Derivatives are a type of mathematical operation, which is extremely relevant for the study of functions and graphs, so many careers related to mathematics, such as physics and all variations of engineering, know about them. Certainly, these operations are known for their great complexity, however, they are really useful, since they facilitate the calculation and development of other mathematical operations.

Derivative benefits

Although these types of operations can be really complicated, they are extremely necessary in the mathematical field, since by using them it is possible to obtain a diverse variety of benefits, among which the following should be mentioned:

They help to know the tangent line of any type of graph

In earlier times, it was extremely difficult to draw the tangent line in certain types of graphs, this due to their shape. That is why, different scientists and mathematicians tried to design an operation that would facilitate this procedure, Apollonius of Perge being the individual who partially resolved this conflict. However, it was Isaac Newton who gave a much more complete solution; which is known today as derived functions.

Derivatives allow you to find the tangent line of any type of graph, with respect to a given point. All this through a mathematical function that reveals the value of a specific coordinate, which corresponds to the place where the tangent of the graph should be.

They are used to measure the speed of change of the value of a function

Derivatives are operations that allow us to calculate the way in which the limit of the speed of change can affect a function in a specific interval; provided that said interval is considered as an independent variable within the operation. All this generates the existence of the value of a derivative of a function, at a given point.

They seek to reduce a function to its maximum expression

One of the most important aspects of derivatives is that these mathematical operations seek to reduce functions to their maximum expression. That is why, they are part of the necessary operations in algebra, since they allow to simplify extremely complex operations, in order to obtain operations much simpler to understand and express (mathematically speaking).

They allow to know the rates of variation of a function

Another of the applications and uses of derivatives is that they allow the study of variation rates; the variation rates are those numbers that represent the rise or fall of a function, when increasing the independent variable from one value to another. Therefore, this benefit is highly relevant in the calculation of maximum and minimum values ​​of a function, as well as its concavity and convexity.

Worked examples of derivative operations

To better understand how to solve derivative operations, practice with exercises is recommended. Bearing this in mind, we proceed with the following example of operations:

This exercise corresponds to that of a logarithmic derived function and is expressed as follows:

To solve this operation it is necessary to apply the chain rule, so, first you must write the derivative of the logarithm, and then multiply it by the derivative of the argument (the derivative of the argument is everything that is inside the parentheses) .

Derivative types

Variables, being so complex operations, have a great variety of qualities and characteristics, which is why it is necessary to classify them. Among the most well-known and important types of derivatives are the following:

Derivative of a constant

This type of operation is governed by a fairly simple principle, which refers to the fact that the derivative of a constant function will always be zero. It is known as the simplest and easiest type of derivative to apply, due to its simplicity.

Derivative of x

This type of derivative reflects that the derivative of x is always equivalent to 1; This means that, in case of obtaining a derivative of x, its result will always be equal to unity.

Derivative of a power

Also known as a potential function, it is an operation in which the derivative is equal to the exponent, times the base raised to the exponent minus one, times the derivative of the base. This means that the base of the derivative will be equal to the exponent, by the base raised to the exponent minus one.

Derived from a root

In this type of derivative, the one obtained from a root is searched. However, for this, it must be taken into account that the derivative of a root is equal to that obtained from the radicand, which is divided by the product of the root, by that root of degree “n”, of the elevated radicand to “n-1”.

Derivative of the square root

This type of derivative is quite similar to the derivative of the function of a root, however, it differs because, in the derivatives of the function of a square root, the result of this is obtained from the radicand divided by the double of the root.

Differences between derivatives and integrals

Both derivatives and integrals are an important part of mathematical calculation tools. However, both types of operations have important differences between them, among which are mentioned:

  • Derivatives are responsible for summarizing a mathematical operation in its maximum expression, while integrals seek to expand the operation.
  • While the derivatives represent the slope of a line, the integrals represent the curved space under a function (this space is also known as an antiderivative).
  • The integrals are used in the calculation of areas that have curved lengths, while the derivatives allow to know the change of the volume of a body, with respect to the area of ​​its surface.

Uses of derivatives in real life

Certainly, the calculation of derivatives is not essential in real life, or at least in certain professional fields. However, these mathematical functions have a great participation in everyday life, a clear example of the use of derivatives, is that they serve to find intervals of growth or decrease of values ​​(only if these can be represented in the form of functions), which which can be very useful in the economic field.

For its part, it is important to bear in mind that derivatives are also used to understand formulas and aspects related to industries, since these operations allow us to know the effectiveness of the systems and for this, it is necessary to express them in the form of a function. Taking this into account, it is possible to summarize that, by using derivatives, it is possible to more accurately identify the performance of a system, and in this way, knowing this aspect makes it easier to decide the factors that should be improved or maintained.

Conclusion: Are derivatives necessary in life?

Derivatives are operations that have a large number of uses in everyday life, since administrative, architectural, chemical, economic and accounting areas benefit from the usefulness of these functions. Therefore, it is necessary to know about these, as well as their intervention in daily life.

Samantha Robson
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Dr. Samantha Robson ( CRN: 0510146-5) is a nutritionist and website content reviewer related to her area of ​​expertise. With a postgraduate degree in Nutrition from The University of Arizona, she is a specialist in Sports Nutrition from Oxford University and is also a member of the International Society of Sports Nutrition.

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