1. Section 2.3 Day 1 Product & Quotient Rules & Higher order Derivatives
AP Calculus AB

2. Learning Targets Define & apply the Product Rule
Define & apply the Quotient Rule
Apply more derivatives of trigonometric functions
Determine higher order derivatives
Recognize & apply the relationship between position, velocity, and acceleration functions

3. Product Rule
𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 =𝑓 𝑥 𝑔 ′ 𝑥 +𝑔 𝑥 𝑓′(𝑥)

4. Example 1: Product Rule Find the derivative of ℎ 𝑥 = 3𝑥− 2𝑥 2 5+4𝑥
Let 𝑓 𝑥 =3− 𝑥 2 and 𝑔 𝑥 =5+4𝑥
ℎ ′ 𝑥 =𝑓 𝑥 𝑔 ′ 𝑥 +𝑔 𝑥 𝑓′(𝑥)
= 3𝑥− 𝑥 (5+4𝑥)(3−4𝑥)
=−24 𝑥 2 +4𝑥+15

5. Example 2: Product Rule Find the derivative of 𝑦=3 𝑥 2 sin 𝑥
Let 𝑓 𝑥 =3 𝑥 2 and 𝑔 𝑥 = sin 𝑥
𝑦 ′ =𝑓 𝑥 𝑔 ′ 𝑥 +𝑔 𝑥 𝑓′(𝑥)
=3 𝑥 2 cos 𝑥 + sin 𝑥 (6𝑥)
=3 𝑥 2 cos 𝑥 +6𝑥 sin 𝑥

6. Example 3: Product Rule Find the derivative of 𝑦=2𝑥 cos 𝑥 −2 sin 𝑥
Let 𝑓 𝑥 =2𝑥, 𝑔 𝑥 = cos 𝑥 , and ℎ 𝑥 =−2 sin 𝑥
𝑦 ′ =𝑓 𝑥 𝑔 ′ 𝑥 +𝑔 𝑥 𝑓 ′ 𝑥 +ℎ′(𝑥)
=2𝑥 − sin 𝑥 + cos 𝑥 2 −2 cos 𝑥
=−2𝑥 sin 𝑥 +2 cos 𝑥 −2 cos 𝑥
=−2𝑥 sin 𝑥

7. Example 4: Product Rule NON Example
1. 𝑑 𝑑𝑥 𝑥 3 ∙ 𝑥 4 = 𝑑 𝑑𝑥 [ 𝑥 7 ]
2. =7 𝑥 6
Non-Example: 𝑑 𝑑𝑥 [ 𝑥 3 ∙ 𝑥 4 ]
1. 𝑑 𝑑𝑥 𝑥 3 ∙ 𝑥 4
= 𝑑 𝑑𝑥 [ 𝑥 3 ]∙ 𝑑 𝑑𝑥 [ 𝑥 4 ]
2. =3 𝑥 2 ∙4 𝑥 3 =12 𝑥 5

8. Quotient Rule
𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑔 𝑥 𝑓 ′ 𝑥 −𝑓 𝑥 𝑔 ′ 𝑥 𝑔 𝑥 2

9. Example 5: Quotient Rule
Find the derivative of 𝑦= 5𝑥−2 𝑥 2 +1
Let 𝑓 𝑥 =5𝑥−2 and 𝑔 𝑥 = 𝑥 2 +1
𝑦 ′ = 𝑔 𝑥 𝑓 ′ 𝑥 −𝑓 𝑥 𝑔 ′ 𝑥 𝑔 𝑥 2 = 𝑥 − 5𝑥−2 2𝑥 𝑥
= 5 𝑥 2 +5 −10 𝑥 2 +4𝑥 𝑥 = −5 𝑥 2 +4𝑥+5 𝑥

10. Example 6: Quotient Rule
Find the derivative of 𝑦= sin 𝑥 cos 𝑥
Let 𝑓 𝑥 = sin 𝑥 and 𝑔 𝑥 = cos 𝑥
𝑦 ′ = 𝑔 𝑥 𝑓 ′ 𝑥 −𝑓 𝑥 𝑔 ′ 𝑥 𝑔 𝑥 2 = cos 𝑥 cos 𝑥 − sin 𝑥 − sin 𝑥 cos 𝑥 2
= cos 2 𝑥 + sin 2 𝑥 cos 2 𝑥 = 1 cos 2 𝑥 = sec 2 𝑥

11. Example 7: Quotient Rule
Find the derivative of 𝑦= 1− cos 𝑥 sin 𝑥
Let 𝑓 𝑥 =1− cos 𝑥 and 𝑔 𝑥 = sin 𝑥
𝑦 ′ = 𝑔 𝑥 𝑓 ′ 𝑥 −𝑓 𝑥 𝑔 ′ 𝑥 𝑔 𝑥 2 = sin 𝑥 sin 𝑥 − 1− cos 𝑥 cos 𝑥 sin 2 𝑥
= sin 2 𝑥 − cos 𝑥 + cos 2 𝑥 sin 2 𝑥 = 1−cos x sin 2 x

12. Exit Ticket for Feedback
1. Find 𝑓′(𝑥) of 𝑓 𝑥 = 𝑥 2 cos 𝑥
2. Find 𝑓′(𝑥) of 𝑓 𝑥 = 4 𝑥 3 5𝑥−2