Principles of Calculus · SHS2 Additional…

Lesson notes

Learning Objectives

Indicator: 2.3.1.LI.1 - Identify the rules of differentiation.

By the end of the lesson, learners can:

State the constant rule, power rule, sum rule, difference rule, and constant multiple rule for differentiation from memory.
Identify which rule or combination of rules is appropriate for differentiating a given function, giving a reason for the choice.
Categorise functions according to the rule(s) needed, distinguishing between functions requiring only basic rules and those requiring the product, quotient, or chain rule.
Justify why a particular rule must be used for a given function by explaining what the structure of the function shows (for example, a product of two expressions requires the product rule).
Differentiate simple functions by applying the correct identified rule, demonstrating the first step of each rule clearly.

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Curriculum details

Strand: Calculus (Strand 3)
Sub-strand: Principles of Calculus (3.1)
Content standard: 2.3.1.CS.1 - Determine the appropriate rule to use in finding the derivative of a function and relations. 2.3.1.LO.1 Determine the appropriate rule and use it to find the derivative of a function. 2.3.1.LO.2 Estimate the area under a curve using the trapezoid rule.
Indicator: 2.3.1.LI.1 - Identify the rules of differentiation.
Suggested placement: Semester 2, Week 6 (Week 26 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

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Curriculum reference: NaCCA curriculum document, p. 378

Exemplars (from the NaCCA curriculum)

Talk for Learning, Think-pair-share, Experiential Learning, and Group Work/Collaborative Learning.
Activity 1: Group Work/Collaborative Learning Work in mixed-ability and gender-balanced or sensitive groups to arrive at situations where differentiation is necessary and distinguish between the rules of differentiation.
Rules for Differentiation
- Derivative of a constant function If 𝛼 is a real number and if 𝑓(𝑥) = 𝛼, then 𝑓 - (𝑥) = 0 for all 𝑥
- Derivative of a sum of two or more functions If 𝑓 - (𝑐) and 𝑔 - (𝑐) exist, then so do (𝑓 + 𝑔) - (𝑐). Moreover(𝑓 + 𝑔) - (𝑐) = 𝑓 - (𝑐) + 𝑔 - (𝑐)
- Derivative of a difference between two functions If 𝑓 - (𝑐) and 𝑔 - (𝑐) exist, then so do (𝑓 − 𝑔) - (𝑐). Moreover (𝑓 − 𝑔) - (𝑐) = 𝑓 - (𝑐) − 𝑔 - (𝑐)
- Derivative of a constant times a function If 𝑓 - (𝑐) exist, and if 𝛼 is any constant, then (𝛼 ∙ 𝑓) - (𝑐) = 𝛼 ∙ 𝑓 - (𝑐).
- Derivative of a linear combination of functions If 𝑓 - (𝑐) and 𝑔 - (𝑐) exist, and if 𝛼 and 𝛽 are any constants, then (𝛼 ∙ 𝑓 + 𝛽 ∙ 𝑔) - (𝑐) exists. Moreover (𝛼 ∙ 𝑓 + 𝛽 ∙ 𝑔) - (𝑐) = 𝛼 ∙ 𝑓 - (𝑐) + 𝛽 ∙ 𝑔 - (𝑐)
- Product rule If 𝑓 - (𝑐) 𝑎 𝑔 - (𝑐) exist, then (𝑓 ∙ 𝑔) - (𝑐) also exist and (𝑓 ∙ 𝑔) - (𝑐) = 𝑓 - (𝑐) ∙ 𝑔(𝑐) + 𝑓(𝑐) ∙ 𝑔 - (𝑐)
- Quotient rule If 𝑓 - (𝑐) 𝑎 𝑔 - (𝑐) exist and if 𝑔(𝑐) ≠ 0, (𝑓/𝑔)'(𝑐) exists. I - J(;)∙I 1 (;)*I(;)∙J-(;) Moreover, i j (𝑐) = J J ! (;)
- The power rule If 𝑦 = [𝑔(𝑥)] % then 𝑦 - = 𝑛[𝑔(𝑥)] %*! 𝑔(𝑥) -
- Chain rule <A <A <X If 𝑦 = 𝑓[𝑔(𝑥)] then <0 = <X × <0 where 𝑢 = 𝑔(𝑥)
Example: Identify which rule(s) of differentiation will be more appropriate for the given functions and why. !
- 𝑓(𝑥) = (0*")
- 𝑓(𝑥) = 4𝑥 − 7𝑥 )
- 𝑓(𝑥) = (2𝑥 + 5𝑥 . )(𝑥 − 2) ()0$.0 ( )(0*))
- 𝑓(𝑥) = (0*")
- 𝑓(𝑥) = (4𝑥 − 7𝑥 ) )(3𝑥 − 9) !(
- 𝑓(𝑥) = (3𝑥 − 4√𝑥) !(
- 𝑓(𝑥) = (3𝑥 − 9) !(
- 𝑓(𝑥) = 𝑠 (2𝑥)
- 𝑓(𝑥) = 𝑠 (2𝑥)
- 𝑓(𝑥) = 𝑠 (2𝑥 − 𝑥 ) )
- 𝑓(𝑥) = 𝑐 (𝑥 ) + 3𝑥) ;PO;PO (0) $;PO ()0)
- 𝑓(𝑥) = (0*")
- 𝑓(𝑥) = 𝑠 ) (𝑥)
- 𝑓(𝑥) = 𝑐 ) (𝑥) + √1 − 𝑥 𝑓(𝑥) = 𝑐 ) (𝑥) + w𝑠 (1 − 𝑥)
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Teaching and Learning Resources:
- Reading resource
- Colour pens
- Notebook
- Graph sheets
- Mathematical sets
- Technological tools
Assessment (2.3.1.AS.1). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 3 Strategic reasoning.