Final work V Unified State Examination form for 11th graders it necessarily contains tasks on calculating limits, intervals of decreasing and increasing derivatives of a function, searching for extremum points and constructing graphs. Good knowledge This topic allows you to correctly answer several exam questions and not experience difficulties in further professional training.
Fundamentals of differential calculus - one of the main topics of mathematics modern school. She studies the use of the derivative to study the dependencies of variables - it is through the derivative that one can analyze the increase and decrease of a function without resorting to a drawing.
Comprehensive preparation of graduates for passing the Unified State Exam on educational portal“Shkolkovo” will help you deeply understand the principles of differentiation - understand the theory in detail, study examples of solutions typical tasks and try your hand at independent work. We will help you close knowledge gaps – clarify your understanding of lexical concepts topics and dependencies of quantities. Students will be able to review how to find intervals of monotonicity, which means the derivative of a function rises or decreases on a certain segment when boundary points are and are not included in the intervals found.
Before you start directly solving thematic tasks, we recommend that you first go to the “Theoretical Information” section and repeat the definitions of concepts, rules and tabular formulas. Here you can read how to find and write each interval of increasing and decreasing function on the derivative graph.
All information offered is presented to the maximum extent possible. accessible form for understanding practically from scratch. The website provides materials for perception and assimilation in several various forms– reading, watching videos and direct training under the leadership experienced teachers. Professional teachers will tell you in detail how to find the intervals of increasing and decreasing derivatives of a function analytically and graphically. During the webinars, you will be able to ask any question you are interested in, both on theory and on solving specific problems.
Having remembered the main points of the topic, look at examples of increasing the derivative of a function, similar to tasks exam options. To consolidate what you have learned, take a look at the “Catalog” - here you will find practical exercises For independent work. The tasks in the section have been selected different levels difficulties taking into account the development of skills. For example, each of them is accompanied by solution algorithms and correct answers.
By choosing the “Constructor” section, students will be able to practice studying the increase and decrease of the derivative of a function on real Unified State Exam options, constantly updated taking into account latest changes and innovations.
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 sense. 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 point 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 are poorly oriented in differential calculus or a wise brain for for many 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 study the listed basic lessons, and maybe it will become 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 came up with it too 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 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 is not a fact that a navigator or even a satellite image will provide reliable data. Therefore, it would be nice to formalize the relief of the path using mathematics.
Let's look at some road (side view): 
Just in case, I remind you of an elementary fact: travel happens from left to right. For simplicity, we assume that the function continuous in the area under consideration.
What are the features of of this schedule?
At intervals 
function increases, that is, each next value of it more previous one. Roughly speaking, the schedule is on from bottom to top(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 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 on 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 on average half a meter of rise.
3) 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 on average by 2 meters. Take care of your clothes at the fifth point.
Now let’s ask ourselves: what is the best “measurement standard” value 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 come to amazing opportunity ideally 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 “techie” students who have 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 just 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's in in a broad sense does the verb “differentiate” mean? 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”.
Decide physical tasks or examples in mathematics is completely impossible without knowledge about the derivative and methods for calculating it. Derivative is one of the most important concepts mathematical analysis. This fundamental topic we decided to dedicate today's article. What is a derivative, what is its physical and geometric meaning, how to calculate the derivative of a function? All these questions can be combined into one: how to understand the derivative?
Geometric and physical meaning of derivative
Let there be a function f(x) , specified in a certain interval (a, b) . Points x and x0 belong to this interval. When x changes, the function itself changes. Changing the argument - the difference in its values x-x0 . This difference is written as delta x and is called argument increment. A change or increment of a function is the difference between the values of a function at two points. Definition of derivative:
The derivative of a function at a point is the limit of the ratio of the increment of the function at a given point to the increment of the argument when the latter tends to zero.
Otherwise it can be written like this:
What's the point of finding such a limit? And here's what it is:
the derivative of a function at a point is equal to the tangent of the angle between the OX axis and the tangent to the graph of the function at a given point.
Physical meaning derivative: the derivative of the path with respect to time is equal to the speed of rectilinear motion.
Indeed, since school days everyone knows that speed is a particular path x=f(t) and time t . Average speed over a certain period of time:
To find out the speed of movement at a moment in time t0 you need to calculate the limit:
Rule one: set a constant
The constant can be taken out of the derivative sign. Moreover, this must be done. When solving examples in mathematics, take it as a rule - If you can simplify an expression, be sure to simplify it .
Example. Let's calculate the derivative:
Rule two: derivative of the sum of functions
The derivative of the sum of two functions is equal to the sum of the derivatives of these functions. The same is true for the derivative of the difference of functions.
We will not give a proof of this theorem, but rather consider a practical example.
Find the derivative of the function:
Rule three: derivative of the product of functions
The derivative of the product of two differentiable functions is calculated by the formula:
Example: find the derivative of a function:
Solution: 
It is important to talk about calculating derivatives of complex functions here. Derivative complex function is equal to the product of the derivative of this function with respect to the intermediate argument and the derivative of the intermediate argument with respect to the independent variable.
In the above example we come across the expression:
In this case, the intermediate argument is 8x to the fifth power. In order to calculate the derivative of such an expression, we first calculate the derivative of the external function with respect to the intermediate argument, and then multiply by the derivative of the intermediate argument itself with respect to the independent variable.
Rule four: derivative of the quotient of two functions
Formula for determining the derivative of the quotient of two functions:
We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it seems, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.
With any questions on this and other topics, you can contact the student service. For short term We will help you solve the most difficult tests and solve problems, even if you have never done derivative calculations before.
Contents of the article
DERIVATIVE– derivative of the function y = f(x), given on a certain interval ( a, b) at point x of this interval is called the limit to which the ratio of the increment of the function tends f at this point to the corresponding increment of the argument when the increment of the argument tends to zero.
The derivative is usually denoted as follows:
Other designations are also widely used:
Instant speed.
Let the point M moves in a straight line. Distance s moving point, counted from some initial position M 0 , depends on time t, i.e. s there is a function of time t: s= f(t). Let at some point in time t moving point M was at a distance s from initial position M 0, and at some next moment t+D t found herself in a position M 1 – at a distance s+D s from the initial position ( see pic.).
Thus, over a period of time D t distance s changed by the amount D s. In this case they say that during the time interval D t magnitude s received increment D s.
The average speed cannot in all cases accurately characterize the speed of movement of a point M at a point in time t. If, for example, the body at the beginning of the interval D t moved very quickly, and at the end very slowly, then average speed will not be able to reflect the specified features of the point’s movement and give an idea of the true speed of its movement at the moment t. To more accurately express the true speed using the average speed, you need to take a shorter period of time D t. Most fully characterizes the speed of movement of a point at the moment t the limit to which the average speed tends at D t® 0. This limit is called the speed of movement in at the moment:
Thus, the speed of movement at a given moment is called the limit of the path increment ratio D s to time increment D t, when the time increment tends to zero. Because
Geometric meaning of the derivative. Tangent to the graph of a function.
The construction of tangent lines is one of those problems that led to the birth of differential calculus. The first published work relating to differential calculus and peruvian Leibniz, had the name New method maxima and minima, as well as tangents, for which neither fractional nor irrational quantities, and a special type of calculus for this, serve as an obstacle.
Let the curve be the graph of the function y =f(x) V rectangular system coordinates ( cm. rice.).
At some value x function matters y =f(x). These values x And y the point on the curve corresponds M 0(x, y). If the argument x give increment D x, then the new value of the argument x+D x corresponds to the new function value y+ D y = f(x + D x). The corresponding point of the curve will be the point M 1(x+D x,y+D y). If you draw a secant M 0M 1 and denoted by j the angle formed by a transversal with the positive direction of the axis Ox, it is immediately clear from the figure that .
If now D x tends to zero, then the point M 1 moves along the curve, approaching the point M 0, and angle j changes with D x. At Dx® 0 the angle j tends to a certain limit a and the straight line passing through the point M 0 and the component with the positive direction of the x-axis, angle a, will be the desired tangent. Its slope is:
Hence, f´( x) = tga
those. derivative value f´( x) at given value argument x equals the tangent of the angle formed by the tangent to the graph of the function f(x) at the corresponding point M 0(x,y) with positive axis direction Ox.
Differentiability of functions.
Definition. If the function y = f(x) has a derivative at the point x = x 0, then the function is differentiable at this point.
Continuity of a function having a derivative. Theorem.
If the function y = f(x) is differentiable at some point x = x 0, then it is continuous at this point.
Thus, the function cannot have a derivative at discontinuity points. The opposite conclusion is incorrect, i.e. from the fact that at some point x = x 0 function y = f(x) is continuous does not mean that it is differentiable at this point. For example, the function y = |x| continuous for everyone x(–Ґ x x = 0 has no derivative. At this point there is no tangent to the graph. There is a right tangent and a left one, but they do not coincide.
Some theorems on differentiable functions. Theorem on the roots of the derivative (Rolle's theorem). If the function f(x) is continuous on the segment [a,b], differentiable in all internal points of this segment and at the ends x = a And x = b goes to zero ( f(a) = f(b) = 0), then inside the segment [ a,b] there is at least one point x= With, a c b, in which the derivative fў( x) goes to zero, i.e. fў( c) = 0.
Finite increment theorem (Lagrange's theorem). If the function f(x) is continuous on the interval [ a, b] and is differentiable at all interior points of this segment, then inside the segment [ a, b] there is at least one point With, a c b that
f(b) – f(a) = fў( c)(b– a).
Theorem on the ratio of the increments of two functions (Cauchy's theorem). If f(x) And g(x) – two functions continuous on the segment [a, b] and differentiable at all interior points of this segment, and gў( x) does not vanish anywhere inside this segment, then inside the segment [ a, b] there is such a point x = With, a c b that
Derivatives of various orders.
Let the function y =f(x) is differentiable on some interval [ a, b]. Derivative values f ў( x), generally speaking, depend on x, i.e. derivative f ў( x) is also a function of x. When differentiating this function, we obtain the so-called second derivative of the function f(x), which is denoted f ўў ( x).
Derivative n- th order of function f(x) is called the (first order) derivative of the derivative n- 1- th and is denoted by the symbol y(n) = (y(n– 1))ў.
Differentials of various orders.
Function differential y = f(x), Where x– independent variable, yes dy = f ў( x)dx, some function from x, but from x only the first factor can depend f ў( x), the second factor ( dx) is the increment of the independent variable x and does not depend on the value of this variable. Because dy there is a function from x, then we can determine the differential of this function. The differential of the differential of a function is called the second differential or second-order differential of this function and is denoted d 2y:
d(dx) = d 2y = f ўў( x)(dx) 2 .
Differential n- of the first order is called the first differential of the differential n- 1- th order:
d n y = d(dn–1y) = f(n)(x)dx(n).
Partial derivative.
If a function depends not on one, but on several arguments x i(i varies from 1 to n,i= 1, 2,… n),f(x 1,x 2,… x n), then in differential calculus the concept of partial derivative is introduced, which characterizes the rate of change of a function of several variables when only one argument changes, for example, x i. 1st order partial derivative with respect to x i is defined as an ordinary derivative, and it is assumed that all arguments except x i, save constant values. For partial derivatives, the notation is introduced
The 1st order partial derivatives defined in this way (as functions of the same arguments) can, in turn, also have partial derivatives, these are second order partial derivatives, etc. Such derivatives taken from different arguments are called mixed. Continuous mixed derivatives of the same order do not depend on the order of differentiation and are equal to each other.
Anna Chugainova
When deciding various tasks geometry, mechanics, physics and other branches of knowledge became necessary using the same analytical process from this function y=f(x) receive new feature which is called derivative function(or just 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 
3) counting x constant and
x0, we find 
, which we denote by f" (x), as if emphasizing that the resulting function depends only on the value x, 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, 
, or 
Note that if for some value x, for example when x=a, attitude 
at
x0 does not tend to finite limit, then in this case they say that the function f(x) at x=a(or at the point x=a) has no derivative or is not differentiable at the point x=a.
2. Geometric meaning of the derivative.
Consider the graph of the function y = f (x), differentiable in the vicinity of the point x 0
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 ∆х, 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:
Derivative of a function at point x 0 equal to slope tangent to the graph of the function drawn at the point with abscissa x 0 .
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 functiony = f(x) at pointx 0 is the rate of change of the functionf(x) at pointx 0
The derivative is used in physics to find speed by known function coordinates versus time, acceleration according to 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 movement:
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,
p = m"(x) - linear density.
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.
