Can be taken out as a sign derivative:
(af(x)" =af " (x).
For example:
Derivative of an algebraic sum several functions (taken in constant numbers) is equal to the algebraic sum of them derivatives:
(f 1 (x) + f 2 (x) - f 3 (x))" = f 1 " (x) + f 2 " (x) - f 3 " (x).
For example:
(0.3 x 2 - 2 x + 0.8)" = (0.3 x 2)" - (2 x)" + (0.8)" = 0.6 x - 2 ( derivative last term equation is zero).
If derivative of a function g is nonzero, then the ratio f/g also has final derivative. This property can be written as:
.
Let functions y = f(x) and y = g(x) have finite derivatives at point x 0 . Then functions f ± g and f g also have finite derivatives in this point. Then we get:
(f ± g) ′ = f ′ ± g ′,
(f g) ′ = f ′ g + f g ′.
Derivative of a complex function.
Let function y = f(x) has finite derivative at a point x 0 , the function z = s(y) has a finite derivative at the point y 0 = f(x 0).
Then complex function z = s (f(x)) also has a finite derivative at this point. The above can be written in the form:
.
Derivative of the inverse function.
Let the function y = f(x) have inverse function x = g(y) on some interval(a, b) and there is a nonzero final derivative this function at point x 0, belonging to domain of definition, i.e. x 0 ∈ (a, b).
Then inverse function It has derivative at point y 0 = f(x 0):
.
Derivative of an implicit function.
If function y = f(x) is given implicitly equation F(x, y(x)) = 0, then its derivative is found from the condition:
.
They say that function y = f(x) is specified implicitly, If she identically satisfies the relation:
where F(x, y) is some function of two arguments.
Derivative of a function defined parametrically.
If function y = f(x) is specified parametrically using the considered
Solving physical problems or examples in mathematics is completely impossible without knowledge of the derivative and methods for calculating it. The derivative is one of the most important concepts in 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 of the 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 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 external function by 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. Behind short term We will help you solve the most difficult tests and solve problems, even if you have never done derivative calculations before.
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 explains in a simple and clear way what a derivative is and why it is needed.. We will not now strive for mathematical rigor in the presentation. The most important thing is to understand the meaning.
Let's remember the definition:
The derivative is the rate of change of a function.
The figure shows graphs of three functions. Which one do you think is growing faster?
The answer is obvious - the third. It has the highest rate of change, that is, the largest derivative.
Here's another example.
Kostya, Grisha and Matvey got jobs at the same time. Let's see how their income changed during the year:
The graph shows everything at once, isn’t it? Kostya’s income more than doubled in six months. And Grisha’s income also increased, but just a little. And Matvey’s income decreased to zero. The starting conditions are the same, but the rate of change of the function, that is derivative, - different. As for Matvey, his income derivative is generally negative.
Intuitively, we easily estimate the rate of change of a function. But how do we do this?
What we're really looking at is how steeply the graph of a function goes up (or down). In other words, how quickly does y change as x changes? Obviously, the same function in different points may have different meaning derivative - that is, it can change faster or slower.
The derivative of a function is denoted .
We'll show you how to find it using a graph.
A graph of some function has been drawn. Let's take a point with an abscissa on it. Let us draw a tangent to the graph of the function at this point. We want to estimate how steeply the function graph goes up. A convenient value for this is tangent of the tangent angle.
The derivative of a function at a point is equal to the tangent of the tangent angle drawn to the graph of the function at this point.
Please note that 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 a tangent to the graph of a function is. This is a straight line that has a single common point with the graph in a given section, and as shown in our figure. It looks like a tangent to a circle.
Let's find it. We remember that the tangent of an acute angle in right triangle equal to the ratio of the opposite side to the adjacent side. From the triangle:
We found the derivative using a graph without even knowing the formula of the function. Such problems are often found in the Unified State Examination in mathematics under the number.
There is another important relationship. 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 the function at a point is equal to slope tangent drawn to the graph of the function at this point.
In other words, the derivative is equal to the tangent of the tangent angle.
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 increases. The tangent to the graph drawn at the point forms sharp corner with positive axis direction. This means that the derivative at the point is positive.
At the point our function decreases. 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.
What will happen at the maximum and minimum points? We see that at the points (maximum point) and (minimum point) the tangent is horizontal. Therefore, the tangent of the tangent at these points is zero, and the derivative is also zero.
Point - maximum point. At this point, the increase in 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 zero, but its sign changes from “minus” to “plus”.
Conclusion: using the derivative, we can learn everything that interests us about the behavior of a function.
If the derivative is positive, then the function increases.
If the derivative is negative, then the function decreases.
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”.
Let's write these conclusions in the form of a table:
| increases | maximum point | decreases | minimum point | increases | |
| + | 0 | - | 0 | + |
Let's make two small clarifications. You will need one of them when solving Unified State Exam problems. Another - in the first year, with a more serious study of functions and derivatives.
It is possible that 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 is the 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 remains 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.
How to find the derivative if the function is given not by a graph, but by a formula? In this case it applies
FIRST DERIVATIVE
FIRST DERIVATIVE
(first derivative) The rate of increase in the value of a function when its argument increases at any point, if the function itself is defined at this point. On the graph, the first derivative of a function shows its slope. If y=f(x), its first derivative at the point x0 is the limit to which it tends f(x0+а)–f(x0)/а as A tends to an infinitesimal value. The first derivative can be denoted dy/dx or y´(x). Function y(x) has a constant value at a point x0, If dy/dx at the point x0 equals zero. Equal to zero the first derivative is a necessary but not sufficient condition for the function to reach its maximum or minimum at a given point.
Economy. Dictionary. - M.: "INFRA-M", Publishing House "Ves Mir". J. Black. General editor: Doctor of Economics Osadchaya I.M.. 2000 .
Economic dictionary. 2000 .
See what "FIRST DERIVATIVE" is in other dictionaries:
- (derivative) The rate at which the value of a function increases when its argument is incremented at any point, if the function itself is defined at this point. On the graph, the first derivative of a function shows its slope. If y=f(x), its first derivative at the point... ... Economic dictionary
This term has other meanings, see Derivative. Illustration of the concept of derivative Derivative ... Wikipedia
Derivative is the basic concept of differential calculus, characterizing the rate of change of a function. Defined as the limit of the ratio of the increment of a function to the increment of its argument when the increment of the argument tends to zero, if such a limit... ... Wikipedia
Boundary value problem special type; consists in finding, in the domain D of variables x = (x1,..., x n), a solution to the differential equation (1) of even order 2m for given values of all derivatives of order not higher than m on the boundary S of the domain D (or its part) ... Mathematical Encyclopedia
- (second derivative) The first derivative of the first derivative of the function. The first derivative measures the slope of the function; The second derivative measures how the slope changes as the argument increases. Second derivative of y = f(x)… … Economic dictionary
This article or section needs revision. Please improve the article in accordance with the rules for writing articles. Fractional about ... Wikipedia
- (cross partial derivative) The effect of changing one argument of a function from two or more variables on the derivative of a given function taken with respect to another argument. If y=f(x,z), then its derivative, or the first derivative of the function y with respect to the argument x, is equal to... ... Economic dictionary
analogue of point speed- The first derivative of the movement of a point along the generalized coordinate of the mechanism...
analogue of link angular velocity- The first derivative of the angle of rotation of the link with respect to the generalized coordinate of the mechanism... Polytechnic terminological explanatory dictionary
generalized speed of the mechanism- The first derivative of the generalized coordinate of the mechanism with respect to time... Polytechnic terminological explanatory dictionary
Books
- Collection of problems on differential geometry and topology, Mishchenko A.S.. This collection of problems is intended to reflect as much as possible the existing requirements for courses in differential geometry and topology, both from new programs and from other courses...
- My scientific articles. Book 3. The method of density matrices in quantum theories of a laser, an arbitrary atom, Bondarev Boris Vladimirovich. This book reviews published scientific papers that use the density matrix method to expound new quantum theories of the laser, the arbitrary atom, and the damped quantum oscillator.…
Solving physical problems or examples in mathematics is completely impossible without knowledge of the derivative and methods for calculating it. The derivative is one of the most important concepts in 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 of the 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.
With any questions on this and other topics, you can contact the 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.
