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

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

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

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 2 4 +(5+4𝑥)(3−4𝑥) =−24 𝑥 2 +4𝑥+15

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

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

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

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

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 2 +1 5 − 5𝑥−2 2𝑥 𝑥 2 +1 2 = 5 𝑥 2 +5 −10 𝑥 2 +4𝑥 𝑥 2 +1 2 = −5 𝑥 2 +4𝑥+5 𝑥 2 +1 2

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

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

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