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 smooth highway. Or along a hilly highway - up and down, up and down. Another road goes only uphill, and another one goes downhill all the time. Extreme enthusiasts will choose a route through a gorge with a steep cliff and a steep climb.

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

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

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

What 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'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 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 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”.

Calculation of the derivative is often found in Unified State Examination tasks. This page contains a list of formulas for finding derivatives.

Rules of differentiation

1. (k⋅ f(x))′=k⋅ f ′(x).
2. (f(x)+g(x))′=f′(x)+g′(x).
3. (f(x)⋅ g(x))′=f′(x)⋅ g(x)+f(x)⋅ g′(x).
4. Derivative of a complex function. If y=F(u), and u=u(x), then the function y=f(x)=F(u(x)) is called a complex function of x. Equal to y′(x)=Fu′⋅ ux′.
5. Derivative of an implicit function. The function y=f(x) is called an implicit function defined by the relation F(x,y)=0 if F(x,f(x))≡0.
6. Derivative of the inverse function. If g(f(x))=x, then the function g(x) is called the inverse function of the function y=f(x).
7. Derivative of a parametrically defined function. Let x and y be specified as functions of the variable t: x=x(t), y=y(t). They say that y=y(x) is a parametrically defined function on the interval x∈ (a;b), if on this interval the equation x=x(t) can be expressed as t=t(x) and the function y=y( t(x))=y(x).
8. Derivative of a power-exponential function. Found by taking logarithms to the base of the natural logarithm.

We advise you to save the link, as this table may be needed many times.

When solving various problems of geometry, mechanics, physics and other branches of knowledge, the need arose using the same analytical process from this function y=f(x) get a new function called derivative function(or simply derivative) of a given function f(x) and is designated by the symbol

The process by which from a given function f(x) get a new feature f" (x), called differentiation and it consists of the following three steps: 1) give the argument x increment  x and determine the corresponding increment of the function  y = f(x+ x) -f(x);

2) make up a relation x 3) counting  x constant and
0, we find f" (x), which we denote by x, as if emphasizing that the resulting function depends only on the value , at which we go to the limit.: Definition Derivative y " =f " (x) given function y=f(x) for a given x
is called the limit of the ratio of the increment of a function to the increment of the argument, provided that the increment of the argument tends to zero, if, of course, this limit exists, i.e. finite.

Thus, x, or Note that if at some value, for example when
x=a  x, attitude f(x) at Note that if at some value0 does not tend to the finite limit, then in this case they say that the function Note that if at some value at Note that if at some value.

(or at the point

) has no derivative or is not differentiable at the point

f(x)

Let's consider an arbitrary straight line passing through a point on the graph of a function - point A(x 0, f (x 0)) and intersecting the graph at some point B(x;f(x)). Such a line (AB) is called a secant. From ∆ABC: ​​AC = ∆x;

BC =∆у; tgβ=∆y/∆x.

Since AC || Ox, then ALO = BAC = β (as corresponding for parallel). But ALO is the angle of inclination of the secant AB to the positive direction of the Ox axis. This means that tanβ = k is the angular coefficient of straight line AB.

Now we will reduce ∆x, i.e. ∆х→ 0. In this case, point B will approach point A according to the graph, and secant AB will rotate. The limiting position of the secant AB at ∆x→ 0 will be a straight line (a), called the tangent to the graph of the function y = f (x) at point A.
If we go to the limit as ∆x → 0 in the equality tgβ =∆y/∆x, we get
ortg =f "(x 0), since
-angle of inclination of the tangent to the positive direction of the Ox axis

, by definition of a derivative. But tg = k is the angular coefficient of the tangent, which means k = tg = f "(x 0).

So, the geometric meaning of the derivative is as follows: 0 Derivative of a function at point x 0 .

equal to the slope of the tangent to the graph of the function drawn at the point with the abscissa x

3. Physical meaning of the derivative.

Consider the movement of a point along a straight line. Let the coordinate of a point at any time x(t) be given. It is known (from a physics course) that the average speed over a period of time is equal to the ratio of the distance traveled during this period of time to the time, i.e.

Vav = ∆x/∆t. Let's go to the limit in the last equality as ∆t → 0.

lim Vav (t) = (t 0) - instantaneous speed at time t 0, ∆t → 0.

and lim = ∆x/∆t = x"(t 0) (by definition of derivative).

So, (t) =x"(t).The physical meaning of the derivative is as follows: derivative of the function = y(xfx 0 ) at pointyis the rate of change of the functionx 0

(x) at point

The derivative is used in physics to find velocity from a known function of coordinates versus time, acceleration from a known function of velocity versus time.

(t) = x"(t) - speed,

a(f) = "(t) - acceleration, or

If the law of motion of a material point in a circle is known, then one can find the angular velocity and angular acceleration during rotational motion:

φ = φ(t) - change in angle over time,

ω = φ"(t) - angular velocity,

ε = φ"(t) - angular acceleration, or ε = φ"(t).

If the law of mass distribution of an inhomogeneous rod is known, then the linear density of the inhomogeneous rod can be found:

m = m(x) - mass,

x  , l - length of the rod,

Using the derivative, problems from the theory of elasticity and harmonic vibrations are solved. So, according to Hooke's law

F = -kx, x – variable coordinate, k – spring elasticity coefficient. Putting ω 2 =k/m, we obtain the differential equation of the spring pendulum x"(t) + ω 2 x(t) = 0,

where ω = √k/√m oscillation frequency (l/c), k - spring stiffness (H/m).

An equation of the form y" + ω 2 y = 0 is called the equation of harmonic oscillations (mechanical, electrical, electromagnetic). The solution to such equations is the function

y = Asin(ωt + φ 0) or y = Acos(ωt + φ 0), where

A - amplitude of oscillations, ω - cyclic frequency,

φ 0 - initial phase.

Application

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The derivative is the most important concept in mathematical analysis. It characterizes the change in the function of the argument x at some point. Moreover, the derivative itself is a function of the argument x

Derivative of a function at a point is the limit (if it exists and is finite) of the ratio of the increment of the function to the increment of the argument, provided that the latter tends to zero.

The most commonly used are the following derivative notation :

Example 1. Taking advantage definition of derivative, find the derivative of the function

Solution. From the definition of the derivative the following scheme for its calculation follows.

Let's give the argument an increment (delta) and find the increment of the function:

Let's find the ratio of the function increment to the argument increment:

Let us calculate the limit of this ratio provided that the increment of the argument tends to zero, that is, the derivative required in the problem statement:

Physical meaning of the derivative

TO concept of derivative led to Galileo Galilei's study of the law of free fall of bodies, and in a broader sense - the problem of the instantaneous speed of non-uniform rectilinear motion of a point.

Let the pebble be lifted and then released from rest. Path s traversed in time t, is a function of time, that is. s = s(t). If the law of motion of a point is given, then the average speed for any period of time can be determined. Let at the moment of time the pebble be in the position A, and at the moment - in position B. Over a period of time (from t to ) point has passed the path . Therefore, the average speed of movement over this period of time, which we denote by , is

.

However, the motion of a freely falling body is clearly uneven. Speed v the fall is constantly increasing. And the average speed is no longer enough to characterize the speed of movement on various sections of the route. The shorter the time period, the more accurate this characteristic is. Therefore, the following concept is introduced: the instantaneous speed of rectilinear motion (or the speed at a given moment in time t) is called the average speed limit at:

(provided that this limit exists and is finite).

So it turns out that the instantaneous speed is the limit of the ratio of the increment of the function s(t) to the increment of the argument t at This is the derivative, which in general form is written as follows:.

.

The solution to the indicated problem is physical meaning of derivative . So, the derivative of the function y=f(x) at point x is called the limit (if it exists and is finite) of the increment of a function to the increment of the argument, provided that the latter tends to zero.

Example 2. Find the derivative of a function

Solution. From the definition of the derivative, the following scheme for its calculation follows.

Step 1. Let's increment the argument and find

Step 2. Find the increment of the function:

Step 3. Find the ratio of the function increment to the argument increment:

Step 4. Calculate the limit of this ratio at , that is, the derivative:

Geometric meaning of derivative

Let the function be defined on an interval and let the point M on the function graph corresponds to the value of the argument, and the point R– meaning. Let's draw through the points M And R straight line and call it secant. Let us denote by the angle between the secant and the axis. Obviously, this angle depends on .

If exists

passing through the point is called the limiting position of the secant MR at (or at ).

Tangent to the graph of a function at a point M called the limit position of the secant MR at , or, which is the same at .

From the definition it follows that for the existence of a tangent it is sufficient that there is a limit

,

and the limit is equal to the angle of inclination of the tangent to the axis.

Now let's give a precise definition of a tangent.

Tangent to the graph of a function at a point is a straight line passing through the point and having a slope, i.e. straight line whose equation

From this definition it follows that derivative of a function is equal to the slope of the tangent to the graph of this function at the point with the abscissa x. This is the geometric meaning of the derivative.