THUYẾT TRÌNH ĐẠO HÀM BẰNG TIẾNG ANH DERIVATIVES

VDERIVATIVESE
Instructors : …
Presentation group 4: ….


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ISAAC NEWTON
and
GOTTFRIED LEIBINZ
were scientist who invented
derivatives, integrals and
differentials

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6. THE SECOND DERIVATIVE

3. DERIVATIVE OF SUM,
1. DERIVATIVE AT A POINT

DIFFERENCE, PRODUCT AND
QUOTIENT

DERIVATIVES

2. DERIVATIVES OF SOME

4. DERIVATIVE OF A

COMMON FUNCTIONS

COMPOSSITE FUNCTION

5. DERIVATIVE OF A
TRIGONOMETRIC FUNCTION
3


I. Definition of a derivative at a point
1. Definition

▹




 
Given
function

defined on the open interval (a;b) and
Derivative : đạo hàm

If there exists (finite) limit

Defined: được xác định

Then the limit is called a derivative of a function

Interval: khoảng
Exists: tồn tại

at point and denoted by

or namely,

Finite: hữu hạn
Point: điểm

.

Namely: tức là

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▹

NOTE




 

Quantity is called the increment of argument at

“

Quantity

is called the corresponding increment of the function. Thus,

Quantity: đại lượng

Corresponding increment: số gia tương ứng

Increment: số gia
Argument: đối số

Thus: như vậy
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2. How to calculate a derivative by definition

Rule:
Step 1: Let be the increment of an argument at
Calculate:
Step 2: Form ratio

⊷Step 3: Find
Calculate: tính
Form: lập
Ratio: tỉ số, tỉ lệ

 

6


Example 1. Calculate the derivative of function
at
  point

▹
Solution:
  Let be the increment of an argument at . We have:
divided by x, we shall show:
Then,
Therefore,

7


Example 2. Calculate the derivative of function
  at point

▹
Solution:
  Let be the increment of an argument at . We have:
divided by x, we shall show:
Then,
Therefore,

8

II. Derivativers of some common functions

 

Theorem 1: Function has
a derivative at every and

Theorem 2: Function has a derivative at every positive x and

9

Theorem: định lí


Example 1. Find the dervative of function

▹

 

Remax

a) Derivative of a constant function equals 0

a)

b) Derivative of function

b)

Remax: nhận xét
Constant function: hàm hằng

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y = x equals 1


III. Derivative of sum, difference, product and quotient
1. Theorem
Theorem 3: Suppose u = u(x) and v = v(x) are functions with derivatives at point x on the defined interval. We have

(u + v )′ = u ′ + v′
(u − v )′ = u ′ − v′
(uv)′ = u ′v + uv′
′
u ′v − uv′
u
(v = v ( x) ≠ 0)
  =
2
v
v
11

1
2

3

4





Quotient: thương
Suppose: giả sử
Defined interval: khoảng xác định


Example 1. Find the dervative of function

 

Solution
 

a) We have

a)

b) Do it yourseft
c) Do it yourseft
b)
c)

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b) We have:

[x (
3

c) We have:

] ( )
′

′
x − x ) = x 3 ( x − x 5 ) + x 3 ( x − x 5 )′
5

 1

= 3 x 2 ( x − 5) + x 3 
− 5x 4 
2 x

 1

=
3x 2 x + x 3 
− 8x 4 
2 x


′


x 
4
3


4
x
−
=
16
x
−x


2 

2

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2. Corollary

 

Corollary 1: If is a constant, then

′
v′
1

=
−
 
v2
v

Corollary 2:

( ku )′ = ku ′

Example 3. Find the dervative of function
a)

y=

1 − 2x
x+3

Solution: We have

′
′
(
1 − 2 x ) ( x + 3) − (1 − 2 x )( x + 3)′
 1− 2x 

 =
( x + 3) 2
 x+3 

=

− 2( x + 3) − (1 − 2 x)
−7
=
( x + 3) 2
( x + 3) 2
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Example 2: Find the derivative of function

a)

b)

3 − 5x
y= 2
x − x +1

2x +1
y=
x4

′
(3 − 5 x)′( x 2 − x + 1) + (3 − 5 x )( x 2 − x + 1)
 3 − 5x 
 2
÷=
( x 2 − x + 1) 2

 x − x +1

2
′
2
−
15
x
8 + (3 − 5 x)( x 2 − x + 1)
3
−
5
x
(3
−
5
x
)′(+x16
−xx −
+ 1)


 2
÷ =
( x 2 − x + 1()x2 2 − x + 1) 2
 x − x +1 

′
2
x

+
1
( 2 x + 1)′ x 4 + ( 2 x + 1)( x 4 )′



 =
4
4 2
x
(
x
)


4 2
3
′
(3 − 5x
x )′( x+
−4
x +x1) + (3 − 5 x)( x 2 − x + 1)
 3 − 5 x  10
 2
÷=
 x − x +1 
x 8 ( x 2 − x + 1)2
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IV. Derivative of a composite function
Theorem 4. If function has derivative at x and function has derivative at , then composite function has a
derivative at
 

y′x = yu′ .u ′x

composite function: hàm hợp

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Example 1: Find the derivative of function

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V. Derivative of function
 

 

Theorem 1: Function y=sinx has derivative at every and

 

 

Theorem 2: Function y=cosx has derivative at every and

 

 

Theorem 3: Function y=tanx has derivative at every and

 

 

Theorem 4: Function y=cotx has derivative at every and


 

Example 1: Find the derivative of function
 

Solution:
Setting , it follows that u’=3 and
We have

 

Example 2: Find the derivative of function
 

Solution:
Setting 5, it follows that and
We have

 

? Exercise 1: Find the derivative of function

 ?

Exercise 2: Find the derivative of function

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VI. THE SECOND DERIVATIVE
1. DEFINITION

▹Assume
  function has a derivative at every point . Then, relation defines a new function on the interval
(a;b). If function has another derivative at , then we call the derivative of function is the second
derivative of function at and denoted by or

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2. NOTE

 

The third derivative of function is defined similarly and denoted by of or .
th
Given function with the n-1 derivative and denoted by.

If has a derivative, then its derivative is called the derivative of and denoted by or
.


th
Example 1: Find the second derivative and the n derivative of the follwing functions:

 

a)

b) y = x

 

Solution

Do it yourseft

If y = then
n
y = 0 with n>5

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GAME RULES

THREE GROUPS WILL PARTICIPATE IN A THREE - PART GAME:

1. Starting up
2. Acceleration
3. To the finish line
At the end of the 3 sessions, the team with the highest total score will win and win a prize.
When one team participates in the game, the other 2 teams will act as judges and the secretary will sum up
the points for each group.


STArTING UP
RULE:
There are 3 question packs, each with 30 seconds of thinking and answering
time. Each correct answer is added 5 points to the group’s score bank. If you
answer incorrectly, you will receive no points.
At the end of the 3 - pack questioning which group is the highest, that group
wins in this game.


STArTING UP
1

2

3

Select the questionnaire package