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
