Presentation on the topic: Derivative. From the history of the creation of the derivative Derivative in chemistry
Function derivative Teacher of GAPOU RO "RKTM" Kolykhalina K.A. Argument increment, function increment Let x be an arbitrary point lying in some neighborhood of a fixed point x0. The difference x-x0 is called the increment of the independent variable (or the increment of the argument) at the point x0 and is denoted ∆x. ∆x = x – x0 – increment of the independent variable. The increment of the function f at the point x0 is the difference between the values of the function at an arbitrary point and the value of the function at a fixed point. f(х) – f(х0)=f(х0+∆х) – f(х0) – function increment f∆f=f(х0+∆х) – f(х0) Determination of the derivative of the derivative function y= f(x) at the point x =x0 is called the limit of the ratio of the increment of the function ∆y at this point to the increment of the argument ∆x, as the increment of the argument tends to zero. Algorithm for calculating the derivative The derivative of the function y= f(x) can be found according to the following scheme: 1. Let's increment ∆x≠0 to the argument x and find the incremented value of the function y+∆y= f(x+∆x). 2. Find the increment of the function ∆y= f(x+∆x) - f(x). 3. Compose the relation 4. Find the limit of this relation at ∆x⇾0, i.e. (if this limit exists). Determination of the derivative of a function at a given point. Its geometric meaning
k - slope of the straight line (secant)
Tangent
The geometric meaning of the derivative
The derivative of a function at a given point is equal to the slope of the tangent drawn to the graph of the function at that point.
The physical meaning of the derivative 1. The problem of determining the speed of a material particle Let a point move along some straight line according to the law s= s(t), where s is the distance traveled, t is the time, and it is necessary to find the speed of the point at the moment t0. By the time t0 the distance traveled is equal to s0 = s(t0), and by the time (t0 +∆t) it is the path s0 + ∆s=s(t0 +∆t). Then for the interval ∆t the average speed will be The smaller ∆t, the better the average speed characterizes the movement of the point at the moment t0. Therefore, under point speed at time t0 one should understand the limit of the average speed for the interval from t0 to t0 +∆t, when ∆t⇾0 , i.e. 2. PROBLEM ON THE RATE OF A CHEMICAL REACTION Let some substance enter into a chemical reaction. The amount of this substance Q changes during the reaction depending on the time t and is a function of time. Let the amount of the substance change by ∆Q during the time ∆t, then the ratio will express the average rate of the chemical reaction over the time ∆t, and the limit of this ratio is the rate of the chemical reaction at a given time t.3. THE PROBLEM OF DETERMINING THE RATE OF RADIOACTIVE DECAY
If m is the mass of the radioactive substance and t is the time, then the phenomenon of radioactive decay at time t, provided that the mass of the radioactive substance decreases over time, is characterized by the function m = m(t).
The average decay rate over time ∆t is expressed by the ratio
and the instantaneous decay rate at time t
The physical meaning of the derivative of a function at a given point
Derivatives of basic elementary functions Basic rules of differentiation Let u=u(x) And v=v(x) - differentiable functions at the point x. 1) (u v) = u v 2) (uv) = uv +uv (cu) = cu 3) , if v 0
The derivative of a function at a point is the basic concept of differential calculus. It characterizes the rate of change of the function at the specified point. The derivative is widely used in solving a number of problems in mathematics, physics, and other sciences, especially in studying the speed of various kinds of processes.
Basic definitions
The derivative is equal to the limit of the ratio of the increment of the function to the increment of the argument, provided that the latter tends to zero:
$y^(\prime)\left(x_(0)\right)=\lim _(\Delta x \rightarrow 0) \frac(\Delta y)(\Delta x)$
Definition
A function that has a finite derivative at some point is called differentiable at a given point. The process of calculating the derivative is called function differentiation.
History reference
The Russian term "derivative of a function" was first used by the Russian mathematician V.I. Viskovatov (1780 - 1812).
The designation of an increment (argument/function) with the Greek letter $\Delta$ (delta) was first used by the Swiss mathematician and mechanic Johann Bernoulli (1667 - 1748). The notation for the differential , the derivative $d x$ belongs to the German mathematician G.V. Leibniz (1646 - 1716). The manner of denoting the time derivative with a dot above the letter - $\dot(x)$ - comes from the English mathematician, mechanic and physicist Isaac Newton (1642 - 1727). The brief designation of the derivative with a stroke - $f^(\prime)(x)$ - belongs to the French mathematician, astronomer and mechanic J.L. Lagrange (1736 - 1813), which he introduced in 1797. The partial derivative symbol $\frac(\partial)(\partial x)$ was actively used in his works by the German mathematician Karl G.Ya. Jacobi (1805 - 1051), and then the outstanding German mathematician Karl T.W. Weierstrass (1815 - 1897), although this designation has already been encountered earlier in one of the works of the French mathematician A.M. Legendre (1752 - 1833). The symbol of the differential operator $\nabla$ was invented by the outstanding Irish mathematician, mechanic and physicist W.R. Hamilton (1805 - 1865) in 1853, and the name "nabla" was proposed by the English self-taught scientist, engineer, mathematician and physicist Oliver Heaviside (1850 - 1925) in 1892.
The branch of mathematics that studies derivatives of functions and their applications is called differential calculus. This calculus arose from solving problems for drawing tangents to curves, for calculating the speed of movement, for finding the largest and smallest values of a function.
A number of problems of differential calculus were solved in ancient times by Archimedes, who developed a method for drawing a tangent. Archimedes built a tangent to the spiral that bears his name. Archimedes (c. 287 - 212 BC) - a great scientist. A pioneer of many facts and methods of mathematics and mechanics, a brilliant engineer.
The problem of finding the rate of change of a function was first solved by Newton. The problem of finding the rate of change of a function was first solved by Newton. He called the function fluent, i.e. the current value. Derivative - flux with and e th. He called the function fluent, i.e. the current value. Derivative - flux with and e th. Newton came up with the concept of a derivative based on questions of mechanics. Isaac Newton (1643 - 1722) - English physicist and mathematician.
Based on Fermat's results and some other conclusions, Leibniz in 1684 published the first article on the differential calculus, which outlined the basic rules for differentiation. Leibniz Gottfried Friedrich (1646 - 1716) - the great German scientist, philosopher, mathematician, physicist, lawyer, linguist
Application of the derivative: Application of the derivative: 1) Power is the derivative of work with respect to time P \u003d A "(t). 2) Current strength is the derivative of charge with respect to time I \u003d g" (t). 3) Force is the derivative of the work of displacement F \u003d A "(x). 4) Heat capacity is the derivative of the amount of heat with respect to temperature C \u003d Q" (t). 5) Pressure - the derivative of the force with respect to the area P \u003d F "(S) 6) The circumference is the derivative of the area of \u200b\u200bthe circle along the radius l env \u003d S" cr (R). 7) The growth rate of labor productivity is the time derivative of labor productivity. 8) Academic success? Derivative of knowledge growth.
Application of the derivative in physics Task: Two bodies move in a straight line, respectively, according to the laws: S 1 (t) \u003d 3.5t 2 - 5t + 10 and S 2 (t) \u003d 1.5t 2 + 3t -6. At what point in time will the speeds of the bodies be equal? Task: Two bodies move in a straight line, respectively, according to the laws: S 1 (t) \u003d 3.5t 2 - 5t + 10 and S 2 (t) \u003d 1.5t 2 + 3t -6. At what point in time will the speeds of the bodies be equal?
Application of the derivative in economics Problem: The enterprise produces X units of some homogeneous product per month. It has been established that the dependence of the financial savings of the enterprise on the volume of output is expressed by the formula Task: The enterprise produces X units of some homogeneous products per month. It has been established that the dependence of the enterprise's financial savings on the volume of output is expressed by the formula Explore the enterprise's potential. Explore the potential of the enterprise. 15
The history of the concept of derivative
Functions, boundaries, derivative and integral are the basic concepts of mathematical analysis studied in the course of high school. And the concept of a derivative is inextricably linked with the concept of a function.
The term "function" was first proposed by a German philosopher and mathematician to characterize different segments connecting the points of a certain curve in 1692. The first definition of a function, which was no longer associated with geometric representations, was formulated in 1718. Student of Johann Bernoulli
in 1748. clarified the definition of the function. Euler is credited with introducing the symbol f(x) to denote a function.A rigorous definition of the limit and continuity of a function was formulated in 1823 by the French mathematician Augustin Louis Cauchy . The definition of the continuity of a function was formulated even earlier by the Czech mathematician Bernard Bolzano. According to these definitions, on the basis of the theory of real numbers, a rigorous substantiation of the main provisions of mathematical analysis was carried out.
The discovery of the approaches and foundations of differential calculus was preceded by the work of a French mathematician and lawyer, who in 1629 proposed methods for finding the largest and smallest values of functions, drawing tangents to arbitrary curves, and actually relied on the use of derivatives. This was also facilitated by the work that developed the method of coordinates and the foundations of analytical geometry. Only in 1666 and a little later, independently of each other, they built the theory of differential calculus. Newton came to the concept of a derivative by solving problems of instantaneous velocity, and , - by considering the geometric problem of drawing a tangent to a curve. and investigated the problem of maxima and minima of functions.
The integral calculus and the very concept of the integral arose from the need to calculate the areas of plane figures and the volumes of arbitrary bodies. The ideas of integral calculus originate in the works of ancient mathematicians. However, this testifies to the "method of exhaustion" of Eudoxus, which he later used in the 3rd century. BC e The essence of this method was that in order to calculate the area of a flat figure and, by increasing the number of sides of the polygon, they found the boundary into which the areas of stepped figures were directed. However, for each figure, the calculation of the limit depended on the choice of a special technique. And the problem of the general method for calculating the areas and volumes of figures remained unresolved. Archimedes did not yet explicitly apply the general concept of boundary and integral, although these concepts were used implicitly.
In the 17th century , who discovered the laws of planetary motion, the first attempt to develop ideas was successfully carried out. Kepler calculated the areas of plane figures and the volumes of bodies, based on the idea of decomposing a figure and a body into an infinite number of infinitely small parts. As a result of the addition, these parts consisted of a figure whose area is known and allows us to calculate the area of \u200b\u200bthe desired one. The so-called "Cavalieri principle" entered the history of mathematics, with the help of which areas and volumes were calculated. This principle was theoretically substantiated later with the help of integral calculus.
The ideas of other scientists became the ground on which Newton and Leibniz discovered the integral calculus. The development of integral calculus continued much later Pafnuty Lvovich Chebyshev
developed ways to integrate some classes of irrational functions.
The modern definition of the integral as the limit of integral sums is due to Cauchy. Symbol
