PPT - Differentiation

``` Page 1

Higher Maths         1            3    Differentiation 1
Page 2

Higher Maths         1            3    Differentiation 1

Higher Maths         1            3    Differentiation
The History of Differentiation
Differentiation is part of the science of Calculus, and was first
developed in the 17
th
century by two different Mathematicians.
Gottfried Leibniz
(1646-1716)
Germany
Sir Isaac Newton
(1642-1727)
England
2
Differentiation, or finding the
instantaneous rate of change, is
an essential part of:
• Mathematics and Physics
• Chemistry
• Biology
• Computer Science
• Engineering
Page 3

Higher Maths         1            3    Differentiation 1

Higher Maths         1            3    Differentiation
The History of Differentiation
Differentiation is part of the science of Calculus, and was first
developed in the 17
th
century by two different Mathematicians.
Gottfried Leibniz
(1646-1716)
Germany
Sir Isaac Newton
(1642-1727)
England
2
Differentiation, or finding the
instantaneous rate of change, is
an essential part of:
• Mathematics and Physics
• Chemistry
• Biology
• Computer Science
• Engineering

Higher Maths         1            3    Differentiation
Calculating Speed
2 4 6 8
4
8
2
6
10
0
0
Time (seconds)
Distance (m)
D
S T ×
÷ ÷
Example
Calculate the speed for each
section of the journey opposite.
A
B
C
speed in A  =
4
3
speed in B  =
5
1
5 m/s

=
speed in C  =
2
5
0.4 m/s

=
average speed  =
9
1.22 m/s

˜
11
˜
1.33 m/s

Notice the following things:
• the speed at each instant is
not the same as the average
• speed is the same as gradient
D
T
S  =

y
x
= m =
3
Page 4

Higher Maths         1            3    Differentiation 1

Higher Maths         1            3    Differentiation
The History of Differentiation
Differentiation is part of the science of Calculus, and was first
developed in the 17
th
century by two different Mathematicians.
Gottfried Leibniz
(1646-1716)
Germany
Sir Isaac Newton
(1642-1727)
England
2
Differentiation, or finding the
instantaneous rate of change, is
an essential part of:
• Mathematics and Physics
• Chemistry
• Biology
• Computer Science
• Engineering

Higher Maths         1            3    Differentiation
Calculating Speed
2 4 6 8
4
8
2
6
10
0
0
Time (seconds)
Distance (m)
D
S T ×
÷ ÷
Example
Calculate the speed for each
section of the journey opposite.
A
B
C
speed in A  =
4
3
speed in B  =
5
1
5 m/s

=
speed in C  =
2
5
0.4 m/s

=
average speed  =
9
1.22 m/s

˜
11
˜
1.33 m/s

Notice the following things:
• the speed at each instant is
not the same as the average
• speed is the same as gradient
D
T
S  =

y
x
= m =
3

Instantaneous Speed
Higher Maths         1            3    Differentiation
Time (seconds)
Distance (m)
Time (seconds)
Distance (m)
In reality speed does not often change instantly. The graph on the
right is more realistic as it shows a gradually changing curve.
The journey has the same average speed, but the instantaneous
speed is different at each point because the gradient of the curve is
constantly changing. How can we find the instantaneous speed?
D
T
S  =

y
x
=
m
=
4
Page 5

Higher Maths         1            3    Differentiation 1

Higher Maths         1            3    Differentiation
The History of Differentiation
Differentiation is part of the science of Calculus, and was first
developed in the 17
th
century by two different Mathematicians.
Gottfried Leibniz
(1646-1716)
Germany
Sir Isaac Newton
(1642-1727)
England
2
Differentiation, or finding the
instantaneous rate of change, is
an essential part of:
• Mathematics and Physics
• Chemistry
• Biology
• Computer Science
• Engineering

Higher Maths         1            3    Differentiation
Calculating Speed
2 4 6 8
4
8
2
6
10
0
0
Time (seconds)
Distance (m)
D
S T ×
÷ ÷
Example
Calculate the speed for each
section of the journey opposite.
A
B
C
speed in A  =
4
3
speed in B  =
5
1
5 m/s

=
speed in C  =
2
5
0.4 m/s

=
average speed  =
9
1.22 m/s

˜
11
˜
1.33 m/s

Notice the following things:
• the speed at each instant is
not the same as the average
• speed is the same as gradient
D
T
S  =

y
x
= m =
3

Instantaneous Speed
Higher Maths         1            3    Differentiation
Time (seconds)
Distance (m)
Time (seconds)
Distance (m)
In reality speed does not often change instantly. The graph on the
right is more realistic as it shows a gradually changing curve.
The journey has the same average speed, but the instantaneous
speed is different at each point because the gradient of the curve is
constantly changing. How can we find the instantaneous speed?
D
T
S  =

y
x
=
m
=
4

Introduction to Differentiation
Higher Maths         1            3    Differentiation
Differentiate means
D
T
speed  =
‘rate of change of
distance with respect to time’
S
T
acceleration  =
‘find out how fast something is changing in comparison
with something else at any one instant’.
y
x
‘rate of change of
speed with respect to time’
‘rate of change of
-coordinate with respect to    -coordinate’ y x
5
```

115 videos|142 docs

## FAQs on PPT - Differentiation - Business Mathematics and Statistics - B Com

 1. What is differentiation in mathematics?
Ans. Differentiation in mathematics is a fundamental concept that involves finding the rate at which a function changes. It is used to calculate the slope or gradient of a curve at any given point, providing information about the function's behavior and the relationship between its variables.
 2. How is differentiation used in real-life applications?
Ans. Differentiation has various real-life applications. For example, it is used in physics to calculate velocities and accelerations, in economics to determine marginal costs and revenues, in biology to study population growth rates, and in engineering to optimize designs and analyze systems' behavior.
 3. What are the basic rules of differentiation?
Ans. The basic rules of differentiation include the power rule, product rule, quotient rule, and chain rule. The power rule states that the derivative of x^n (where n is a constant) is nx^(n-1). The product rule is used to differentiate the product of two functions, while the quotient rule is used for differentiating the quotient of two functions. The chain rule is applied when differentiating composite functions.
 4. How can differentiation be used to find maximum and minimum points of a function?
Ans. Differentiation is used to find maximum and minimum points of a function by analyzing its critical points. Critical points occur where the derivative of the function is either zero or undefined. By setting the derivative equal to zero and solving for the variable, we can identify potential maximum and minimum points. Further analysis, such as the second derivative test, can help determine whether these points are maximum or minimum.
 5. Can differentiation be used to find the area under a curve?
Ans. No, differentiation cannot be directly used to find the area under a curve. Differentiation focuses on finding the rate of change of a function. However, integration, which is the reverse process of differentiation, can be used to find the area under a curve. Integration involves summing up infinitely small areas under the curve and is commonly used in calculus to solve problems related to areas, volumes, and accumulation.

115 videos|142 docs

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