Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 2nd Term
Week: 2
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Calculus
Previous Lesson: Differentiation: Meaning, Limits and First Principles.
Topic: DIFFERENTIATION
Subject Matter: Differentiation of transcendental function (sin x, e^ax, log 3x)

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

Define transcendental functions.
State the standard derivatives of trigonometric, exponential, and logarithmic functions.
Apply the chain rule to differentiate transcendental functions.

Affective Domain:

Appreciate the importance of differentiation in solving mathematical problems.
Develop interest in solving problems involving transcendental functions.

Psychomotor Domain:

Correctly differentiate functions involving sin(ax+b), e^(ax+b), and log_a(bx+c).
Solve problems involving the differentiation of transcendental functions accurately.

Collaborate with peers to solve differentiation problems.
Share and discuss solutions to differentiation exercises.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
Further Mathematics for Senior Secondary Schools by P.N. Chikezie
https://www.mathsisfun.com/calculus/derivatives-transcendental.html
https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-rules/ab-diff-rules-review/v/derivatives-of-transcendental-functions

Instructional Materials

The teacher will teach this lesson with the aid of:

Chart showing standard derivatives of transcendental functions.
Whiteboard and markers.
Further Mathematics textbooks.
Calculators.

Rationale for the Lesson

This lesson enables pupils to understand how to differentiate functions that are not algebraic. It helps pupils to solve more complex problems in calculus, which is important for advanced studies in science and engineering.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of basic differentiation rules (power rule, product rule, quotient rule) and an understanding of different types of functions.

Lesson Content/Board Summary

Differentiation of Transcendental Functions

Introduction to Transcendental Functions

Transcendental functions are functions that cannot be expressed as a finite combination of algebraic operations (addition, subtraction, multiplication, division, raising to a power, and extracting roots). Common examples include trigonometric functions (sin x, cos x, tan x), exponential functions (e^x, a^x), and logarithmic functions (ln x, log_a x).

Standard Derivatives of Transcendental Functions

The following are the standard derivatives for common transcendental functions:

d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (tan x) = sec² x
d/dx (e^x) = e^x
d/dx (e^(ax)) = ae^(ax)
d/dx (ln x) = 1/x
d/dx (log_a x) = 1/(x ln a)

Differentiation of Trigonometric Functions (e.g., sin(ax+b))

To differentiate functions like sin(ax+b), we use the chain rule: d/dx [f(g(x))] = f'(g(x)) * g'(x).

Example 1: Differentiate y = sin(4x+3)

Step 1: Identify the outer function f(u) = sin u and inner function u = 4x+3.
Step 2: Differentiate the outer function: d/du (sin u) = cos u.
Step 3: Differentiate the inner function: d/dx (4x+3) = 4.
Step 4: Apply the chain rule: dy/dx = cos(4x+3) * 4 = 4cos(4x+3).

Example 2: Find the derivative of y = cos(5x)

Step 1: Identify outer function f(u) = cos u and inner function u = 5x.
Step 2: Differentiate outer function: d/du (cos u) = -sin u.
Step 3: Differentiate inner function: d/dx (5x) = 5.
Step 4: Apply chain rule: dy/dx = -sin(5x) * 5 = -5sin(5x).

Differentiation of Exponential Functions (e.g., e^(ax))

For functions like e^(ax+b), the chain rule is also applied.

Example 1: Differentiate y = e^(5x)

Step 1: Identify outer function f(u) = e^u and inner function u = 5x.
Step 2: Differentiate outer function: d/du (e^u) = e^u.
Step 3: Differentiate inner function: d/dx (5x) = 5.
Step 4: Apply chain rule: dy/dx = e^(5x) * 5 = 5e^(5x).

Example 2: Find dy/dx if y = e^(-2x+1)

Step 1: Identify outer function f(u) = e^u and inner function u = -2x+1.
Step 2: Differentiate outer function: d/du (e^u) = e^u.
Step 3: Differentiate inner function: d/dx (-2x+1) = -2.
Step 4: Apply chain rule: dy/dx = e^(-2x+1) * (-2) = -2e^(-2x+1).

Differentiation of Logarithmic Functions (e.g., log_a(bx))

For functions like ln(f(x)) or log_a(f(x)), the chain rule is used.

Example 1: Differentiate y = ln(x² + 1)

Step 1: Identify outer function f(u) = ln u and inner function u = x² + 1.
Step 2: Differentiate outer function: d/du (ln u) = 1/u.
Step 3: Differentiate inner function: d/dx (x² + 1) = 2x.
Step 4: Apply chain rule: dy/dx = (1/(x² + 1)) * 2x = 2x/(x² + 1).

Example 2: Find dy/dx if y = log_3(3x)

Step 1: Use the change of base formula if needed: log_a b = ln b / ln a. So, log_3(3x) = ln(3x) / ln 3.
Step 2: Differentiate y = (1/ln 3) * ln(3x). (1/ln 3) is a constant.
Step 3: Differentiate ln(3x) using the chain rule: d/dx (ln(3x)) = (1/3x) * 3 = 1/x.
Step 4: Combine: dy/dx = (1/ln 3) * (1/x) = 1/(x ln 3).

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher reviews basic differentiation rules and introduces the concept of transcendental functions, asking pupils if they recall any non-algebraic functions.
Pupils’ Activity: Pupils recall basic differentiation rules and mention examples of trigonometric, exponential, or logarithmic functions.
Learning Point: Pupils connect prior knowledge to the new topic and understand what transcendental functions are.

Step 2: Presentation of Trigonometric Functions

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher writes down the standard derivatives for sin x, cos x, and tan x on the board and demonstrates how to differentiate functions like sin(ax+b) using the chain rule with worked examples.
Pupils’ Activity: Pupils observe, listen, and copy the formulas and examples into their notebooks, asking questions for clarification.
Learning Point: Pupils learn the standard derivatives of trigonometric functions and how to apply the chain rule.

Step 3: Presentation of Exponential Functions

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher presents the standard derivatives for e^x and e^(ax), then demonstrates how to differentiate functions like e^(ax+b) using the chain rule with worked examples on the board.
Pupils’ Activity: Pupils pay attention, take notes, and work through the examples shown by the teacher.
Learning Point: Pupils understand how to differentiate exponential functions, including those with a linear exponent.

Step 4: Presentation of Logarithmic Functions

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the standard derivatives for ln x and log_a x, explaining the use of the chain rule for functions like ln(f(x)) and log_a(bx). The teacher provides worked examples, including log_3(3x).
Pupils’ Activity: Pupils listen, write down the formulas, and follow the steps in the examples, attempting to solve them alongside the teacher.
Learning Point: Pupils learn to differentiate logarithmic functions and apply the chain rule correctly.

Step 5: Practice Problems

Time: 5 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher assigns a few practice problems involving a mix of trigonometric, exponential, and logarithmic functions for pupils to solve individually or in pairs.
Pupils’ Activity: Pupils attempt to solve the practice problems in their exercise books.
Learning Point: Pupils apply the learned rules and techniques to solve new problems.

Step 6: Class Discussion/Correction

Time: 5 minutes
Teaching Skill: Feedback/Correction
Teacher’s Activity: The teacher asks pupils to share their solutions and guides a discussion, correcting any errors and clarifying misconceptions.
Pupils’ Activity: Pupils present their solutions, participate in the discussion, and make necessary corrections in their notes.
Learning Point: Pupils consolidate their understanding and correct any errors in their problem-solving.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

State the derivative of sin(ax).
Differentiate y = e^(7x).
Find dy/dx if y = log_5(x).
Differentiate y = sin(2x-5).
Differentiate y = ln(4x).

Pupils’ Activity: Pupils answer orally and in writing.
Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points of the lesson, emphasizing the importance of standard formulas and the chain rule for transcendental functions. The teacher then assigns homework.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Pupils have a clear recap of the lesson and are aware of further practice.

Lesson Keywords

Differentiation – The process of finding the derivative of a function.
Transcendental Function – A function that cannot be expressed as a finite combination of algebraic operations.
Trigonometric Function – Functions involving angles, such as sine, cosine, and tangent.
Exponential Function – Functions where the variable is in the exponent, like e^x or a^x.
Logarithmic Function – The inverse of an exponential function, like ln x or log_a x.
Chain Rule – A formula to compute the derivative of a composite function.

Differentiation

For pupils who grasp the concepts quickly, the teacher will provide more complex problems involving combinations of transcendental functions and other rules (product/quotient rule). For pupils who need more support, the teacher will provide additional guided practice with simpler examples and one-on-one assistance.

Note for teachers using this lesson plan

Ensure pupils have a solid understanding of basic differentiation rules before introducing transcendental functions. Emphasize the chain rule as it is fundamental for differentiating composite transcendental functions. Encourage pupils to memorize the standard derivative formulas for efficiency.