Chapter 2
Calculus Differentiation
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The Derivative and the Tangent Line Problem(Section: 2.1)
Objective:
• Find the slope of the tangent line to a curve at a
point (Read page 98, work examples 1, 2).
Video link:
https://www.youtube.com/watch?v=Uataz2hvNks
• Use the limit definition to find the derivative of a
function (Read pages 99-100, work examples 3, 4).
Video link:
https://www.youtube.com/watch?v=vzDYOHETFlo
• Understand the relationship between differentiability
and continuity (Read pages 101, work examples 6, 7).
Video link:
https://www.youtube.com/watch?v=-kKLO3qtJJU
Assignment: Sec. 2.1 /Page103: 1, 5, 7, 11, 14, 17, 18, 25, 26, 29, 33, 34, 39-42all, 45-48all, 64, 73, 83-87odd
Basic Differentiation Rules and Rates of Change(Section: 2.2)
Objective:
• Find the derivative of a function using the Constant
Rule (Read page 107, work examples 1).
Video link:
https://www.youtube.com/watch?v=YdxeLS2qvp4
• Find the derivative of a function using the Power
Rule (Read page 108, work examples 2).
Video link:
https://www.youtube.com/watch?v=ujcC47TwVjE
• Find the derivative of a function using the Constant
Multiple Rule (Read page 110, work examples 5.
Video link:
https://www.youtube.com/watch?v=HzR3XvMjfO4
• Find the derivative of a function using the Sum and
Difference Rules (Read page 111, work examples 7).
Video link:
https://www.youtube.com/watch?v=ZxBbN0mo4tI
• Find the derivatives of the sine function and of the
cosine function (Read page 112, work examples 8).
Video link:
https://www.youtube.com/watch?v=aFcYYT26tCo
• Use derivatives to find rates of change (Read page
113, work examples 9).
Video link:
https://www.youtube.com/watch?v=jlihNi_Mkos
Assignment: Sec.2.2/Page 115: 1, 3, 7, 9, 17-29odd, 31, 37,38, 41-53odd, 55-57all, 59-63odd, 65, 66, 72, 93, 95, 97, 99, 117
Product and Quotient Rules and Higher-Order Derivatives (Section: 2.3)
Objective:
• Find the derivative of a function using the Product
Rule(Read page 119, work examples 1,2,3).
Video link:
https://www.youtube.com/watch?v=uPCjqfT0Ixg
• Find the derivative of a function using the Quotient
Rule(Read page 121, work examples 4,5).
Video link:
https://www.youtube.com/watch?v=mMqKXxz-6AY
• Find the derivative of a trigonometric function(Read
page 123, work examples 8,9).
Video link:
https://www.youtube.com/watch?v=bL_4STAOGbE
• Find a higher-order derivative of a function (Read
page 125, work examples 10).
Video link:
https://www.youtube.com/watch?v=-8Dwx2a5z50
Assignment: Sec.2.3/Page 126: 1-31odd, 39-51odd, 63, 65, 69-75odd, 93, 99, 101, 103, 105, 107, 109, 113
The Chain Rule (Section. 2.4)
Objective:
• Find the derivative of a composite function using the
Chain Rule (Read page 131, work examples 3).
Video link:
https://www.youtube.com/watch?v=6kScLENCXLg
• Find the derivative of a function using the General
Power Rule (Read page 132, work examples 4,5,6).
Video link:
https://www.youtube.com/watch?v=ujcC47TwVjE
• Simplify the derivative of a function using
algebra(Read page 134, work examples 7,8,9).
Video link:
https://www.youtube.com/watch?v=PkdYCDA0CWU
• Find the derivative of a trigonometric function using
the Chain Rule (Read page 135, work examples
10,11,12).
Video link:
https://www.youtube.com/watch?v=Rw4NdMRa5bU
Assignment: Page 137: 7-25odd, 37, 45-65odd, 67, 71, 73, 75, 79, 81, 87, 99, 101-103all, 108
Implicit Differentiation (Section. 2.5)
Objective:
• Distinguish between functions written in implicit
form and explicit form (Read page 141, work
examples 1,2).
Video link:
https://www.youtube.com/watch?v=JIixsGDbyEY
• Use implicit differentiation to find the derivative of a
function form (Read page 142, work examples 2,3).
Video link:
https://www.youtube.com/watch?v=CFvVbAxwPd0
Assignment: Sec.2.5/Page 146: 1-15odd, 29, 31, 33, 35, 45, 74
Related Rates (Section. 2.6)
Objective:
• Find a related rate(Read page 149, work examples 1,2).
Video link:
https://www.youtube.com/watch?v=wTYvMpVITg8
• Use related rates to solve real-life problems(Read page 151,
work examples 3,4).
Video link:
https://www.youtube.com/watch?v=89IlCWG9SVk
Assignment: Sec.2.6/Page 154: 1, 3, 5, 13, 15, 20, 22, 25, 33, 34, 43
Chapter 2 Flash Cards
Sec 2.1: The Derivative and the Tangent Line Problem
Definition of a Derivative:
f ( x  h)  f ( x )
f ( x )  lim
h
h 0
Alternative form of derivative:
f (c )  lim
x c
f ( x )  f (c )
xc
Theorem 2.1: Differentiability Implies Continuity
If f is differentiable at x = c, then f is continuous at x = c.
Sec2.2: Basic Differentiation Rules and Rates of Change
d
c   0
dx
The Constant Rule:
The Constant Multiple Rule:
The Power Rule:
d n
 x   nx n 1
dx
d
cf ( x )  cf '( x )
dx
The Sum and Difference Rules
d
 f ( x)  g ( x)  f '( x)  g '( x )
dx
d
 f ( x)  g ( x)  f '( x)  g '( x )
dx
Derivative of Sine and Cosine Functions:
d
sin( x )  cos( x )
dx
d
cos( x )   sin( x )
dx
Sec2.3 Product and Quotient Rules and Higher-Order Derivatives
The Product Rule:
The Quotient Rule:
d
 f ( x) g ( x)  f ( x) g '( x)  g ( x) f '( x)
dx
d  f ( x )  g ( x ) f '( x )  f ( x ) g '( x )

2
dx  g ( x ) 
 g ( x )
Derivatives of Trigonometric Functions
d
tan( x)  sec2 ( x)
dx
d
cot( x)   csc2 ( x)
dx
d
sec( x)  sec( x) tan( x)
dx
d
csc( x )   csc( x )cot( x)
dx
Sec2.4: The Chain Rule
The Chain Rule:
d
 f ( g ( x))  f '( g ( x)) g '( x)
dx
Sec2.5: Implicit Differentiation
Sec2.6: Related Rates
Trig Identities
Double Argument
sin 2 x  2sin x cos x
cos 2 x  cos2 x  sin 2 x  1  2sin 2 x
Pythagorean
sin 2 x  cos2 x  1
(others are easily derivable by dividing by sin2x or cos2x)
1  tan 2 x  sec2 x
cot 2 x  1  csc2 x
Reciprocal
1
sec x 
or cos x sec x  1
cos x
1
csc x 
or sin x csc x  1
sin x
Odd-Even
sin(–x) = – sin x
cos(–x) = cos x
(odd)
(even)
Higher-Order Derivatives
Functions Not Differentiable at a Point
Trigonometric Formulas
1. sin 2   cos 2   1
2. 1  tan 2   sec 2 
3. 1  cot 2   csc 2 
4. sin(  )   sin 
5. cos(  )  cos
6. tan(  )   tan 
7. sin 2  2 sin  cos
8. cos 2  cos 2   sin 2   2 cos 2   1  1  2 sin 2 
sin 
1

cos  cot 
cos 
1

10. cot  
sin  tan 
1
11. sec  
cos 
1
12. csc  
sin 
9.
tan  
Differentiation Formulas
1.
2.
3.
4.
5.
6.
7.
8.
d n
( x )  nx n 1
dx
d
( fg )  fg   gf 
dx
d f
gf   f g 
( )
dx g
g2
d
f ( g ( x))  f ( g ( x)) g ( x)
dx
d
(sin x)  cos x
dx
d
(cos x)   sin x
dx
d
(tan x)  sec 2 x
dx
d
(cot x)   csc 2 x
dx
d
(sec x)  sec x tan x
dx
d
(csc x)   csc x cot x
10.
dx
9.