The document discusses differentiation and its applications. It provides a brief history of differentiation and introduces concepts such as the derivative and reverse process of integration. Some key applications of differentiation discussed include using it to determine maximum/minimum values, in subjects like physics, chemistry, and economics, and in devices like odometers, speedometers, and radar guns. Two surveys were conducted on the awareness and uses of differentiation. In conclusion, differentiation can help improve devices and make tomorrow better by finding how one variable changes with respect to another.

1. SHREE VAISHNAV ACADEMY
BALVIGYAN
2016-17

2. Topic - Differentiation
Presented by-
Theme - Design for better tomorrow
Sub Theme - Mathematical solution of real life

3. Acknowledgements
I would like to express my special thanks of gratitude to my
teacher as well as our principal H.C.Tiwari who gave me the
golden opportunity to do this wonderful project on the topic
DIFFERENTIATION, which also helps me to know about so many
things. I am really thankful to them.
Secondly I would also like to thank my parents and friends who
helped me a lot in finalizing this project within the limited time
frame.

4. Content
1.History Differentiation
2. Introduction of Differentiation
3. Reverse of Differentiation
4. Basic Formulas of Differentiation
5. Application of Differentiation
6.Differentiation in Economics
7. Differentiation in physics
8. Radar gun
9. Differentiation in Chemistry
10. Differentiation in Biology
11. Use of Differentiation in odometer & speedometer & Vector Derivative
12. Differentiation in society
13. Results of survey
14. Conclusion

5. Modern Differentiation and derivative usually
credited to Sir Issac Newton and Gottfried Leibniz
The concept of a derivative in the sense of a tangent
line is a very old one, familiar to Greek geometers
such as Euclid (c. 300 BC), Archimedes (c. 287–212
BC) and Apollonius of Perga (c. 262–190 BC).[1]
Archimedes also introduced the use of infinitesimals,
although these were primarily used to study areas
and volumes rather than derivatives and tangents;
see Archimedes' use of infinitesimals
History of Derivative

6. Introduction of Differentiation -
Differentiation is the action of computing a derivative. The
derivative of a function y = f(x) of a variable x is a measure of
the rate at which the value y of the function changes with
respect to the change of the variable x. It is called
the derivative of f with respect to x. If x and y are real
numbers, and if the graph of f is plotted against x, the
derivative is the slope of this graph at each point.
Slope of graph = Δy/Δx

7. Reverse of Differentiation (Integration)
Integration is a way of adding slices to find the whole.Integration can be used
to find areas, volumes, central points and many useful things.
finding an Integral is the reverse of finding a Derivative.
Applications of the Indefinite Integral-Displacement from Velocity, and Velocity
from AccelerationA very useful application of calculus is displacement, velocity
and acceleration.Recall (from Derivative as an Instantaneous Rate of Change)
that we can find an expression for velocity by differentiating the expression for
displacement:
v=​ds/dt

8. Similarly, we can find the expression for the acceleration by differentiating
the expression for velocity, and this is equivalent to finding the second
derivative of the displacemen.
a=dv/dt=d^2s/dt^2
It follows (since integration is the opposite process to differentiation) that to
obtain the displacement s of an object at time t (given the expression for
velocity v) we would use:
s=∫vdt
Similarly, the velocity of an object at time t with acceleration a, is given by:
v=∫a dt
Integration -

9. 1.d(constant) /dx=0 2. d(x) /dx=1
3.d(log x) /dx=1/x 4. d(ax) /dx=axlog a
5. d(ex) /dx=e x 6. d(sin x)/dx =cos x
7.d(cos x) /dx = - sin x 8.d(tan x) /dx =sec^2 x
9.d(sec x) /dx =sec x. tan x
10.d(cosec x) /dx = - cosec x.cot x 11.d(cot x) /dx= - cosec^2 x
12.d(f).(g)/dx= f dg/dx +g df/dx
13.d[(f) /(g)] /dx =( g df/dx – f dg/dx ) /g^2
Basic Formulas of Differentiation

10. Application of Differentiation -
We use the derivative to determine the maximum and minimum values of particular
functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.).
Derivatives are met in many engineering and science problems, especially when
modeling the behavior of moving objects.
Other use of Differentiation are -
1.It is used in history, for predicting the life of a stone.
2.It is mainly used in daily by pilots to measure the pressure n the air.
3.It is also use in many subjects like Physics, Chemistry, Economics, etc.
4.It is used to solve problems including limits.
5.It is used to find local maxima and local minima

11. Most undergrad level core micro
and macro involves fairly simple
differentiation, you will do a lot of
optimisation and use the chain rule
and product rules a lot. One thing
you will have to get used to in
economics is seeing things written
as functions and differentiating
them.
We always use differentiation to
find Marginal Cost.
Differentiation in Economics -

12. Mathematical use of Differentiation in
Economics
The concept of ‘marginals’ (marginal revenue, marginal product, marginal
cost) etc is about the most important concept in microeconomics, because all
decisions are taken ‘at the margin’. Do you increase production by another
unit or just produce at the level you are doing? Well if your marginal revenue
(the amount of revenue you will earn by producing another unit of output) is
higher than your marginal cost (the amount it will cost you to produce
another unit) then go for it. If your marginal cost is higher then you don’t. As
you produce more your MR will fall and your MC will rise so you will maximise
profits by producing where MR = MC. Basic golden rule of microeconomics.

13. Because MR is basically the ‘change in revenue over the change in output’
you find it by differentiating total revenue with respect to output. Total
revenue is price x quantity.
So we have TR=PQ
MR=d(TR) /dQ
So MR=d(PQ) /dQ
PQ is P times Q, and TR and MR are ‘total revenue’ and ‘marginal
revenue’
Use of Differentiation -

14. Differentiation in physics
1.Velocity: It is the derivative of position with respect to time.
v=ds/dt
2.Acceleration: It is the derivative of velocity with respect to time
a=dv/dt
3.Momentum and Force: Momentum (usually denoted p) is mass
times velocity,
p=mv

15. 4.Total Energy: For so-called "conservative" forces, there is a function V(x)
such that the force depends only on position and is minus the derivative of V,
namely F(x)=−dV(x)/dx.The function V(x) is called the potential energy. For
instance, for a mass on a spring the potential energy is 1/2kx^2where k is a
constant, and the force is −kxThe kinetic energy is 1/2mv^2. Using the chain
rule we find that the total energy
d/dt(1/2mv^2+V(x))=mvdv/dt+V′(x)dx/dt
Force (F) is mass times acceleration, sothe derivative of
momentum is
dp/dt=d(mv)/dt=mdv/dt=ma

16. Radar gun -
When a radar gun is pointed and fired at your care on the highway.The
gun is able to determine the speed 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 that the car was from the radar gun.

17. Radar speed guns, like other types of radar, consist of a radio transmitter and receiver.
They send out a radio signal in a narrow beam, then receive the same signal back after
it bounces off the target object. Due to a phenomenon called the Doppler effect , if the
object is moving toward or away from the gun, the frequency of the reflected radio
waves when they come back is different from the transmitted waves. From that
difference, the radar speed gun can calculate the speed of the object from which the
waves have been bounced. This speed is given by the following equation:
where c is the speed of light , f is the emitted frequency of the radio waves and Δf is the
difference in frequency between the radio waves that are emitted and those received
back by the gun. This equation holds precisely only when object speeds are low
compared to that of light, but in everyday situations, this is the case and the velocity of
an object is directly proportional to this difference in frequency.
Working of Radar gun

18. Use of differentiation in chemistry
In chemistry derivative are used to calculate instantaneous rate of reaction
The instantaneous rate of reaction i.e. rate of reaction at any instant of
time is the rate of change of concentration of any one of reactant or
product at that particular instant.
Instantaneous rate of reaction = dx/dt
Here dx is small change in concentration in small interval of time dt
For example , for the reaction R P
Instantaneous rate of reaction =-d[R]/dt=+d[P]/dt

19. Use of differentiation in biology
Growth of Bacteria: Suppose a droplet of bacterial suspension is
introduced into a flask containing nutrients for the bacteria. The
bacteria undergo cell divisions and the bacterial density is observed at
intervals of time during a short period. The data is then fit to a model
describing the bacterial density, N(t), observed at time t. Assume for
three different types of bacteria, the growth rates are described
by the following differential equations,
1. dN1/dt = 2N(t)/t, has solution N(t) = t
2. dN2/dt = 2N(t), has solution N(t)= e2 t
3. dN3/dt = 0.2[1 + cos(0.5t)]N(t), has solution N(t) = e0.2t+0.4 sin(0.5t)

20. Drug Sensitivity: It is extremely important for doctors to understand the
characteristics of the drugs they prescribe to patients. The strength of the
drug is given by R(M) where M measures the dosage, i.e. the amount of
medicine absorbed in the blood, and the sensitivity of the patient’s body to
the drug is the derivative of R with respect to M. For a certain drug, the drug
strength is described by R(M) = 2M√(10 + 0.5M) where M is given in
milligrams. Find R0’(50), the sensitivity to a dose of 50mg.
Muscle Contraction: In 1938 Hill hypothesized the relationship
between the rate at which a muscle contracts, v, under a
given load, p. The Hill Equation is given by
(p + a)v = b(p0 − p)
where a, b, p0 are positive constants.

21. Use of differentiation in odometer and
speedometer -
In an automobile there is always a odometer and
speedometer. These two gauges work in tandem
and allow to determine his speed and his
distance that he has traveled .Electronic version
of this gauges simply use derivatives to transform
the data sent to the electronic motherboard
from the tires to mile per hour (MPH) and
distance (KM).
Speedometer is used to find instantaneous velocity

22. A vector derivative is a derivative taken with respect to a vector
field. Vector derivatives are extremely important in physics, where
they arise throughout fluid mechanics, electricity and magnetism,
elasticity, and many other areas of
theoretical and applied physics.
Vector derivatives
Types of vector derivative and
their symbol

23. DIFFERENTIATION IN SOCIETY
Basically Differentiation is use in society to compare
cost or margin of different things , as well as to
compare marks for example – Let cost of car 1 is
increased by 10000 INR and cost of car 2 is
increased by 20000 INR , then
Δ =10000 INR and Δ =20000 INR
therefore Δ /Δ =1/2
Then change of cost of car 1 w.r.t. car 2 is 1/2

24. Survey 1st
In survey one we meet shopkeeper who sell various devices which
work on the concept of differentiation and we get that maximum
no. of shopkeeper do not know any thing about concept of it.
Devices like odometer , speedometer etc .
Survey 2nd
In our second Survey we went to meet different teacher of
mathematics and science and we ask them that how
differentiation can help to make tomorrow better and we get many
different answers.
1.It is use in science therefore it make our tomorrow better.
2.

25. Conclusion
With the help of previous slides we can conclude that
Differentiation is use in almost everything . With the use of
Differentiation, equipment like odometer, speedometer etc
can be improved
So, Differentiation is use to find change of one thing with
respect to other , So Differentiation can help to make
tomorrow better.

26. Bibliography
1.wikipedia.com
2.WolFarm.World.com
3.Fundamental Chemistry
4.Fundamental physics
5. R.D. Sharma
6. www.Meritnation.com