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 for some value, for example when 
x=a
x, attitude f(x) at Note that if for some value0 does not tend to the finite limit, then in this case they say that the function Note that if for some value at Note that if for some value.
(or at the point
) has no derivative or is not differentiable at the point
f(x)
2. Geometric meaning of the derivative.
Consider the graph of the function y = f (x), differentiable in the vicinity of the point x 0
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;
ВС =∆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 slope 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. 
, 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 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,
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.
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 knowledge of differential calculus or a wise brain for long years successfully got rid of this baggage, please start with function limits. At the same time, master/remember their solution.
The same practical sense dictates that it is advantageous first learn to find derivatives, including derivatives of complex functions. Theory is theory, but, as they say, you always want to differentiate. In this regard, it is better to 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 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 flat highway. Or along a hilly highway - up and down, up and down. Another road goes only uphill, and another one goes downhill all the time. Extreme enthusiasts will choose a route through a gorge with a steep cliff and a steep climb.
But whatever your preferences, it is advisable to know the area or at least locate it topographic map. What if such information is missing? After all, you can choose, for example, a smooth path, but as a result stumble upon a ski slope with cheerful Finns. It 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 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 one more important feature: at 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 our way: 
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! Designation are ONE symbol, that is, you cannot “tear off” the “delta” from the “X” and consider these letters separately. Of course, the comment also concerns the function increment symbol.
Let's explore the nature of the resulting fraction more meaningfully. Let us initially be at a height of 20 meters (at the left black point). Having covered the distance of meters (left red line), we will find ourselves at an altitude of 60 meters. Then the increment of the function will be 
meters (green line) and: . Thus, on every meter this section of the road height increases average by 4 meters...forgot your climbing equipment? =) In other words, the constructed relationship characterizes the AVERAGE RATE OF CHANGE (in this case, growth) of the function.
Note : 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: how less value , the more accurately we will describe the road topography. Moreover, the following facts are true:
– For anyone lifting points 
you can select a value (even if very small) that fits within the boundaries of a particular rise. This means that the corresponding height increment will be guaranteed to be positive, and the inequality will correctly indicate the growth of the function at each point of these intervals.
- Likewise, for any slope point there is a value that will fit completely on this slope. Consequently, the corresponding increase in height is clearly negative, and the inequality will correctly show the decrease in the function at each point of the given interval.
– A particularly interesting case is when the rate of change of the function is zero: . Firstly, zero height increment () is a sign of a smooth path. And secondly, there are other interesting situations, examples of which you see in the figure. Imagine that fate has brought us to the very top of a hill with soaring eagles or the bottom of a ravine with croaking frogs. If you take a small step in any direction, the change in height will be negligible, and we can say that the rate of change of the function is actually zero. This is exactly the picture observed at the points.
Thus, we have come to an amazing opportunity to perfectly accurately characterize the rate of change of a function. After all, mathematical analysis 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 the body, depending on time, and the function of the speed of movement 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”.
Date: 11/20/2014
What is a derivative?
Table of derivatives.
Derivative is one of the main concepts higher mathematics. In this lesson we will introduce this concept. Let's get to know each other, without strict mathematical formulations and proofs.
This acquaintance will allow you to:
Understand the essence of simple tasks with derivatives;
Successfully solving these very problems difficult tasks;
Prepare for more serious lessons on derivatives.
First - a pleasant surprise.)
The strict definition of the derivative is based on the theory of limits and the thing is quite complicated. This is upsetting. But the practical application of derivatives, as a rule, does not require such extensive and deep knowledge!
To successfully complete most tasks at school and university, it is enough to know just a few terms- to understand the task, and just a few rules- to solve it. That's all. This makes me happy.
Let's start getting acquainted?)
Terms and designations.
There are many different mathematical operations in elementary mathematics. Addition, subtraction, multiplication, exponentiation, logarithm, etc. If you add one more operation to these operations, elementary mathematics becomes higher. This new operation called differentiation. The definition and meaning of this operation will be discussed in separate lessons.
It is important to understand here that differentiation is simply a mathematical operation on a function. We take any function and, according to certain rules, transform it. The result will be new feature. This new function is called: derivative.
Differentiation- action on a function.
Derivative- the result of this action.
Just like, for example, sum- the result of addition. Or private- the result of division.
Knowing the terms, you can at least understand the tasks.) The formulations are as follows: find the derivative of a function; take the derivative; differentiate the function; calculate derivative and so on. This is all same. Of course, there are also more complex tasks, where finding the derivative (differentiation) will be just one of the steps in solving the problem.
The derivative is indicated by a dash at the top right of the function. Like this: y" or f"(x) or S"(t) and so on.
Reading igrek stroke, ef stroke from x, es stroke from te, well, you understand...)
A prime can also indicate the derivative of a particular function, for example: (2x+3)", (x 3 )" , (sinx)" etc. Often derivatives are denoted using differentials, but we will not consider such notation in this lesson.
Let's assume that we have learned to understand the tasks. All that’s left is to learn how to solve them.) Let me remind you once again: finding the derivative is transformation of a function according to certain rules. Surprisingly, there are very few of these rules.
To find the derivative of a function, you need to know only three things. Three pillars on which all differentiation stands. Here they are these three pillars:
1. Table of derivatives (differentiation formulas).
3. Derivative complex function.
Let's start in order. In this lesson we will look at the table of derivatives.
Table of derivatives.
There are an infinite number of functions in the world. Among this variety, there are functions that are most important for practical application. These functions are found in all laws of nature. From these functions, like from bricks, you can construct all the others. This class of functions is called elementary functions. It is these functions that are studied at school - linear, quadratic, hyperbola, etc.
Differentiation of functions "from scratch", i.e. Based on the definition of derivative and the theory of limits, this is quite a labor-intensive thing. And mathematicians are people too, yes, yes!) So they simplified their (and us) life. They calculated the derivatives of elementary functions before us. The result is a table of derivatives, where everything is ready.)
Here it is, this plate for the most popular functions. On the left is an elementary function, on the right is its derivative.
| Function y |
Derivative of function y y" |
|
| 1 | C (constant value) | C" = 0 |
| 2 | x | x" = 1 |
| 3 | x n (n - any number) | (x n)" = nx n-1 |
| x 2 (n = 2) | (x 2)" = 2x | |
 |
||
 |
 |
|
| 4 | sin x | (sin x)" = cosx |
| cos x | (cos x)" = - sin x | |
| tg x |  |
|
| ctg x |  |
|
| 5 | arcsin x |  |
| arccos x |  |
|
| arctan x |  |
|
| arcctg x |  |
|
| 4 | a x |  |
| e x | ||
| 5 | log a x |  |
| ln x ( a = e) |
I recommend paying attention to the third group of functions in this table of derivatives. Derivative power function- one of the most common formulas, if not the most common! Do you understand the hint?) Yes, it is advisable to know the table of derivatives by heart. By the way, this is not as difficult as it might seem. Try to solve more examples, the table itself will be remembered!)
Finding the table value of the derivative, as you understand, is not the most difficult task. Therefore, very often in such tasks there are additional chips. Either in the wording of the task, or in the original function, which doesn’t seem to be in the table...
Let's look at a few examples:
1. Find the derivative of the function y = x 3
There is no such function in the table. But there is a derivative of the power function in general view(third group). In our case n=3. So we substitute three instead of n and carefully write down the result:
(x 3) " = 3 x 3-1 = 3x 2
That's it.
Answer: y" = 3x 2
2. Find the value of the derivative of the function y = sinx at the point x = 0.
This task means that you must first find the derivative of the sine, and then substitute the value x = 0 into this very derivative. Exactly in that order! Otherwise, it happens that they immediately substitute zero into the original function... We are asked to find not the value of the original function, but the value its derivative. The derivative, let me remind you, is a new function.
Using the tablet we find the sine and the corresponding derivative:
y" = (sin x)" = cosx
We substitute zero into the derivative:
y"(0) = cos 0 = 1
This will be the answer.
3. Differentiate the function:
What, does it inspire?) There is no such function in the table of derivatives.
Let me remind you that to differentiate a function is simply to find the derivative of this function. If you forget elementary trigonometry, looking for the derivative of our function is quite troublesome. The table doesn't help...
But if we see that our function is double angle cosine, then everything gets better right away!
Yes Yes! Remember that transforming the original function before differentiation quite acceptable! And it happens to make life a lot easier. Using the double angle cosine formula:
Those. our tricky function is nothing more than y = cosx. And this is a table function. We immediately get:
Answer: y" = - sin x.
Example for advanced graduates and students:
4. Find the derivative of the function:
There is no such function in the derivatives table, of course. But if you remember elementary mathematics, operations with powers... Then it is quite possible to simplify this function. Like this:
And x to the power of one tenth is already a tabular function! Third group, n=1/10. We write directly according to the formula:
That's all. This will be the answer.
I hope that everything is clear with the first pillar of differentiation - the table of derivatives. It remains to deal with the two remaining whales. In the next lesson we will learn the rules of differentiation.
Solving physical problems or examples in mathematics is completely impossible without knowledge of the derivative and methods for calculating it. Derivative is one of the most important concepts mathematical analysis. We decided to devote today’s article to this fundamental topic. 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. The derivative of a 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.
If you have any questions on this or other topics, you can contact student service. In a short time, we will help you solve the most difficult test and understand the tasks, even if you have never done derivative calculations before.
