Derivatives of numbers: calculation methods and examples

Probably, the concept of a derivative is familiar to each of us from school. Usually, students have difficulty understanding this, undoubtedly, very important thing. It is actively used in various areas of people's lives, and many engineering developments have been based on mathematical calculations obtained with the help of a derivative. But before we go over to the analysis of what are the derivatives of numbers, how to calculate them, and where they will be useful to us, we shall plunge a little into history.

History

The concept of the derivative, which is the basis of mathematical analysis, was open (it is even better to say "invented", because in nature it as such did not exist) 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 foundation of an entire field of mathematics called "mathematical analysis." Whether at that time the Nobel Prize, Newton would most likely have received it several times.

Not without the 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. It is thanks to them that we have received the theory of differential calculus in the form in which it exists to this day. Incidentally, this Leibniz discovered the geometric meaning of the derivative, which turned out to be nothing more than the tangent of the angle of inclination of the tangent to the graph of the function.

What are the derived numbers? We will repeat a little that passed at school.

What is a derivative?

You can define this concept in several different ways. The simplest explanation: the derivative is the rate of change of a function. We represent the graph of some function y of x. If this is not a straight line, then it has some bends in the graph, periods of increasing and decreasing. If we take an 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 coordinate y to the size of the coordinate x will be the derivative of the given function at a given point. If we consider the function as a whole, and not at a particular point, then we get a 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 geometrical meaning. About him, we now talk.

Geometrical meaning

Derivatives of numbers in themselves represent 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 us examine it in more detail. Suppose 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 a minimum. Through any such point, you can draw a line that is perpendicular to the graph of the function at that point. Such a line will be called a tangent. Suppose we ran it to the intersection with the OX axis. So, the angle between the tangent and the OX axis will be determined by the derivative. Or rather, the tangent of this angle will be equal to it.

Let's talk a little about particular cases and analyze the derived numbers.

Special cases

As we have already 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. 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 following conditions are present:

1) There must exist partial derivatives of the first order from the real and imaginary parts in the game and in the x.

2) The Cauchy-Riemann conditions are fulfilled, connected with the equality of the partial derivatives described in the first paragraph.

Another interesting case, although not as complex as the previous one, is the derivative of a negative number. In fact, any negative number can be represented as positive, multiplied by -1. But 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 now discussing.

Application

Probably, each of us at least once in life catches himself thinking that mathematics is hardly useful to him. And such a complex thing as the 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 gave rise to a mathematical analysis that enabled us to draw conclusions from the graphs of functions, and we learned to interpret the laws of nature and turn them in our favor thanks to it.

Conclusion

Of course, not everyone may need a derivative in real life. But mathematics develops a logic that 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.