, there are intervals with a positive derivative.

There is no contradiction.

Extrema are included in the intervals of increase and decrease, but the points at which the derivative is equal to zero are not included in the intervals at which the derivative is positive.

A Z 28.01.2019 19:09

Colleagues, there is a concept of increasing at a point

(see Fichtenholtz for example)

and your understanding of the increase at the point x=2 is contrary to the classical definition.

Increasing and decreasing is a process and I would like to adhere to this principle.

In any interval that contains the point x=2, the function is not increasing. Therefore, the inclusion of the given point x=2 is a special process.

Usually, to avoid confusion, the inclusion of the ends of the intervals is said separately.

Alexander Ivanov

The function y=f(x) is called increasing on some interval if the larger value of the argument from this interval corresponds to the larger value of the function.

At the point x = 2, the function is differentiable, and on the interval (2; 6) the derivative is positive, which means that on the interval )