derivatives math

SUBMITTED TO:
Ma”m SADIA FIRDUS
SUBMITTED BY:
GROUP NO. 03

BS-MECHANICAL TECHNOLOGY (1ST SEMESTER)
NAME ROLL NO.
1. AHSAN WAQAR 17093386-032
2. ALI HASSAN 17093386-015
3. ALI RAZA 17093386-031
4. USMAN ZAFAR 17093386-018
5. ABDULLAH 17093386-035
6. Muhammad Umer 17093386-036

Contents
Definition of derivatives .
History of derivatives.
Application of derivatives.
Derivatives rules & examples.

 DEFINATION OF DERIVATION?
1. The derivatives is the exact rate at which one quantity changes
with respect to another.
2. Geometrically, the derivatives is the slope of curve at point on
curve.
3. The derivatives is often called the instantaneous rate of
change.
4. The derivatives of a function represents an infinitely small
change the fuction with respect to one of its variable.
Its is written as
dy
dx

NOTE:
does not mean dy dx
(except when it is convenient to think of it as division.)
df
dx
does not mean df dx
(except when it is convenient to think of it as division.)

Note:
dx does not mean d times x !
dy does not mean d times y !

APPLICATIONS OF DERIVATIVES
in real life.

 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.

Applications of Derivatives in Various
fields/Sciences:
Such as in:
 –Physics
 –Biology
 –Economics
 –Chemistry
 –Mathematics

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:
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 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.

Rules & example of derivation:
Rules of derivative are as under:

Power Rule:
Example: What is
𝑑
𝑑𝑥
𝑥3 ?
The question is asking "what is the derivative of 𝑥3
?"
We can use the Power Rule, where n=3:
𝑑
𝑑𝑥
𝑥 𝑛
= nxn−1
𝑑
𝑑𝑥
𝑥3 = 3 𝑥3−1 = 3 𝑥2
(In other words the derivative of 𝑥3is3𝑥2)

Multiplication by constant:
Example: What is
𝑑
𝑑𝑥
5 𝑥3
?
the derivative of cf = cf’
the derivative of 5f = 5f’
We know (from the Power Rule):
𝑑
𝑑𝑥
𝑥3 = 3 𝑥3−1=3 𝑥2
So:
𝑑
𝑑𝑥
5𝑥3 = 5
𝑑
𝑑𝑥
𝑥3
= 5 × 3𝑥2 = 15 𝑥2

Sum Rule:
Example: What is the derivative of 𝑥2+𝑥3 ?
The Sum Rule says:
the derivative of f + g = f’ + g’
So we can work out each derivative separately and then add them.
Using the Power Rule:
𝑑
𝑑𝑥
𝑥2
= 2x
𝑑
𝑑𝑥
𝑥3
= 3 𝑥2
And so:
the derivative of 𝑥2+ 𝑥3 = 2x + 3𝑥2

Difference Rule:
Example: What is
𝑑
𝑑𝑣
( 𝑣3 − 𝑣4 ) ?
The Difference Rule says
the derivative of f − g = f’ − g’
So we can work out each derivative separately and then subtract them.
Using the Power Rule:
𝑑
𝑑𝑣
𝑣3
= 3 𝑣2
𝑑
𝑑𝑣
𝑣4 = 4 𝑣3
And so:
the derivative of 𝑣3 − 𝑣4 = 3𝑣2−4𝑣3

Product Rule:
Example: What is the derivative of cos(x)sin(x) ?
The Product Rule says:
the derivative of fg = f g’ + f’ g
In our case:
f = cos
g = sin
We know (from the table above):
𝑑
𝑑𝑥
cos(x) = −sin(x)
𝑑
𝑑𝑥
sin(x) = cos(x)
So:
the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x)
= 𝑐𝑜𝑠2(x) − 𝑠𝑖𝑛2(x)

Quotient rule:
Examples: The quotient rule can be used to find the derivative of
As follow:

Chain rule:
Example: What is
𝑑
𝑑𝑥
sin(x2) ?
sin(x2) is made up of sin() and x2:
f(g) = sin(g)
g(x) = x2
The Chain Rule says:
the derivative of f(g(x)) = f'(g(x))g'(x)
The individual derivatives are:
f'(g) = cos(g)
g'(x) = 2x
So:
𝑑
𝑑𝑥
sin(x2) = cos(g(x)) (2x)
= 2x cos(x2)

derivatives math

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derivatives math