AP Calculus BCMonday, 21 September 2015

• OBJECTIVES TSW find (1) derivatives using the product and quotient rules, and (2) higher-order derivatives.

• ASSIGNMENT DUE (wire basket)

– Sec. 3.3: pp. 160-161 (7-17 odd, 19-24 all omit 21, 35, 36)

• TODAY’S ASSIGNMENT (due W/Th, 09/23-24/15)

– Sec. 3.4: pp. 170-171 (7, 8, 11, 12, 15-19 odd, 20, 25, 26, 40. 41)

– Sec. 3.3: p. 161 (44, 45, 47)

Week of 09/21/15 – 09/25/15

• MONDAY Sec. 3.4: Product & Quotient RulesSec. 3.3: Higher-Order Derivatives

• TUESDAY Sec. 3.7: The Chain Rule

• WED/THUR Sec. 3.8: Implicit Differentiation

• FRIDAY Sec. 3.8: continuedQUIZ: Product, Quotient, & Chain Rules

• M, 09/28/15 Review: Derivatives

• Tu, 09/29/15 TEST: Derivatives

Product and Quotient Rules and Higher-Order Derivatives

Product and Quotient Rules and Higher-Order Derivatives

The Product Rule

(first) (derivative of the second) + (second) (derivative of the first)

Product and Quotient Rules and Higher-Order Derivatives

Ex: Find the derivative.

22 5 4a) 3f x x x x

22 5 3 4 3 2 10f x x x x x

2 26 15 8 34 30x x x x 245 28 8x x

Product and Quotient Rules and Higher-Order Derivatives

Ex: Find the derivative.

cb) osd

x xdx

sin cos 1x x x

cos sinx x x

Product and Quotient Rules and Higher-Order Derivatives

Ex: Find the derivative.

3 sin 4sic) ny x x x

3 cos sin 3 4cosy x x x x

3 cos 3sin 4cosx x x x

Product and Quotient Rules and Higher-Order Derivatives

The Quotient Rule

2

( ) ( ) ( ) ( )

( )

denominator derivative of numerator numerator derivative of denominator

denominator

Product and Quotient Rules and Higher-Order Derivatives

Example: Find the derivative.

2

3 1

1a)

d x

dx x

2

22

1 3 3 1 2

1

x x x

x

2 2

22

3 3 6 2

1

x x x

x

2

22

3 2 3

1

x x

x

Do NOT square the denominator

Product and Quotient Rules and Higher-Order Derivatives Ex: Find the derivative.

b)

15

3xy

x

x

x

First, eliminate the complex fraction.2

5 1

3

x

x x

Now find the derivative

2

22

3 5 5 1 2 3

3

x x x xy

x x

2 2

22

5 15 10 13 3

3

x x x x

x x

2

22

5 2 3

3

x xy

x x

Product and Quotient Rules and Higher-Order Derivatives

When not to use the Quotient Rule:3 5

8

x xy

Ex :

3rewrite as

and then differ

1

entiate.

5

8 8y x x

24

7

xy Ex :

2rewrite as

and then differenti

4

t7

a e.

y x

24 5 3

9

x xy

x

Ex : rewrite as

and then diff

20 12

erentiate.9 9

y x

Product and Quotient Rules and Higher-Order Derivatives

Derivatives of Trigonometric Functions

Memorize MemoriZEMemorizE MemorIZE MemoRIZE

MemORIZEMeMORIZEMEMORIZE

Product and Quotient Rules and Higher-Order Derivatives

Higher-Order Derivatives

• • •

First Derivative: ,y xD y ,f x ,dy

dx ,d

f xdx

nth Derivative: ,ny ,nf x ,n

n

d y

dx ,

n

n

df x

dx n

xD y

Fourth Derivative: 4 ,y 4 ,f x4

4,

d y

dx

4

4,

df x

dx 4

xD y

Third Derivative: ,y ,f x3

3,

d y

dx

3

3,

df x

dx 3

xD y

Second Derivative: ,y ,f x2

2,

d y

dx

2

2,

df x

dx 2

xD y