Probably, the concept of derivative is familiar to each of us since school. Typically, students have difficulty understanding this, undoubtedly, 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 an analysis of what derivatives are, how to calculate them and where they will be useful to us, we will plunge a little into history.
History
The concept of derivative, which is the basis of mathematical analysis, was discovered (it’s better to even say “invented”, because in nature it did not exist as such) by Isaac Newton, whom we all know by the discovery of the law of universal gravitation. It was he who first applied this concept in physics to link the nature of 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 field of mathematics called "mathematical analysis". If the Nobel Prize at that time, Newton would most likely have received it several times.
Not without other great minds. In addition to Newton, such eminent geniuses of mathematics as Leonard Euler, Louis Lagrange and Gottfried Leibniz worked on the development of the derivative and integral. Thanks to them, we got 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 happened in school.
What is a derivative?
There are several different ways to define this concept. The simplest explanation: the derivative is the rate of change of the function. Imagine a graph of some function y versus x. If it is not straight, then it has some bends in the graph, periods of increase and decrease. If you take any infinitely small interval of this graph, it will be a straight line segment. So, the ratio of the size of this infinitesimal 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 get the function of the derivative, that is, a certain dependence of the game on x.
In addition to the physical meaning of the derivative as the rate of change of the function, there is also a geometric meaning. We’ll talk about him now.
Geometric meaning
Derived numbers in themselves are a certain number, which without proper understanding does not make any sense. 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 quite clear definition. Let's analyze it in more detail. Suppose we have a graph of a function (for interest, we take a curve). There are an infinite number of points on it, but there are areas where only one single point has a maximum or minimum. Through any such point, you can draw a straight line that would be perpendicular to the graph of the function at this point. Such a line will be called a tangent. Suppose we held it to the intersection with the axis OX. 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 the derivatives of numbers.
Special cases
As we have said, the derivatives of numbers are the values of the derivative at a particular point. For example, take the function y = x 2 . The derivative x is a number, and in the general case, a function equal to 2 * x. If we need to calculate the derivative, say, at the point x 0 = 1, then we get y '(1) = 2 * 1 = 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. We will only 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 following conditions are met:
1) There must be partial derivatives of the first order from the real and imaginary parts of the game and X.
2) The Cauchy-Riemann conditions related to the equality of the partial derivatives described in the first paragraph are satisfied.
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 are going to discuss now.
Application
Probably, each of us at least once in his life catches himself with the thought that mathematics is unlikely to be useful to him. And such a complicated thing as a derivative probably has no use 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 laid the foundation for mathematical analysis, which gave us the opportunity 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 certainly will be needed. It is not in vain that mathematics is called the queen of sciences: from it the foundations of understanding other areas of knowledge are formed.