Derivatives

Submitted To: Ma’m Sapna MakhdoomSubmitted By:

• Irum GulBahar 02• Hajrah Majeed 14• Humera Yousaf 19• Amna Ayub 21

Topic: DerivativesSession: 2013-17Department: Mathematics

Mirpur University Of Science & Technology(MUST)

• Dedication:We dedicate this project named “ Derivatives”

to our parents and Family members.

• AbstractIn this project we have discussed about

Derivatives such as Definition , History, Real life applications, and Application of

derivatives in different sciences.

• AcknowledgementFirst of all we would like to thank Allah almighty

for making this project possible for us. Then special thanks to Ma’m Sapna Makhdoom for helping us in completing this project

Contents:

1. Definition of Derivative2. History3. Real life Applications4. Applications in Sciences

Definition of Derivative:

1. The Derivative is the exact rate at which one quantity changes with respect to another.

2. Geometrically, the derivative is the slope of curve at the point on the curve.

3. The derivative is often called the “instantaneous “ rate of change.

4. The derivative of a function represents an infinitely small change the function with respect to one of its variables.

• The Process of finding the derivative is called “differentiation.”

History:• Modern differentiation and derivatives are usually cradited to “Isaac Newton” and

“Gottfried Leibniz”.• They developed the fundamental theorem of calculus in the 17th century. This

related differentiation and integration in ways which revolutionized the methods for computing areas and volumes.

• However , Newton’s work would not have been possible without the efforts of Isaac Borrow who began early development of the derivative in the 16th century.

Real life Applications of Derivatives

Automobiles• In an automobile there is always an odometer and a

speedometer. These two gauges work in tandem and allow the driver to determine his speed and his distance that he has traveled. Electronic versions of these gauges simply use derivatives to transform the data sent to the electronic motherboard from the tires to miles per Hour(MPH) and distance(KM).

Radar Guns• Keeping with the automobile theme from the previous slide ,

all police officers who use radar guns are actually taking advantage of the easy use of derivatives. When a radar gun is pointed and fired at your care on the highway. The gun is able to determine the time and distance at which the radar was able to hit a certain section of your vehicle. With the use of derivative it is able to calculate the speed at which the car was going and also report the distance that the car was from the radar gun.

Business• In the business world there are many applications for

derivatives. One of the most important application is when the data has been charted on graph or data table such as excel. Once it has been input, the data can be graphed and with the applications of derivatives you can estimate the profit and loss point for certain ventures.

Graphs: • The most common application of derivative is to analyze

graphs of data that can be calculated from many different fields. Using derivative one is able to calculate the gradient at any point of a graph.

Applications of Derivatives in Various fields/Sciences:

Such as in:–Physics –Biology –Economics–Chemistry–Mathematics–Others(Psychology,

sociology & geology)

Derivatives in Physics• 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 W.R.T time is acceleration.

• Newton’s second law of motion states that the derivative of the momentum of a body equals the force applied to the body.

Derivatives in Biology• Population growth is another instance of the

derivative used in the sciences.• Suppose n=f(t) is the number of individuals

in some animal or plant population at time t. the change in the population size between time t1 and t2 ∆n=f(t2)-f(t1).

• The average rate of growth is then is:Average rate of growth is = (∆n/ ∆t)=(f(t2)-

f(t1))/(t2-t1)• The instantaneous rate of growth is the

derivative of the function n with respect to t, i.e.

growth rate=lim(∆t→0) (∆n/ ∆t)=(dn/dt)

Derivatives in Biology:• The instantaneous rate of change does not make exact

sense in the previous example because the change in population is not exactly a continuous process. However, for large population we can approximate the population function by a smooth(continuous) curve.– Example: Suppose that a population of bacteria

doubles its population , n, every hour. Denote by n0 the initial population i.e. n(0)=n0. In general then,

n(t)=2t no

– Thus the rate of growth of the population at time t is (dn/dt)=no2tln2

Derivatives in Economics:• Use of derivatives in Economics is as follows:• Let x represent the number of units of a certain commodity produced

by some company. Denote by C(x) the cost the company incurs in producing x units. Then the derivative of C(x) is what’s called the marginal cost:

Marginal cost =(dC/dx)

• Furthermore, suppose the company knows that if it produces x units, they can expect the revenue to be R(x),i.e. the revenue is a function of the number of units produced. Then the derivative of R(x) is what’s called the marginal revenue.

Marginal revenue= (dR/dx)

• If x units are sold, then total profit is given by the formula:P(x)=R(x)-C(x)

• The derivative of profit function is the marginal profit:Marginal profit=(dP/dx)= (dR/dx)-(dC/dx).

Derivatives in Chemistry• One use of derivatives in chemistry is

when you want to find the concentration of an element in a product.

• Derivative is used to calculate rate of reaction and compressibility in chemistry.

Derivatives in Mathematics: The most common use of the derivatives

in Mathematics is to study functions such as:

• Extreme values of function• The Mean Value theorem • Monotonic functions• Concavity & curve sketching• Newton’s Method etc.

Connecting Derivatives

Derivatives

Continuity

Inflection Points

Motion Proble

msVectors

Mean Value

Theorem

Implicit Function Theorem

Estimating

Graphs

Some other Applications of Derivatives• Derivatives are also use to

calculate:1. Rate of heat flow in Geology.2. Rate of improvement of

performance in psychology3. Rate of the spread of a rumor in

sociology.

Conclusion:• Derivatives are constantly used in

everyday life to help measure how much something is changing. They're used by the government in population censuses, various types of sciences, and even in economics. Knowing how to use derivatives, when to use them, and how to apply them in everyday life can be a crucial part of any profession, so learning early is always a good thing.