Probably, the concept of a derivative is familiar to each of us since school. Usually students have difficulty understanding this, no doubt, very important thing. It is actively used in various areas of people's lives, and many engineering developments were based precisely on mathematical calculations obtained using the derivative. But before proceeding to the analysis of what derivatives of numbers are, how to calculate them and where they are useful to us, let's plunge into history.
History
The concept of the derivative, which is the basis of mathematical analysis, was discovered (it would be better to say "invented", because it did not exist in nature as such) by Isaac Newton, whom we all know from the discovery of the law of universal gravitation. It was he who first applied this concept in physics to link the nature of the speed and acceleration of bodies. And many scientists still praise Newton for this magnificent invention, because in fact he invented the basis of differential and integral calculus, in fact, the basis of a whole area of mathematics called "calculus". If at that time the Nobel Prize, Newton would have received it with a high probability several times.
Not without other great minds. Except Newtonsuch eminent mathematical geniuses as Leonhard Euler, Louis Lagrange and Gottfried Leibniz worked on the development of the derivative and the integral. It is thanks to them that we have received the theory of differential calculus in the form in which it exists to this day. By the way, it was Leibniz who discovered the geometric meaning of the derivative, which turned out to be nothing more than the tangent of the slope of the tangent to the graph of the function.
What are derivatives of numbers? Let's repeat a little what we went through at school.
What is a derivative?
This concept can be defined in several different ways. The simplest explanation is that the derivative is the rate of change of the function. Imagine a graph of some function y of x. If it is not straight, then it has some curves in the graph, periods of increase and decrease. If we take some infinitely small interval of this graph, it will be a straight line segment. So, the ratio of the size of this infinitely small segment along the y coordinate to the size along the x coordinate will be the derivative of this function at a given point. If we consider the function as a whole, and not at a specific point, then we will get a derivative function, that is, a certain dependence of y on x.
Besides, in addition to the physical meaning of the derivative as the rate of change of a function, there is also a geometric meaning. We will talk about him now.
Geometric sense
The derivatives of numbers themselves represent a certain number, which, without proper understanding, does not carryno point. It turns out that the derivative not only shows the rate of growth or decrease of the function, but also the tangent of the slope of the tangent to the graph of the function at a given point. Not a very clear definition. Let's analyze it in more detail. Let's say we have a graph of a function (for interest, let's take a curve). It has an infinite number of points, but there are areas where only one single point has a maximum or minimum. Through any such point it is possible to draw a line that would be perpendicular to the graph of the function at that point. Such a line will be called a tangent. Let's say we spent it to the intersection with the OX axis. So, the angle obtained between the tangent and the OX axis will be determined by the derivative. More precisely, the tangent of this angle will be equal to it.
Let's talk a little about special cases and analyze derivatives of numbers.
Special cases
As we have already said, derivatives of numbers are the values of the derivative at a particular point. For example, let's take the function y=x2. The derivative x is a number, and in the general case, a function equal to 2x. If we need to calculate the derivative, say, at the point x0=1, then we get y'(1)=21=2. Everything is very simple. An interesting case is the derivative of a complex number. We will not go into a detailed explanation of what a complex number is. Let's just say that this is a number that contains the so-called imaginary unit - a number whose square is -1. The calculation of such a derivative is possible only if the followingconditions:
1) There must be first-order partial derivatives of the real and imaginary parts with respect to Y and X.
2) The Cauchy-Riemann conditions associated with the equality of partial derivatives described in the first paragraph are met.
Another interesting case, although not as complicated as the previous one, is the derivative of a negative number. In fact, any negative number can be represented as a positive number multiplied by -1. Well, the derivative of the constant and the function is equal to the constant multiplied by the derivative of the function.
It will be interesting to learn about the role of the derivative in everyday life, and this is what we will discuss now.
Application
Probably, each of us at least once in his life catches himself thinking that mathematics is unlikely to be useful to him. And such a complicated thing as a derivative, probably, has no application at all. In fact, mathematics is a fundamental science, and all its fruits are developed mainly by physics, chemistry, astronomy, and even economics. The derivative was the beginning of mathematical analysis, which gave us the ability to draw conclusions from the graphs of functions, and we learned to interpret the laws of nature and turn them to our advantage thanks to it.
Conclusion
Of course, not everyone may need a derivative in real life. But mathematics develops logic, which will certainly be needed. It is not for nothing that mathematics is called the queen of sciences: it forms the basis for understanding other areas of knowledge.
