Differentiation review

MCS 122 Chapter 1

Review of Derivatives

Some of the material in these slides is from Calculus 9/E by

Howard Anton, Irl Bivens, and Stephen Davis

Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.

Calculus 9/E by Howard Anton, Irl Bivens, and Stephen Davis


Derivative = Slope of Tangent Line



Derivative = Slope of Tangent Line

Definition 2.1.1 (p. 132)



Alternate Derivative Formulations



Alternate Derivative Formulations – Independent Time Variable



Derivative Function Definition

Definition 2.2.1 (p. 143)



Differentiability

Definition 2.2.2 (p. 146)



Derivative Rules

Theorem 2.3.2 (p. 156)

The Power Rule



Derivative Rules

Theorem 2.3.3 (p. 157)

Extended Power Rule



Derivative Rules

Theorem 2.3.4 (p. 157)

Constant Multiple Rule



Derivative Rules

Theorem 2.3.5 (p. 158)

Sum and Difference Rules



Derivative Rules

Theorem 2.4.1 (p. 164)

The Product Rule



Derivative Rules

Theorem 2.4.2 (p. 165)

The Quotient Rule



Derivative Rules



Derivatives of Trig Functions

Formula (3) (p. 169)

Formula (4)



Derivatives of Trig Functions

Formula (5) (p. 170)

Formula (6)

Formula (7)

Formula (8)



Derivatives of Composite Functions – Chain Rule

Theorem 2.6.1 (p. 174)

The Chain Rule



Derivatives of Logarithmic Functions

Equations 2 - 5 (p. 193)



Derivatives of Exponential Functions

Equations 5 - 8 (p. 199)



Derivatives of Inverse Trig Functions

Equations 9 - 11 (p. 201)



Implicit Differentiation

Find dy/dx by implicit differentiation: 3 xy



4 y

3 

8 x





Select the best answer for the following question.

• During the first 50 seconds of a missile launch, the missile is propelled straight up so that in t seconds, it s

  

0.6

t

3 velocity of the missile during the first 50 seconds?

• 12,500 ft/sec



• 1,500 ft/sec

• 38,250 ft/sec

• 75,000 ft/sec

1 n o i t s e u

Q




f x 

   lim h



0

2

 x

 h



2 

5

 x

 h

 

2 x

2 

5 x h











•

•

•

•



  

4 x



5

   

2 x

2 

5 x

  

2 x

2 

5 x

  

2

 x

 h



2 

5

 x

 h

 i t s e o u

2 n

Q




3. Which of the following equations has derivative dy dx



12 x

3 

24 x



6 ?











•

•

•

• y



3 x

4 

12 x

2 

6 x



11 y



36 x

2 

24 y



3 x

4 

12 x

2 

6 y



12 x

4 

24 x

2 

6 x



11 i t s e o u

3 n

Q













•

•

•

•


   

9 x



4 

16

  

36 x



3 

1

2 x

3

4

  

36 x



5 

1

2 x



5

4

   

36 x



5 

1

2 x



3

4

  

36 x



5

4 n o ti s e u

Q












•

•

• y

 x

3 

6

’ . x



3 y

  y

 

2 x

3 

9 x

2 

6

 x



3



2

4 x

3 

9 x

2 

6

 x



3



2 y

 

2 x

• y

 



2 x

3 

9 x

2 

6

 x



3



2

6 n e u

Q st io















•

•

•

•


  

3tan x

 x cos x

  

3sec

2 x

 sin x



  

3sec x

 x sin x

 cos x

  

3sec

2 x

 x sin x

 cos x

  

3sec

2 x

 x sin x

 cos x

 

7 n e u

Q st io



Answers

• 1.b

• 2.c

• 3.a

• 4.b

5. a

6. d



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