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How to Differentiate – Calculus and Mathematics

How to differentiate – meanings and history

We all remain in constant motion and change because we are supposed to behave naturally, hence, you can say that constant movement and change is a natural process. People who observed this process deeply are famous for their work and known as mathematicians, astronomers, and astrologers. Besides the topic – how to differentiate – we will also talk about some well-known people in the following lines.

The word differentiate comes in from Medieval Latin. It stands for, to recognize or determine what makes someone or something different from others. In Mathematics, it is all about transforming a function into its derivative. It is about finding the rates of change by comparing two values or quantities. We usually differentiate when the rate of change is not constant.

The branch of math which provides ways for voluminous investigation of a number of processes of motion, change and reliance of one value on another is known as calculus. The study of the first module of calculus gives us an insight of some of the very basic techniques of mathematical analysis or simply analysis.

The development of Mathematics

The development of methods in mathematics to know how to differentiate or analyze is stirred by the problems faced during the study of physics in the course of the sixteenth century. This is the period when the investigations related to motion were mostly done.

On the basis of his own experiments and the study of Archimedes, Galileo Galilee laid the foundations of the new mathematical mechanics, which later on proved to be an indispensable science for the newly emerging technology.

Revolutionary work started

The lust to expand the trade and land explorations made it significant to make improvements in the navigation-related techniques. This resulted in turn to a large extent of developments in astronomy. In 1543, a renaissance mathematician and astronomer namely Nicolaus Copernicus took the initiative and published his revolutionary piece of writing with the title “On the revolution of the heavenly bodies “. A German mathematician, astronomer and astrologer Johannes Kepler followed his footsteps and blasted his famous work “New Astronomy” in 1609, telling about his first and second laws of motion of planets around the sun. Kepler published his “Harmony of the world” in 1618 containing his third law of motion.

In the course of a renaissance, Europeans became familiar with Greek mathematics through the Arabic translations. Astonishingly, this occurred after a scientific stagnation of almost one thousand years. The theories offered by Euclid, Ptolemy and al-Khwarizmi were transformed into Latin only in the twelfth century as Latin remains the widely used language of Western Europe for expressing scientifically. At the same time, the ancient Greek and Roman techniques of calculation were progressively taken over by the vastly better Indian sub-continental method. This method paved its way into Europe through the Arabs, as well.

The study of Calculus

The study of calculus, a branch of mathematics, comprises of three key aspects; differentiation, integration and the relation between these two. If someone wants to know how to differentiate between two values, he/she first needs to find out the rate of change between these two values or quantities, while integration is something that calculates the area of a figure. The relationship between differentiation and integration guides us to the “Fundamental Theorem of Calculus” or “Newton-Leibniz Formula” (as it is called) with which accordingly, integration and differentiation are reverse processes of each other.

During the seventeenth century there was a great breakthrough in knowing differential calculus when Fermat wrote a letter to Gilles and told him how he found the maxima and minima of a function. Nowadays, it is used as a fundamental way to find out the critical points by setting the derivative as zero. Barrow used a triangle (having dx, dy, ds as its sides) to find tangents. This is how we apply derivatives to find tangents in modern days. Hence, we see that humans mastered the art of how to differentiate and integrate by the seventeenth century.

Dispute in Invention of Calculus

As you may know, the calculus was known as infinitesimal in its early times. It is regarded as the branch of mathematics which primarily deals with functions, limits, derivatives, infinite series and integrals. Isaac Newton and Gottfried Leibniz are widely accepted as the independent inventors of calculus. Though both had a bitter dispute and claimed that other one had pilfered his work. This dispute remained alive, in the mid of seventeenth century, until the end of their lives. However, both have laid the foundation of differential and integral calculus.

Apart from mathematics we always differentiate between two things or people in our daily lives. Especially teachers always work with different students having a wide range of backgrounds. They differ in ages, abilities and skills. Teachers always find themselves teaching a mixed level of classes consisting of learners who have a variety of needs. Some students have to struggle to complete their mathematical assignments. Some have problems with very basic skills. Due to their problems with solving mathematical questions they fall behind in grade level. Some of them are advanced and they only need enrichment. Challenges make them sharp and if they are not challenged they may start losing their interest in studies.

There is still a category of students who fall in the middle. This ‘middle group’ makes the whole lot of the majority amongst the others. Teachers may feel stressed to teach this majority. They feel pressured to gear both the core curriculum and the process of learning towards this ‘middle group’. Mathematics needs to be taught in a way so that all the students get the benefit and understand why these skills are much needed. All students should be taught about the important mathematical concepts and skills. They all should receive the right to use the same core content. Hitherto a uniform approach to the process of learning cannot exhibit results in the case of struggling, middling and advanced students. In this context, students receive differentiated instructions related to their learning process and can benefit from it through their specific readiness level, absorbing the style and interest. Students should also be introduced to an enjoyable way to solve mathematical challenges and learn calculating skills that can engage their attention by offering an instant feedback facility.

Assessing The Students’ Understanding Levels

Ascertain whether your students are likely to be advanced, middling or struggling. For this, you need to look over their recent grades and test scores. Go through their sample assignments. Measure the range of your students’ present skills and knowledge. Design your own goals for reading to teach your students. Gather the content that your students are going to learn. Think about both the short and long-term projects that can assist them to achieve their goals. Devise your own overall goals for the class.

Talk to your students about mathematics. Ask questions about the kinds of mathematical tasks students do. Do your students like calculating? Why or why not? Do they feel it easy to understand calculus or difficult? What they like or do not like to learn about? What new concepts they recently learned and what new ideas they do not understand?

The importance of calculus in our lives

Let your students realize that calculus is just not a vocational training adventure. In some ways, students should understand and learn calculus with the same thoughts that they have while studying Darwin, Marx, and Voltaire. These ideas belong to a basic part of our society; these ideas have influenced us to perceive the world correctly and to perceive our right place in the world.


Through assessment build your students’ math learning skills. Make your students to apply the newly learned skills. Check for the responses of advanced, middling or struggling students separately and make a future line of action for them.

Assess what your students learned as without assessment you cannot achieve your goals so as your students. Check for whether students’ self-assessments are accurate or not related to their learning and what they marked as easy or difficult. What are their understanding levels about their learning process? Assess yourself as well and predict the needs of the students. How many of them need more practice regarding certain mathematical skills? How many of them can master new skills comparatively faster?

Break students into small mixed-level units. Have each unit think they are making a mathematical puzzle for some database. Inspire them to create the puzzle by choosing the questions from classroom material. Urge them to work collectively and design, draw and write their own puzzles from the mathematical calculations they have collected.

Focus on struggling students to make sure that they understand what a polynomial function is. Ask them to find the derivative by putting the correct values in the given formula or equation.

After the completion of each assigned task do not forget to praise the achievements to embolden the abilities of your students. Ask them to go over all the scores. How do you feel about the learning of your students? Have your students learned a lot? Have they just sharpened their skills? Is teaching struggling, middling and advanced students getting easier and faster?

Through your efforts your students have learnt the basic skills of mathematical calculation and differential and integral calculus. Ask them to learn through various available resources and go through a case study format and gather information on writing an apa essay on the history of calculus

Encourage them to apply their skills and provide the solutions related to:

  • how to write a coursework related to modern geometry.
  • writing a descriptive essay on calculus and
  • how to critique an article, which is written in the perspective of mathematical analysis

We started studying the basic concepts of differentiation and integration in the curriculum since the school phase. In the course of study of mathematics in engineering college, we come across advanced concepts of differentiation and integration.

These concepts come with their application in the form of differential equations. Then they find their way towards implementation in key subjects of engineering. In Electrical engineering, we find differential equations playing their role in solving the basis of power flow problems and system related controls of the machines. Heat-mass transfer problems are solved through differential equations in Mechanical engineering. Structural engineering and stress-strain problems also need differential and integral calculus. Even computer engineering, which seems to be totally isolated from these typical analog concepts demands the use of differential and integral calculus for the programming design and its subsequent application. It is, therefore, pertinent to have your concepts clear about how to differentiate to get an efficient grasp over the advanced studies.

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