Product & Quotient Rules Higher Order Derivatives Lesson 2.3.
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Transcript of Product & Quotient Rules Higher Order Derivatives Lesson 2.3.
![Page 1: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/1.jpg)
Product & Quotient RulesHigher Order Derivatives
Lesson 2.3
![Page 2: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/2.jpg)
After this lesson, you should be able to:
• Find the derivative of a function using the Product Rule
• Find the derivative of a function using the Quotient Rule
• Find the derivative of a trig function
• Find a higher-order derivative of a function
![Page 3: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/3.jpg)
Basic Rules
• Product Rule
( ) ( ) ' ( ) '( ) '( ) ( )f x g x f x g x f x g x
2
( ) ( ) '( ) ( ) '( )'
( ) ( )
f x g x f x f x g x
g x g x
How would you put this rule into words?
How would you put this rule into words?
![Page 4: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/4.jpg)
The derivative of a product of 2 functions is equal to the first
times the derivative of the second plus the second
function times the derivative of the first.
![Page 5: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/5.jpg)
Theorem 2.7 The Product Rule
![Page 6: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/6.jpg)
Try Some
• Use additional rules to determine the derivatives of the following function
3 2) ( ) 2 3 6 3b p x x x
) ( ) 3 cos 4sina h x x x x
![Page 7: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/7.jpg)
Try Some More
• Use additional rules to determine the derivatives of the following function
• a)
'
( ) 3 cos 4sin
( ) 3cos 3 sin 4cos
h x x x x
h x x x x x
![Page 8: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/8.jpg)
Try Some More
• Use additional rules to determine the derivatives of the following function
• b)
3 2
' 2 3
( ) 2 3 6 3
( ) 6 6 3 2 3 (6 )
p x x x
p x x x x x
![Page 9: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/9.jpg)
Try Some More
• Use additional rules to determine the derivatives of the following function
• b)
3 2
' 2 3 2 3
( ) 2 3 6 3
( ) 6 6 3 2 3 (6 ) 6 (6 3 2 3)
p x x x
p x x x x x x x x
![Page 10: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/10.jpg)
Try Some More
• Use additional rules to determine the derivatives of the following function
• b)
3 2
' 2 3 2 3
3 2
( ) 2 3 6 3
( ) 6 6 3 2 3 (6 ) 6 (6 3 2 3)
6 (2 3 3)
p x x x
p x x x x x x x x
x x x
NOTE!!! YOU MUST FACTOR COMPLETELY!!!!!!!!!!
GET IT DOWN TO THE POINT WHERE YOU COULD SOLVE FOR THE ZEROS OF THE EQUATION!!!!!
![Page 11: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/11.jpg)
Basic Rule
• Quotient Rule
2
( ) ( ) '( ) ( ) '( )'
( ) ( )
f x g x f x f x g x
g x g x
How would you put this rule into words?
How would you put this rule into words?
![Page 12: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/12.jpg)
The derivative of a quotient of 2 functions is equal to the
denominator times the derivative of the numerator
minus the numerator times the derivative of the denominator
all over the square of the denominator.
![Page 13: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/13.jpg)
Theorem 2.8 The Quotient Rule
![Page 14: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/14.jpg)
Just Checking . . .
• Find the derivatives of the given functions
sin xy
x
4 2( ) 1
1f x x
x
2
7 4( )
5
xq x
x
![Page 15: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/15.jpg)
Just Checking . . .
• Find the derivatives of the given functions
2'
2 2 2
7 4 7(5 ) ( 2 )(7 4)( ) ( )
5 (5 )
x x x xq x q x
x x
![Page 16: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/16.jpg)
Just Checking . . .
• Find the derivatives of the given functions
2'
2 2 2
2 2
2 2
7 4 7(5 ) ( 2 )(7 4)( ) ( )
5 (5 )
35 7 14 8
(5 )
x x x xq x q x
x x
x x x
x
![Page 17: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/17.jpg)
Just Checking . . .
• Find the derivatives of the given functions
2'
2 2 2
2 2 2
2 2 2 2
7 4 7(5 ) ( 2 )(7 4)( ) ( )
5 (5 )
35 7 14 8 7 8 35
(5 ) (5 )
x x x xq x q x
x x
x x x x x
x x
![Page 18: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/18.jpg)
4 2( ) 1
1f x x
x
![Page 19: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/19.jpg)
4
' 3 42
2( ) 1
1
2 ( 0)( 1) ( 2)(1)( ) 4 1
1 ( 1)
f x xx
xf x x x
x x
![Page 20: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/20.jpg)
4
' 3 42
3 42
2( ) 1
1
2 ( 0)( 1) ( 2)(1)( ) 4 1
1 ( 1)
2 1 2 (2)4
1 ( 1)
f x xx
xf x x x
x x
xx x
x x
![Page 21: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/21.jpg)
4
' 3 42
3 42
3 42
2( ) 1
1
2 ( 0)( 1) ( 2)(1)( ) 4 1
1 ( 1)
2 1 2 (2)4
1 ( 1)
8 4 (2)
1 ( 1)
f x xx
xf x x x
x x
xx x
x x
xx x
x x
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4
' 3 42
3 42
33 4
2
2( ) 1
1
2 ( 0)( 1) ( 2)(1)( ) 4 1
1 ( 1)
2 1 2 (2)4
1 ( 1)
8 4 (2) 8 4 2
1 ( 1) 1 1
f x xx
xf x x x
x x
xx x
x x
x x x xx x
x x x x
![Page 23: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/23.jpg)
4
' 3 42
3 42
33 4
2
3 3
2( ) 1
1
2 ( 0)( 1) ( 2)(1)( ) 4 1
1 ( 1)
2 1 2 (2)4
1 ( 1)
8 4 (2) 8 4 2
1 ( 1) 1 1
10 4 2 (5 2)
1 1 1 1
f x xx
xf x x x
x x
xx x
x x
x x x xx x
x x x x
x x x x
x x x x
![Page 24: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/24.jpg)
4
' 3 42
3 42
33 4
2
3 3 3
2( ) 1
1
2 ( 0)( 1) ( 2)(1)( ) 4 1
1 ( 1)
2 1 2 (2)4
1 ( 1)
8 4 (2) 8 4 2
1 ( 1) 1 1
10 4 2 (5 2) 2 (5 2
1 1 1 1
f x xx
xf x x x
x x
xx x
x x
x x x xx x
x x x x
x x x x x x
x x x x
2
)
( 1)x
![Page 25: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/25.jpg)
sin xy
x
![Page 26: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/26.jpg)
'2
2
sin cos ( sin )
cos sin
x x x xy y
x xx x x
x
![Page 27: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/27.jpg)
Other Trig Derivatives
• Now try it out
2 2tan sec cot csc
sec sec tan csc csc cot
d dx x x x
dx dxd d
x x x x x xdx dx
) 4 tan ' ?a f f
) sec tand
b x xdx
![Page 28: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/28.jpg)
Theorem 2.9 Derivatives of Trigonometric Function
![Page 29: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/29.jpg)
Other Trig Derivatives
2
4 tan
' 4(tan ) 4 (sec )
f
f
![Page 30: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/30.jpg)
2
2
4 tan ' 4(tan ) 4 (sec )
4(tan sec )
f f
![Page 31: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/31.jpg)
2
22
2
4 tan ' 4(tan ) 4 (sec )
sin4(tan sec ) 4
cos cos
cos sin4
cos
f f
![Page 32: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/32.jpg)
2
22
2 2 2
4 tan ' 4(tan ) 4 (sec )
sin4(tan sec ) 4
cos cos
cos sin 1/ 2(sin 2 ) sin 2 24 4 4
cos cos 2cos
f f
![Page 33: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/33.jpg)
2
22
2 2 2
2
4 tan ' 4(tan ) 4 (sec )
sin4(tan sec ) 4
cos cos
cos sin 1/ 2(sin 2 ) sin 2 24 4 4
cos cos 2cos
sin 2 22
cos
f f
JUST AN EXAMPLE OF HOW COMPLICATED THEY CAN GET!!!! CAN WE SIMPLIFY FURTHER???
![Page 34: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/34.jpg)
Derivatives of the remaining trig functions can be determined the same way.
sin cosd
x xdx
cos sind
x xdx
2tan secd
x xdx
2cot cscd
x xdx
sec sec tand
x x xdx
csc csc cotd
x x xdx
![Page 35: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/35.jpg)
Find Those High Orders
• Find the requested derivatives
2
2 32
) 4 1 ?d y
a y x xdx
4 3 2) ( ) 2 9 6 5 '''( ) ?b p x x x x p x
![Page 36: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/36.jpg)
Find each derivative.
2 21. ( ) 3 2 5 4f x x x x
2. ( ) sinf x x x
'( )f x
'( )f x
2 23 2 8 5 4 3 4x x x x x 2 3 2 324 16 15 20 12 16x x x x x
3 232 36 20 15x x x
cos sin 1x x xcos sinx x x
![Page 37: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/37.jpg)
Find the derivative.
33 4 4 13. ( )
2 3
x xg x
x
'( )g x
3 2 4 3
2
2 3 48 48 3 12 16 3 4 2
2 3
x x x x x x
x
4 312 16 3 4
2 3
x x x
x
4 3 3 2 4 3
2
96 96 6 144 144 9 24 32 6 8
2 3
x x x x x x x x
x
4 3 2
2
72 80 144 17
2 3
x x x
x
![Page 38: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/38.jpg)
Find the derivative.
1 cos1. ( )
sin
xf x
x
'( )f x
2
sin sin 1 cos cos
sin
x x x x
x
2 2
2
sin cos cos
sin
x x x
x
2
1 cos
sin
x
x
2
1 cos
1 cos
x
x
1 cos
1 cos 1 cos
x
x x
1
1 cos x
2 2
2
sin cos cos
sin
x x x
x
![Page 39: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/39.jpg)
( )( )
( )
g xf x
h x
'( )f x
2
' '
( )
h x g x g x h x
h x
'(2) 2, '(2) 4, (2) 4, (2) 3g h g h
2. Find '(2).f
'(2)f
2
2 ' 2 2 ' 2
(2)
h g g h
h
6 16
9
10
9
![Page 40: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/40.jpg)
Equation of a Horizontal Tangent Line
Find the equations of the horizontal tangent lines for 2( )
1
xf x
x
![Page 41: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/41.jpg)
Equation of a Horizontal Tangent Line
Find the equations of the horizontal tangent lines for 2( )
1
xf x
x
2 2'2 2 2 2
(1)( 1) (2 )( ) 1( )( 1) ( 1)
which is 0, when x = 1. And when x = 1, in the original equation, we have (1, 1/2) and (-1, -1/2). So, we have two tangent lines:
x x x xf xx x
1 1 and 2 2
y y
![Page 42: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/42.jpg)
Example: ( ) tanf x x xWrite the equation of the tangent line at
4x
Equation of the Tangent Line
![Page 43: Product & Quotient Rules Higher Order Derivatives Lesson 2.3.](https://reader035.fdocuments.us/reader035/viewer/2022081422/5516d9d655034643088b45f6/html5/thumbnails/43.jpg)
Example: ( ) tanf x x xWrite the equation of the tangent line at
4x
Equation of the Tangent Line
' 2 2 1 3( ) sec tan x = , sec ( ) tan( ) 1
4 4 4 2 2
, from the original equation, at , have ( , ), so the equation of the4 4 4
tangent line is:
3 3
4 2 4 2 8
f x x x x at
So we
b y x