Product rule (AASL/HL)
Keywords
[br][table][br][tr][br][td]Product rule[/td][br][td]積の法則[/td][br][td]곱의 법칙[/td][br][td]乘积规则[/td][br][/tr][br][tr][br][td]Differential calculus[/td][br][td]微分計算[/td][br][td]미분 계산[/td][br][td]微分学[/td][br][/tr][br][tr][br][td]Derivative[/td][br][td]導関数[/td][br][td]도함수[/td][br][td]导数[/td][br][/tr][br][tr][br][td]Functions[/td][br][td]関数[/td][br][td]함수[/td][br][td]函数[/td][br][/tr][br][tr][br][td]Differentiating products[/td][br][td]積の微分[/td][br][td]곱의 미분[/td][br][td]乘积的微分[/td][br][/tr][br][/table][br]
[table][br][tr][br][td][b]Factual Inquiry Questions[/b][/td][br][td][b]Conceptual Inquiry Questions[/b][/td][br][td][b]Debatable Inquiry Questions[/b][/td][br][/tr][br][tr][br][td]1. What is the product rule in differential calculus?[/td][br][td]1. Why is the product rule necessary for differentiating products of functions, as opposed to simply multiplying their individual derivatives?[/td][br][td]1. Is the product rule more foundational to calculus than the quotient rule, given its applications and the way it's taught?[/td][br][/tr][br][tr][br][td]2. How is the product rule applied to find the derivative of the product of two functions?[/td][br][td]2. How does the product rule facilitate the understanding and computation of derivatives in more complex functions?[/td][br][td]2. Can the conceptual understanding of the product rule be extended to enhance approaches to teaching other rules of differentiation?[/td][br][/tr][br][tr][br][td][/td][br][td][/td][br][td]3. How do advancements in symbolic computation impact the practical importance of manually applying the product rule in calculus?[/td][br][/tr][br][/table][br]
Mini-Investigation: The Tale of Two Derivatives
Greetings, Math Apprentice! Today, we embark on a quest to uncover the secrets of derivatives, both through the mystical Product Rule and without its aid. Prepare your wits and let's begin our exploration.[br]
Chapter 1: The Classic Approach[br][br]We have before us a magical expression y = (x + 3)(x + 2). The ancient wizards derived this using the Product Rule. But what if the Product Rule had not yet been discovered?[br][br]1. If you were a mathemagician without the Product Rule at your disposal, how would you find the derivative of y = (x + 3)(x + 2)?[br]2. Perform this ancient technique to calculate dy/dx. Show your process step by step.[br][br]
Chapter 2: The Product Rule Revealed[br][br]Now, let's use the Product Rule, a spell that combines the powers of u(x) and v(x) to find the derivative of their product.[br][br]1. Apply the Product Rule to find the derivative of y = (x + 3)(x + 2). Record each incantation and compare it to your previous method.[br]2. Which method was quicker? Describe the advantages of using the Product Rule over the classic approach.[br][br]
Chapter 3: The Duel of Derivatives[br][br]It's time to test your newfound skills in a friendly duel against the time-tested classic method.[br][br]1. Using both methods, find the derivative of a new spell z = (x+1)(2x + 4)^2). Which method leads you to the answer more efficiently?[br]2. Reflect on your findings. How does each method's result help us understand the behavior of the spell z as x changes?[br]
Chapter 4: Beyond the Textbook[br][br]The real world is filled with complexities far beyond the boundaries of a textbook. Sometimes, the Product Rule is a necessity, not a choice.[br][br]1. Can you think of a real-world scenario where the Product Rule would be essential in finding the rate of change?[br]2. Propose a situation where the classic method might be more insightful or useful, despite its lengthier process.[br][br]## Epilogue: The Wisdom Gained[br][br]As the stars of knowledge align, let's reflect on the lessons learned from our investigative journey.[br][br]1. Discuss how the Product Rule and the classic method each contribute to your understanding of calculus.
Part 2
Checking your understanding
Bonus - Geometric interpretation of the product rule
Question 1[br]If [math]f(x)=x^2[/math] and [math]g(x)=e^x[/math], what is the derivative of [math]h(x)=f(x)g(x)[/math]?[br][br][br][br][br]
Question 2[br]Given [math]u(x)=ln(x)[/math] and [math]v(x)=cos(x)[/math], what is the derivative of [math]w(x)=u(x)v(x)[/math]?[br][br]
Question 3[br]If [math]p(t)=t^3[/math] and [math]q(t)=\sqrt{(}t)[/math], what is the derivative of [math]r(t)=p(t)q(t)[/math]?[br][br][br][br]
Question 4[br]Consider [math]f(x)=sin(x)[/math] and [math]g(x)=ln(x)[/math]. What is the derivative of the product [math]h(x)=f(x)g(x)[/math]?[br]
Question 5[br]If [math]m(x)=e^{(2x)}[/math] and [math]n(x)=x^2[/math], what is the derivative of [math]o(x)=m(x)n(x)[/math]?[br]
Lesson plan - Understanding the Product Rule in Differential Calculus
Product rule - Intuition pump (thought experiments and analogies)
