DIFFERENTIATION AND IT’S APPLICATIONS
ABSTRACT
The project is written simply to illustrate on differentiations and their applications. The formation and classification of differentiation, the basic techniques of differentiations, list of derivatives and the basic applications of differentiation, which include motion, economic and chemistry.
TABLE OF CONTENT
CHAPTER ONE
1.1Background Of Study
1.2 Statement Of The Problem
1.3 Aim And Objectives
1.4 Definitions Of Terms
1.5 Differentiation rules
CHAPTER TWO
2.1 Introduction
2.2 Roberual Method Of Tangent Lines Using Instantaneous Motion
2.3 The Elusive InversesThe Differential
2.4 Newton and Leibniz
2.5 The Ellusive Inverses
CHAPTER THREE
3.0 Differential Calculus
3.1 List of Derivatives
3.1.1 Simple Functions
3.1.2 Exponential And Logarithmic Functions
3.1.3 Trigonometric Functions
3.1.4 Inverse Trigonometric Functions
3.2 Techniques Of Differentiation
3.2.1 The Power Rule
3.2.2 The Product Rule
3.2.3 The Quotient Rule
3.2.4 The Chain Rule
3.2.5 The Implicit Differentiation
3.2.6 The Higher Order Derivation
CHAPTER FOUR
4.0 Applications of differentiation
4.1 Introduction
4.2 Application To Motion
4.3 Application To Economics
4.4 Application To Chemistry
CHAPTER FIVE
5.0 Summary and Conclusion
5.1 Summary
5.2 Conclusion
CHAPTER ONE
Differentiation is a process of looking at the way a function changes from one point to another. Given any function we may need to find out what it looks like when graphed. Differentiation tells us about the slope (or rise over run, or gradient, depending on the tendencies of your favorite teacher). As an introduction to differentiation we will first look at how the derivative of a function is found and see the connection between the derivative and the slope of the function.
Given the function f (x), we are interested in finding an approximation of the slope of the function at a particular value of x. If we take two points on the graph of the function which are very close to each other and calculate the slope of the line joining them we will be approximating the slope of f (x) between the two points. Our xvalues are x and x + h, where h is some small number. The yvalues corresponding to x and x + h are f (x) and f (x + h). The slope m of the line between the two points is given by
Where and are the two points.
Hence m is called the slope or change which is the differentiation.
The primary objects of study in differentiation are the derivative of a function, related notions such as the differential and their applications. The derivate of a function at a chosen input value.
1.1 BACKGROUND OF THE STUDY
Historically, the primary motivation for the study of differentiation was the tangent line problem: for a given curve, find the slope of the straight line that is tangent to the curve at a given point. The word tangent comes from the Latin word “tangens”, which means touching. Thus, to solve the tangent line problem, we need to find the slope of a line that is “touching” a given curve at a given point, or, in modern language, that has the same slope. But what exactly do we mean by “slope” for a curve?
The solution is obvious in some cases: for example, a line is its own tangent; the slope at any point is . For the parabola the slope at the point is 0; the tangent line is horizontal.
In mathematics, differential calculus (differentiation) is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus (integration).
The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a realvalued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.
Differentiation and integration are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.
Differentiation has applications to nearly all quantitative disciplines. For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. The derivative of the momentum of a body equals the force applied to the body; rearranging this derivative statement leads to the famous F = maequation associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories.
Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.
1.2 STATEMENT OF THE PROBLEM
Differentiation is a technique which can be used for analyzing the way in which functions change. In particular, it measures how rapidly a function is changing at any point. This research intends to examine the differential calculus and its various applications in various fields, solving problems using differentiation. This work is to show the important of differentiation, that it is not limited to mathematics alone, it is applied in our day to day life, it has its own share in our sciences * motion, economic, chemistry. e.t.c).
1.3 AIM AND OBJECTIVES
· To show that differentiation is not limited to mathematics alone.
· To relate differentiation to velocity and acceleration in motion.
· To relate differentiation in calculating rate of change of chemical reactions.
· How differentiation affects performance of demand and supply between buyers and sellers in economic.
1.4 DEFINITIONS OF TERMS
Differentiations have a lot of terms (in terms of identifications) which we use in identifying what type of differentiation is it or what type of differential equation are we working on, which are called The Notations For Differentiation. There are two main types of notation used to denote the derivative of a function.
Lagrange’s Notation is to write the derivative of the function as
Leibniz’s Notation is to write the derivative of the function as
Two other notations are worth mentioning
Newton’s Notation is to write the derivative of using a dot
Euler’s Notation is to use a capital D i.e.
The Lagrange and Leibniz notation will be considered in some situations involving differentiation.

Lagrange 
Leibniz 
Function 

Derivative 


2^{nd} Derivative 


Higher Derivative 


Integral 

1.5 Differentiation rules
Product Rule 

Lagrange 

Leibniz 
Chain Rule 

Lagrange 

Leibniz 
Implicit Differentiation 
(say of 
Lagrange 

Leibniz 
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