SHS2 Additional Mathematics · Semester 2 Week 8: Principles of Calculus

Lesson notes

Learning Objectives

Indicator: 2.3.1.LI.4 - Identify and apply techniques of differentiation to solve problems involving transcendental functions.

By the end of the lesson, learners can:

Define exponential and natural logarithmic functions and state the derivative rules for e^x, e^u, ln x, and ln u.
Apply the chain rule to differentiate composite functions involving e^x and ln x, such as e^(3x) and ln(x² + 1).
Differentiate products and quotients that combine transcendental functions with polynomial functions.
Use the derivative of transcendental functions to solve practical problems involving growth and decay models.
Justify the choice of differentiation technique (chain rule, product rule, quotient rule) for a given transcendental function.

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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.4 - Identify and apply techniques of differentiation to solve problems involving transcendental functions.
Suggested placement: Semester 2, Week 8 (Week 28 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. 386

Exemplars (from the NaCCA curriculum)

Talk for Learning, Think-pair-share, Experiential Learning, and Group Work/Collaborative Learning.
Activity 1: Derivative of specific transcendental functions Each group identifies suitable techniques for finding a derivative of a specific transcendental function and shares their techniques with other groups. Learners to research how exponential and
logarithmic functions are applied in technology, business, biology, chemistry, physics and engineering and present findings to the whole class.
NB: Growth and decay models are investigated using such functions.
Activity 2: Derivative of Exponential and Logarithm functions Learners work in convenient groups (ability, mixed-ability and mixed gender) to find the derivative of exponential and natural logarithm functions. Hint:
- If 𝑔(𝑥) = 𝑒 I(0) then <(J(0) = 𝑓'(𝑥)𝑒 I(0) <0 and
- if 𝑓(𝑥) then <0 = I(0) <J(0) I-(0)
Example 1: Find the derivative of the following; 𝑔(𝑥) = 𝑒 0
-
- 𝑔(𝑥) = 𝑒 "0
- 𝑔(𝑥) = 𝑥 0
- 𝑔(𝑥) = 𝑒 0 */0 !
- 𝑔(𝑥) = 𝑙 (𝑥)
- 𝑔(𝑥) =𝑙 𝑙 (𝑥 + 2)
- 𝑔(𝑥) = 𝑙 (𝑥 + 4𝑥 ) ) 𝑔(𝑥) =𝑙 𝑙 (𝑥 + 4𝑥 ) ) + 𝑒 "0
-
- 𝑔(𝑥) = 𝑥 (𝑥 + 4𝑥 ) )
Example 2: Find the derivative of the following: i. 𝑦 = √𝑙 "
Solution: $ 𝑦 = (𝑙) " <A ! ! ! = " (𝑙) * " ∙ 0 <0 ! ! = "0 ∙ " u(_%0) ! ! = "0 u(_%0) ! " N ) $N .) ii. 𝑦 = N ) *N .)
Solution: 𝑑 (𝑒 0 −𝑒 *0 )(𝑒 0 −𝑒 *0 ) − (𝑒 0 + 𝑒 *0 )(𝑒 0 +𝑒 *0 ) = 𝑑 (𝑒 0 −𝑒 *0 ) ) (N ) *N .) ) ! *(N ) $N .) ) ! = (N ) *N .) ) ! (N ) *N .) *N ) *N .) )(N ) *N .) $N ) $N .) ) = (N ) *N .) ) ! (*)N .) )()N ) ) = (N ) *N .) ) ! , = − (N ) *N .) ) !
Teaching and Learning Resources:
- Reading resource
- Colour pens
- Notebook
- Graph sheets
- Mathematical sets
- Technological tools
Assessment (2.3.1.AS.4). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 4 Extended critical thinking and reasoning.