Section 2.3 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net Product Rule Section 2.3 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net 2/24/2019 5:48 AM §2.3: Product Rule
Derivative (not being simplified) Guess The Rule Equation Derivative (not being simplified) 1) 𝒅 𝒅𝒙 𝟐 𝒙 𝟐 𝟓+𝟒𝒙 2) 𝒅 𝒅𝒙 𝟑𝒙−𝟐 𝒙 𝟐 (𝟓+𝟒𝒙) 3) 𝒅 𝒅𝒙 𝒙 ( 𝒙 𝟐 +𝟐) 4) 𝒅 𝒅𝒙 𝒙 𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 5) 𝒅 𝒅𝒙 𝒇 𝒙 𝒈 𝒙 2/24/2019 5:48 AM §2.3: Product Rule
The Product Rule “First d-second plus second d-first” 2/24/2019 5:48 AM §2.3: Product Rule
The Product Rule 𝒅 𝒅𝒙 𝒇 𝒙 𝒈 𝒙 =𝒇 𝒙 𝒈 ′ 𝒙 +𝒈 𝒙 𝒇 ′ 𝒙 𝒅 𝒅𝒙 𝒇 𝒙 𝒈 𝒙 =𝒇 𝒙 𝒈 ′ 𝒙 +𝒈 𝒙 𝒇 ′ 𝒙 “First times the derivative of the second plus the second times the derivative of the first” Many times, there can be more than one derivative can be taken for a differentiable function. These derivatives imply continued continuity (like the first derivative): First Derivative: 𝒚′=𝒇′(𝒙) = 𝒅𝒚 𝒅𝒙 = 𝒅 𝒅𝒙 𝒇 𝒙 Second Derivative: 𝒚′′=𝒇′′(𝒙) = 𝒅 𝟐 𝒚 𝒅 𝒙 𝟐 = 𝒅 𝟐 𝒅 𝒙 𝟐 𝒇 𝒙 Third Derivative: 𝒚′′′=𝒇′′′(𝒙) = 𝒅 𝟑 𝒚 𝒅 𝒙 𝟑 = 𝒅 𝟑 𝒅 𝒙 𝟑 𝒇 𝒙 Fourth Derivative: 𝒚′′′= 𝒇 (𝟒) (𝒙)= 𝒅 𝟒 𝒚 𝒅 𝒙 𝟒 = 𝒅 𝟒 𝒅 𝒙 𝟒 𝒇 𝒙 nth Derivative: 𝒚 (𝒏) = 𝒇 (𝒏) (𝒙) = 𝒅 𝒏 𝒚 𝒅 𝒙 𝒏 = 𝒅 𝒏 𝒅 𝒙 𝒏 𝒇 𝒙 2/24/2019 5:48 AM §2.3: Product Rule
Proof of Product Rule 2/24/2019 5:48 AM §2.3: Product Rule
Proof of Product Rule 2/24/2019 5:48 AM §2.3: Product Rule
Example 1 Solve for the derivative of 𝒇 𝒙 = 𝒙 𝟐 𝟐𝒙−𝟑 2/24/2019 5:48 AM §2.3: Product Rule
Example 2 Solve for the derivative of 𝒇 𝒙 = (−𝟐𝒙 𝟒 +𝟓 𝒙 𝟐 +𝟒) −𝟑 𝒙 𝟐 +𝟐 2/24/2019 5:48 AM §2.3: Product Rule
Your Turn Solve for the derivative of 𝒇 𝒙 = 𝒙 𝟐 +𝟏 𝒙 𝟑 +𝟑 2/24/2019 5:48 AM §2.3: Product Rule
Example 3 Solve for the derivative of 𝒇 𝒙 =𝟑 𝒙 𝟑 𝐬𝐢𝐧 𝒙 §2.3: Product Rule
Example 4 Solve for the derivative of 𝒇 𝒙 =𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 §2.3: Product Rule
Your Turn Solve for the derivative of 𝒇 𝒙 =𝟐𝒙 𝐬𝐢𝐧 𝒙 by using the Product Rule 2/24/2019 5:48 AM §2.3: Product Rule
Acceleration Acceleration can be defined as the second derivative of the position function or the first derivative of the velocity of the function Rate of Change of Velocity 𝒇 𝒕 = 𝒗 ′ 𝒕 = 𝒂 ′′ 𝒕 2/24/2019 5:48 AM §2.3: Product Rule
Example 5 The position of an object is defined by the equation, 𝒚= 𝟏 𝟒 𝒕 𝟒 + 𝒕 𝟐 . What is the acceleration of the object at 𝒕=𝟐? 2/24/2019 5:48 AM §2.3: Product Rule
Example 6 Given 𝒅𝒚 𝒅𝒙 =𝟓 𝒙 𝟒 −𝟑𝒙, solve for 𝒅 𝟒 𝒚 𝒅 𝒙 𝟒 §2.3: Product Rule
Your Turn Given 𝒇′′(𝒙)=𝟔𝒙+𝟐𝟕 𝒙 −𝟐 , solve for 𝒇 𝟒 𝒙 2/24/2019 5:48 AM §2.3: Product Rule
AP Multiple Choice Practice Question 1 (non-calculator) If 𝒇 ′′ 𝒙 =𝟑 𝒙 𝟐 +𝟔𝒙+𝟒, solve for 𝒇 (𝟒) 𝒙 (A) 𝟎 (B) 𝟔 (C) 𝟐𝒙+𝟔 (D) 𝟔𝒙+𝟔 2/24/2019 5:48 AM §2.3: Product Rule
AP Multiple Choice Practice Question 1 (non-calculator) If 𝒇 ′′ 𝒙 =𝟑 𝒙 𝟐 +𝟔𝒙+𝟒, solve for 𝒇 (𝟒) 𝒙 Vocabulary Connections and Process Answer and Justifications 2/24/2019 5:48 AM §2.3: Product Rule
AP Multiple Choice Practice Question 2 (non-calculator) Find the derivative of 𝒙 𝟐 𝒇 𝒙 . (A) 𝒙 𝒙 𝒇 ′ 𝒙 +𝟐 𝒇 𝒙 (B) 𝟐𝒙 𝒇 ′ 𝒙 (C) 𝒙 𝒙 𝒇 𝒙 +𝟐 𝒇 ′ 𝒙 (D) 𝒙 𝟐 𝒇 ′ 𝒙 2/24/2019 5:48 AM §2.3: Product Rule
AP Multiple Choice Practice Question 2 (non-calculator) Find the derivative of 𝒙 𝟐 𝒇 𝒙 . Vocabulary Connections and Process Answer and Justifications 2/24/2019 5:48 AM §2.3: Product Rule
Assignment Page 125 1-6 all, 13, 17, 31, 39, 63, 67A, 91, 93, 97 2/24/2019 5:48 AM §2.3: Product Rule