### DIFFERENTIATION AND ITS APPLICATION

**CHAPTER ONE**

**1.1 ****INTRODUCTION**

Man has always been fascinated by the rate at which physical and nonphysical things change. Astronomers, physicists, chemists, engineers, business enterprises, and industries all strive for accurate values of these time-varying parameters.

As a result, the mathematician devotes his time to studying rate of change concepts. The concept of rate of change gave rise to the concept of DIFFERENTIATION in calculus.

Another topic is known as INTEGRATION.

CALCULUS is a broad subject that combines integration and differentiation.

As a result, in order to provide an excellent work that is of great utilitarian value to students of science and social science, the research project is divided into four chapters, with each of these chapters broken down further.

up into sub units.

The first chapter includes an introduction, scope of study, purpose of study, review of related literature, and limitations.

The second chapter focuses on the fundamentals of calculus, specifically functions of a single real variable and their graph, limits, and continuity.

Chapter three properly deals with differentiation, which also includes the gradient of a line and a curve, as well as the gradient function, also known as the derived function.

The fourth chapter discusses the use of differentiation, summary, and conclusion.

1.2 **Scope Of The Study And Limitation**

This research will provide a detailed examination of differentiation and its application.

It will cover the fundamentals of calculus as well as limits and continuity.

For this work to be completed effectively, time, important related text books, and financial considerations must be available.

1.3 **Purpose Of The Study**

This project’s goal is to teach the operational principles of differentiation in calculus. In addition, many problems that have long been considered by mathematicians and scientists will be examined.

1.4 **Significance Of The Study**

The importance of this study cannot be overstated, particularly in this modern era when everything in the entire world is changing with respect to time, because the rate of change is an integral part of operation in science and technology, and thus there is a need to ascertain the origin of calculus and its application.

Finally, the purpose of this work is to examine the use of differentiation in calculus.

**1.5 LITERATURE REVIEW**

Calculus, also known as infinitesimal calculus in the past, is a mathematical discipline that deals with limits, functions, derivations, integrals, and infinite series.

Throughout the 17th century, ideas leading up to the concept of function, derivatives, and integrals were developed, but Isaac Newton and Gottfried Leibniz made the decisive step.

Ancient Greek Calculus Precursors (Forerunners)

The Greek mathematicians are credited with making extensive use of infinitesimals.

Democritus is the first recorded person to seriously consider the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone’s smooth slope prevented him from accepting the idea at around the same time.

Zeno of Elea further discredited infinitesimals by articulating the paradoxes that they generate.

Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which allowed for the computation of the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes.

Archimedes of Syracuse expanded on this method while also inventing heuristic methods that resemble modern-day concepts in some ways. These methods were not incorporated into a general framework of integral calculus until Newton’s time.

However, it should not be assumed that infinitesimals were put on a firm footing during this period.

Only when it was accompanied by a suitable

Greek mathematicians would accept a proposition as true if it had geometric proof.

Modern calculus pioneers

European mathematicians Isaac Barrow, Rene Descartes, and Pierre deferment invented the concept of a deferment in the 17th century.

The concept of a derivative was discussed by Blaise Pascal, John Wallis, and others. Fermat developed an adequality method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation in method sad disquirendam maximum et minima and De tangetibus linearism Curvarum.

Later, Isaac Newton would write that his own early ideas about calculus stemmed directly from formats method of drawing tangents.

On the integral side, in the 1630s and 40s, Cavalieri developed his method of in divisibles, which provided a modern version of the ancient Greek method of exhaustion and division.

Computing the area under the curves Xn of higher degree using Cavalierr’s quadrate formula, which Archimedes had previously only computed for the parabola.

Torricili extended this work to other curves such as the cycloid, and Wallis generalized the formula to fractional and negative powers in 1656.

In a 1659 treatise, fermat is credited with devising an ingenious method for directly calculating the integral of any power function.

Fermat also discovered a method for determining the centers of gravity of various plane and solid figures, which influenced subsequent quadrature work.

In the mid-1700s, James Gregory, influenced by Fermat’s contributions to both tangency and quadrature, was able to prove a restricted version of the second fundamental theorem of calculus. The

Isaac Barrow provided the first complete proof of the fundamental theorem of calculus.

In the late 17th century, Newton and Leibniz independently developed the surrounding theory of infinitesimal calculus.

In addition, Leibniz worked hard to develop consistent and useful notation and concepts.

Newton made some of the most significant contributions to physics, particularly integral calculus.

Prior to Newton and Leibniz, the term “calculus” was used to refer to any body of mathematics, but in the years that followed, “calculus” became a specific term. Based on their insight, it became a popular term for a field of mathematics. Leibniz invented the integral symbol and wrote the derivative of a function y of the variable x, both of which are still in use today.

## Leave a Comment