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Derivative calculator

Derivative calculator

In mathematical analysis and physics, the derivative is widely used, describing complex functions and variables. The latter may include electrical voltage, chemical reactions, and speed of movement.

That is, any quantity that is difficult or impossible to describe as a constant value. For example, the speed of a moving car that accelerates and decelerates many times while driving. The mathematical derivative of a function is intended to describe, systematize and analyze such quantities.

Derivative of a function

According to the official definition, the derivative is the limit of the ratio of the increment of a function to the increment of its argument when the latter tends to zero. The process of calculating the derivative is called differentiation. And a function is called differentiable only if it has a finite derivative.

A function can be described as the dependence of one quantity on another, and depicted in the coordinate plane as a line. To differentiate it:

  • Take the x value on the x-axis.
  • Substitute the selected x value into the formula y = f(x).
  • Get the coordinates of the point in x, y format.
  • Construct a point with coordinates x, y.
  • We repeat this procedure, substituting all other x values.

The derivative will show how many times the increment in the y value is greater or less than the increment in the x value. The ratio of these increments is described as dy/dx, and the derivative as f(x).

A little history

Derivatives began to be used in mathematics back in the 15th century - to determine the dependence of the flight range of projectiles on the inclination of guns. The first to use this technique was the Italian mathematician Niccolo Fontana Tartaglia.

And in the 17th century, the Bernoulli brothers from Switzerland began to study derivatives in earnest. The younger brother, Johann Bernoulli, first published a systematic presentation of differential calculus, which became the basis for “Infinitesimal Analysis” in 1687. By 1742, the scientist also completed the development of a course on integral calculus and proposed new methods for solving ordinary differential equations.

Johann's older brother, Jacob Bernoulli, used the derivative to find the curvature of a flat curved line, and also used it to study the logarithmic spiral. It was Jacob Bernoulli who was the author of the name “integral”, which, in fact, is the opposite of a differential.

The Bernoulli brothers at the turn of the 17th-18th centuries made a huge contribution to the study of derivatives, and laid the foundation for the mathematical calculus of variations.

In the period from the 17th to the 19th centuries in Europe, other eminent scientists were also involved in the study of derivatives: Leibniz, Newton, Lagrange, Jacobi, Weierstrass, Legendre. For example, the modern notation for a differential - d(x) - was introduced by Gottfried Wilhelm Leibniz, and the notation for a derivative with a prime - f'(x) - by Joseph Louis Lagrange.

The term “derivative” itself was first used by Lagrange in 1797. This word is a translation of the French derivee, which comes from derive - “derived.”

Subsequently, many European mathematicians used the notation introduced in France, and the notation “delta” (∇) appeared only in 1853, thanks to the Irish mathematician William Rowan Hamilton.

Roller coaster analogy

To make it easier to understand functions and find their derivatives, you can use a simple analogy with the world-famous attraction - a roller coaster. If you look at them from the side, you can even by eye, without complex calculations, determine the main features of the movement of the trolley: in what areas it will rise/descend, where it will accelerate/slow down, how many times it will cross the boundaries between ascents/descents.

The function depicted on the plane can be described in exactly the same way. In different areas it will increase and decrease in different ways - this process can be described and determined using a derivative. To do this, we introduce the following definitions:

  • Function increment is the difference between the values on the y-axis.
  • The argument increment is the difference between the values on the x-axis.
  • The rate of change of a function is the ratio of its increment to the increment of the argument: dy/dx.

The smaller the increment of the argument x, the higher the accuracy of the calculations. The highest accuracy is achieved when the increment of the argument tends to zero. In this case, finding derivatives will require a number of calculations tending to infinity (adjusted for accuracy/gradation).

If this task is too difficult for a person, then a modern computer can handle it in a split second. It is enough to use a special online application that will find the derivative of a function using the entered data, even if they are included in complex formulas with sines, cosines, roots, and exponents.

Differentiation calculator

Differentiation calculator

In mathematics there are many types of derivatives: partial and complete, one-sided and variational, in measure and direction, derivatives of Gateau, Lie, Frechet.

It makes no sense to list and describe their differences, but it is worth understanding what a derivative is in its generalized form and how it can be used in everyday life.

According to the official definition, the derivative is the ratio of the increment of a function to the increment of the argument when the latter tends to zero. The formula looks like this:

  • f'(x) = Δf / Δx, with Δx → 0.

Accordingly, f'(x) is the designation of the derivative, Δf is the increment of the function, and Δx is the increment of the argument. The expression is valid when considering a function on a plane: with the abscissa (x) and ordinate (y) axis.

Practical application of derivative

It is a mistake to believe that functions and their derivatives are purely mathematical concepts that cannot be useful in everyday life. On the contrary, these components of mathematical analysis are widely used not only in algebra and physics, but also in geography, biology, economics, IT, and electrical engineering.

Let's look at an illustrative example of how the derivative can be used to assess the growth of video popularity on social networks.

For example, you uploaded a video that received 1 view during the first hour, 3 during the second hour, 9 during the third hour, and 16 during the fourth hour. This can be represented as functions, where the hours will be located on the x-axis (x), and the number of views will be on the ordinate (y) axis.

Having placed the corresponding view values in front of each hour, we connect them with a line originating at point 0. If we draw a line parallel to the x-axis from the point corresponding to the second hour, and a line parallel to the y-axis from the point corresponding to the third hour, they will intersect under the graph of the function. Subtracting 2 hours from 3 hours, and 4 views from 9 views (corresponding to the point of intersection of the lines), we get:

  • f'(x) = Δf / Δx = 5 / 1 = 5.

This is the desired derivative, meaning that the average effectiveness of your video over the period from 0 to 4 hours is 5 views per hour. A similar calculation can be made for any other interdependent changing quantities: speed and distance traveled, current and voltage, time and frequency.

Application of derivatives in various fields

It is difficult to find at least one area of human activity where, in one form or another, mathematical laws would not be used.

In the case of functions and their derivatives, which are part of mathematical analysis, the most common areas are electrical engineering, economics, physics, chemistry, biology, geography and sociology.

It is difficult to imagine them not only without integration, but also without differentiation, which is the finding of derivatives of functions. With the help of derivatives today they determine:

  • Expected decrease/increase in income when prices for goods increase.
  • Dependence of budget revenues on the introduction of certain customs duties.
  • Energy efficiency of electrical equipment operating on alternating current.
  • The speed and dynamics of chemical reactions.
  • The influence of biological populations on each other within common habitat halos.
  • Features of the Earth's electromagnetic field.
  • Nuclear geophysical indicators.
  • Expected population in individual cities and countries.

Most often, the argument (x) in such studies is time (t), and the function is a quantity that changes in relation to it: quantity, speed, voltage, mass. The more often it is measured in dynamics, the smaller the argument (x) will be, and the higher the differentiation accuracy will be.

For example, if you take speedometer readings every 30 seconds, you will get a much more accurate graph of the function than if you take measurements every 7-10 minutes.

The derivative is also indispensable in sociological research. At one time, it formed the basis of the social model of Thomas Malthus and objectively reflected population growth in proportion to the size of the population at the current time.

This model was used in the United States from the 1790s to the 1860s, but was then abandoned due to fundamental changes in demographic dynamics: not only in the United States, but throughout the world. But this does not mean that the derivative has become useless in this area. It’s just that one function of two interdependent quantities was replaced by others that better reflect modern realities.