**DIFFERENTIAL CALCULUS AND ITS APPLICATION TO EVERY DAY LIFE**

**ABSTRACT**

In this project we review the work of some authors on differential calculus. Different types of functions and the method for finding their derivatives were also considered the application of differential calculus was death with to show the importance of this work.

**TABLE OF CONTENT**

**Chapter One **

Introduction

1.1Â Â Â Â Â Â Â Â Background of the study

1.2Â Â Â Â Â Â Â Â Differential calculus

1.3Â Â Â Â Â Â Â Â Statement of the problem

1.4Â Â Â Â Â Â Â Â Purpose of the study

1.5Â Â Â Â Â Â Â Â Scope of the study

1.6Â Â Â Â Â Â Â Â Â Importance of the study

1.7Â Â Â Â Â Â Â Â The limitation of the study

1.8Â Â Â Â Â Â Â Â Definition of terms

**Chapter Two **

Literature Review

2.1Â Â Â Â Â Â Â Â Introduction

2.2Â Â Â Â Â Â Â Â Limits of a functions

2.3Â Â Â Â Â Â Â Â Gradient

**Chapter Three **

3.1Â Â Â Â Â Â Â Â Introduction

3.2Â Â Â Â Â Â Â Â Differentiation of a constant

3.3Â Â Â Â Â Â Â Â Differentiation of the sum of difference of function

3.4Â Â Â Â Â Â Â Â Differentiation of a product

3.5Â Â Â Â Â Â Â Â Differentiation of a quotient

3.6Â Â Â Â Â Â Â Â Differentiation of trigonometric function

3.7Â Â Â Â Â Â Â Â Differentiation of function of a function

3.8Â Â Â Â Â Â Â Â Differentiation of inverse function

3.9Â Â Â Â Â Â Â Â Second and higher derivatives

3.10Â Â Â Â Â Differentiation of implicit function

3.11Â Â Â Â Â Differentiation of parametric equation

**Chapter Four**

Application of differentiation

4.1Â Â Â Â Â Â Â Â Introduction

4.2Â Â Â Â Â Â Â Â Maximum points

4.3Â Â Â Â Â Â Â Â Minimum points

4.4Â Â Â Â Â Â Â Â Point of inflexion

4.5Â Â Â Â Â Â Â Â Maximum and minimum problem

4.6Â Â Â Â Â Â Â Â Tangent and normal to a curve

4.7Â Â Â Â Â Â Â Â Velocity and acceleration

Chapter Five

Summary

Conclusion

Recommendation

**References **

**CHAPTER ONE**

**INTRODUCTION**

**1.1Â BACKGROUND OF THE STUDY**

Â This topic differential calculus, a field in mathematics is the study of how functions change when their inputs change. The primary object of study in differential calculus is the derivative. A closely related notion is the differential. The derivative of a function at a chosen input value describe the behaviour of a function near the input value. For a real value function of a single real variable, the derivative at a point equal the slope of the tangent line to the graph of the function at a point. In general the derivative of a function at a point determine the best linear approximation to the function at that point. The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration. The word differentiation has to do with the gradient of a line and point. Calculus an aspect of mathematics, which provide method for solving two large classes of problem. The first of these involves finding the rate at which a variable quantity is changed. The second type of problem calculus deal with is that of finding a function when its rate of change is given.

Â Calculus as an aspect of mathematics was invented to provide a tool for solving problems involving motion. In other to give a precise meaning to the motions of velocity and acceleration, it is necessary to use one of the fundamental concepts of calculus, the derivative. Although calculus was introduced to help solve problem in physics, it has been applied to many different fields. One of the reasons for its wide usage is the fact that the derivatives are useful in the study of rates of change of many entities other than objects in motion. For example a biologist may exmploy it in the investigation of the rate of growth of bacterial in a culture.

**1.2Â Â Â Â Â Â Â Â DIFFERENTIAL CALCULUSÂ **

One of the fundamental concepts of calculus is the derivatives or differential calculus. Although calculus was introduced to help solve problems in physics but it has been applied to different field the reason for this wide usage is the study of rates of changes of many entities other than object in motion such as velocity and acceleration. For example chemist may use derivative to forecast the outcome of various chemical reaction.

**1.3Â Â Â Â Â Â Â Â STATEMENT OF PROBLEMÂ **

Differential calculus is a topic in mathematics, which is taught in most institution. Students find it difficult in solving differential calculus because of the way they are been taught or students low level of understanding or the time given to them is not enough for them to be properly grained in differential calculus but if they on their own can spend extra time in studying, they will not find it difficult in knowing how to solve differential calculus.

**1.4Â Â Â Â Â Â Â Â PURPOSE OF THE STUDY**

The purpose of this study is for the researcher to expose students to differential calculus in order for them to stand a ground when they come across differential calculus, if the student can be deeply rooted in differential calculus; I believe they will find no problem in differential calculus in general. However, the researcher will want to simplify the system of approaching differential calculus, hence get rid of the tension students once get in differential calculus and create confidence in them and they can do better than their predeccessor in this field.

**1.5Â Â Â Â Â Â Â Â SCOPE OF THE STUDY**

The study is limited to textbooks on calculus written by various authors, dictionaries and possible encyclopedia.

**1.6Â Â Â Â Â Â Â Â IMPORTANCE OF THE STUDY**

At the end of this study differential calculus would have been successfully simplified for the average learners to understand. Some of the problems students fails to understand in differential calculus are function, continuous function, explicit function, implicit function, dependent variable, independent variable, limit of a function and so on will be analysed for a perfect understanding.

**1.7Â Â Â Â Â Â Â Â DEFINITION OF TERMS**

**Mathematics**: Is the branch of science concerned with number, quantity and space.

**Calculus**: Calculus is the mathematics of change and motion. This implies that calculus is the type of mathematics that deals with rate of change.

**Delta**: Means change of variable it could be written as âˆ‚ orÂ Î”.

**Variable**: A variable is a symbol such as x that may take any value in some specified set of number,.

**Function**: A function is a set of ordered pairs of number (x,y) such that to each value of the first variable (x) the corresponds a unique value of the second variable (y).

**Continuous function**: A function f which is defined in some neigbhourhood of c is said to be continuous at C provided.

Lim f(x) = f (c)

XÂ Â Â X

**Explicit functions**: If Y = X^{2}-4X + 2, Y is completely defined interms of X and Y is called an explicit function.

**Implicit function**: This is a kind of equation with More than one variable that is having two variables known as â€˜Yâ€™ and â€˜Xâ€™ e.g.

Y = 4Â X Y + Y + Y + X or Y = 4 + XY or 4 = X + Y.

**Independent variable**: The variable x which yield the first of two number in the ordered pair (X, Y) is often called independent variable or argument of the function f.

**Dependent variable**Â The second variable Y is called the dependent variable.