Mathematician integrally founding presentation

1. Derivative
2020-2021
Group members:
Khalat khalil
Shara mohameed
Saya majeed

2. OUTLINE
 Definition of derivatives
 History of derivatives
 Application of derivatives
 Derivatives rules & examples

3. Definition of derivatives
• in mathematics, the derivative of a function of a real variable
measures the sensitivity to change of the function value (output
value) with respect to a change in its argument (input value).
Derivatives are a fundamental tool of calculus. For example, the
derivative of the position of a moving object with respect to time
is the object's velocity: this measures how quickly the position of
the object changes when time advances.
• It is written as
𝒅𝒚
𝒅𝒙

4. History of derivation
• The modern development of calculus is usually
credited to Isaac Newton (1643–1727) and
Gottfried Wilhelm Leibniz (1646–1716), who
provided independent and unified approaches
to differentiation and derivatives.

5. Application of derivatives in real life
• Automobiles
In automobiles 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).

7. Application of derivatives in real life
• Business
In the business world there are many application for
derivatives, one of the most important application in
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 application of derivatives you can
estimate the profit and loss point for certain ventures.

8. Derivatives application in real life
• Derivative in mathematics
The most common use of the derivatives in
mathematics is to study function such as:
Extreme values of function
The mean value theorem
Monotonic function
Concavity & curve sketching
Newton’s method

9. Differences between level types
Sum rule:
Example: what is the derivative of 𝑥2
+ 𝑥3
=?
The sum rule says: f’ + g’
So we can work out each derivatives separately and
then add them
𝑑
𝑑𝑥
x2
= 2x
𝑑
𝑑𝑥
𝑥3
= 3𝑥2
𝑡ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑥2
+ 𝑥3
= 2𝑥 + 3𝑥2

10. quotient rule:
•
𝑓 𝑥
𝑔 𝑥
′
=
𝑔 𝑥 𝑓′ 𝑥 −𝑓 𝑥 𝑔′(𝑥)
𝑔 𝑥 2
𝑓 𝑥 =
6𝑥−5
6𝑥2+1
𝑓 𝑥 ′
=
(𝑥2+1)(6)−(6𝑥−5)(2𝑥)
𝑥2+1 2
𝑓′
𝑥 =
6𝑥2+6−12𝑥2+10𝑥
𝑥2+1 2
𝑓′
𝑥 =
6𝑥2+10𝑥+6
𝑥2+1 2

11. Second derivative ′′
f′′(𝑥) =
𝑥2+1
2
−12𝑥+10 −(−6𝑥2+10𝑥+6)(2 𝑥2+1 2𝑥 )
𝑥2+1 4
𝑓′′
(𝑥) =
(𝑥2+10) 𝑥2+1 −12𝑥+10 −(−6𝑥2+10𝑥+6)(4𝑥)
𝑥2+1 4
𝑓′′
𝑥 =
−12𝑥3+10𝑥2−12𝑥+10+24𝑥3−40𝑥2−24𝑥
𝑥2+1 3
f′′(x) =
12𝑥3−30𝑥2−36𝑥+10
𝑥2+1 3

12. ln rule:
𝑦 = ln 𝑓 𝑥
𝑑𝑦
𝑑𝑥
=
𝑓′
(𝑥)
𝑓(𝑥)
• 𝑓 𝑥 = 𝑙𝑛𝑥2
𝑓′
𝑥 =
2𝑥
𝑥2 =
2
𝑥

13. • 𝑓 𝑥 = 𝑥𝑙𝑛𝑥
𝑓′
𝑥 = 𝑥
1
𝑥
+ 𝑙𝑛𝑥 1
𝑓′
𝑥 = 1 + 𝑙𝑛𝑥

14. sin,cos,e rule
• 𝑦 = sin 𝑓 𝑥
𝑑𝑦
𝑑𝑥
= 𝑓′ 𝑥 𝑐𝑜𝑠(𝑓 𝑥 )
• 𝑦 = cos 𝑓 𝑥
𝑑𝑦
𝑑𝑥
= −𝑓′ 𝑥 sin 𝑓 𝑥
• 𝑦 = 𝑒𝑓 𝑥 𝑑𝑦
𝑑𝑥
= 𝑓′ 𝑥 𝑒𝑓 𝑥
𝑓(𝑥) = 𝑥 𝑠𝑖𝑛𝑥
𝑓′
𝑥 =
1
2 𝑥
𝑠𝑖𝑛𝑥 + 𝑥𝑐𝑜𝑠𝑥
𝑓′
𝑥 =
𝑠𝑖𝑛𝑥
2 𝑥
+ 𝑥𝑐𝑜𝑠𝑥

15. • 𝑓 𝑥 = 𝑒𝑥
𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥
𝑓′
𝑥 = 𝑒𝑥
𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥 + (𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥)(𝑒𝑥
)
𝑓′
𝑥 = 𝑒𝑥
𝑐𝑜𝑠𝑥 − 𝑒𝑥
𝑠𝑖𝑛𝑥 + 𝑒𝑥
𝑠𝑖𝑛𝑥 + 𝑒𝑥
𝑐𝑜𝑠𝑥
𝑓′
𝑥 = 2𝑒𝑥
𝑐𝑜𝑠𝑥