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Section 2.3 Calculus AP/Dual, Revised Β©2017 viet.dang@humbleisd.net
Product Rule Section 2.3 Calculus AP/Dual, Revised Β©2017 2/24/2019 5:48 AM Β§2.3: Product Rule
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Derivative (not being simplified)
Guess The Rule Equation Derivative (not being simplified) 1) π
π
π π π π π+ππ 2) π
π
π ππβπ π π (π+ππ) 3) π
π
π π ( π π +π) 4) π
π
π π π π¬π’π§ π ππ¨π¬ π 5) π
π
π π π π π 2/24/2019 5:48 AM Β§2.3: Product Rule
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The Product Rule βFirst d-second plus second d-firstβ
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The Product Rule π
π
π π π π π =π π π β² π +π π π β² π
π
π
π π π π π =π π π β² π +π π π β² π βFirst times the derivative of the second plus the second times the derivative of the firstβ 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: πβ²=πβ²(π) = π
π π
π = π
π
π π π Second Derivative: πβ²β²=πβ²β²(π) = π
π π π
π π = π
π π
π π π π Third Derivative: πβ²β²β²=πβ²β²β²(π) = π
π π π
π π = π
π π
π π π π Fourth Derivative: πβ²β²β²= π (π) (π)= π
π π π
π π = π
π π
π π π π nth Derivative: π (π) = π (π) (π) = π
π π π
π π = π
π π
π π π π 2/24/2019 5:48 AM Β§2.3: Product Rule
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Proof of Product Rule 2/24/2019 5:48 AM Β§2.3: Product Rule
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Proof of Product Rule 2/24/2019 5:48 AM Β§2.3: Product Rule
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Example 1 Solve for the derivative of π π = π π ππβπ 2/24/2019 5:48 AM
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Example 2 Solve for the derivative of π π = (βππ π +π π π +π) βπ π π +π 2/24/2019 5:48 AM Β§2.3: Product Rule
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Your Turn Solve for the derivative of π π = π π +π π π +π
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Example 3 Solve for the derivative of π π =π π π π¬π’π§ π
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Example 4 Solve for the derivative of π π =π¬π’π§ π ππ¨π¬ π
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Your Turn Solve for the derivative of π π =ππ π¬π’π§ π by using the Product Rule 2/24/2019 5:48 AM Β§2.3: Product Rule
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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 π π = π β² π = π β²β² π 2/24/2019 5:48 AM Β§2.3: Product Rule
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Example 5 The position of an object is defined by the equation, π= π π π π + π π . What is the acceleration of the object at π=π? 2/24/2019 5:48 AM Β§2.3: Product Rule
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Example 6 Given π
π π
π =π π π βππ, solve for π
π π π
π π
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Your Turn Given πβ²β²(π)=ππ+ππ π βπ , solve for π π π 2/24/2019 5:48 AM
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AP Multiple Choice Practice Question 1 (non-calculator)
If π β²β² π =π π π +ππ+π, solve for π (π) π (A) π (B) π (C) ππ+π (D) ππ+π 2/24/2019 5:48 AM Β§2.3: Product Rule
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AP Multiple Choice Practice Question 1 (non-calculator)
If π β²β² π =π π π +ππ+π, solve for π (π) π Vocabulary Connections and Process Answer and Justifications 2/24/2019 5:48 AM Β§2.3: Product Rule
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AP Multiple Choice Practice Question 2 (non-calculator)
Find the derivative of π π π π . (A) π π π β² π +π π π (B) ππ π β² π (C) π π π π +π π β² π (D) π π π β² π 2/24/2019 5:48 AM Β§2.3: Product Rule
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AP Multiple Choice Practice Question 2 (non-calculator)
Find the derivative of π π π π . Vocabulary Connections and Process Answer and Justifications 2/24/2019 5:48 AM Β§2.3: Product Rule
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Assignment Page all, 13, 17, 31, 39, 63, 67A, 91, 93, 97 2/24/2019 5:48 AM Β§2.3: Product Rule
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