Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 2nd Term
Week: 1
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Calculus
Previous Lesson: .
Topic: DIFFERENTIATION
Subject Matter: Meaning of differentiation (rate of change of a function), limits of a function, differentiation from first principle, differentiation of polynomials.

Specific Objectives

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

Cognitive Domain:

Define differentiation as the rate of change of a function.
Evaluate the limits of a given function.
Differentiate simple functions from first principle.
Differentiate polynomial functions using basic rules.

Affective Domain:

Appreciate the importance of differentiation in various fields.
Show interest in solving problems involving differentiation.

Psychomotor Domain:

Apply the first principle to find the derivative of a function.
Accurately use differentiation rules to find the derivatives of polynomial functions.

Collaborate with peers to solve problems related to limits and differentiation.
Communicate their mathematical solutions clearly and logically.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
New Further Mathematics for Senior Secondary Schools 2

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing rules of differentiation
Whiteboard and markers
Textbook examples

Rationale for the Lesson

This lesson helps pupils understand the fundamental concept of differentiation, which is essential for studying rates of change in science, engineering, and economics. It enables pupils to develop problem-solving skills necessary for advanced mathematical concepts and real-world applications.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of algebraic manipulation, functions, and basic concepts of limits from introductory calculus or general mathematics.

Lesson Content/Board Summary

DIFFERENTIATION

Meaning of Differentiation

Differentiation is a process in calculus used to find the rate at which a function changes with respect to one of its variables. It represents the slope of the tangent line to the graph of the function at any given point.

Limits of a Function

The limit of a function f(x) as x approaches a specific value ‘a’ is the value that f(x) gets closer and closer to as x gets closer and closer to ‘a’, without necessarily being equal to ‘a’. It is written as lim (x→a) f(x).

Methods of evaluating limits:

Direct Substitution: Substitute the value ‘a’ into the function if it yields a defined real number.
Factorization/Simplification: If direct substitution results in an indeterminate form (e.g., 0/0), factorize the numerator and denominator to cancel out common factors, then substitute.

Example 1: Evaluate lim (x→2) (3x + 1)

Step 1: Substitute x = 2 into the function.

lim (x→2) (3x + 1) = 3(2) + 1

Step 2: Simplify.

= 6 + 1 = 7

Example 2: Evaluate lim (x→3) (x^2 – 9) / (x – 3)

Step 1: Direct substitution gives (3^2 – 9) / (3 – 3) = 0/0, which is indeterminate.

Step 2: Factorize the numerator using difference of two squares (x^2 – 9 = (x – 3)(x + 3)).

lim (x→3) [(x – 3)(x + 3)] / (x – 3)

Step 3: Cancel out the common factor (x – 3).

lim (x→3) (x + 3)

Step 4: Substitute x = 3.

= 3 + 3 = 6

Differentiation from First Principle

Differentiation from first principle involves using the definition of the derivative based on limits. For a function y = f(x), its derivative dy/dx or f'(x) is given by:

f'(x) = lim (h→0) [f(x + h) – f(x)] / h

Example 1: Differentiate f(x) = x^2 from first principle.

Step 1: Find f(x + h).

f(x + h) = (x + h)^2 = x^2 + 2xh + h^2

Step 2: Find f(x + h) – f(x).

f(x + h) – f(x) = (x^2 + 2xh + h^2) – x^2 = 2xh + h^2

Step 3: Divide by h.

[f(x + h) – f(x)] / h = (2xh + h^2) / h = 2x + h

Step 4: Take the limit as h → 0.

f'(x) = lim (h→0) (2x + h) = 2x + 0 = 2x

So, d/dx (x^2) = 2x.

Example 2: Differentiate f(x) = 3x + 5 from first principle.

Step 1: Find f(x + h).

f(x + h) = 3(x + h) + 5 = 3x + 3h + 5

Step 2: Find f(x + h) – f(x).

f(x + h) – f(x) = (3x + 3h + 5) – (3x + 5) = 3h

Step 3: Divide by h.

[f(x + h) – f(x)] / h = 3h / h = 3

Step 4: Take the limit as h → 0.

f'(x) = lim (h→0) (3) = 3

So, d/dx (3x + 5) = 3.

Differentiation of Polynomials

A polynomial function is a function of the form f(x) = ax^n + bx^(n-1) + … + cx + d, where a, b, c, d are constants and n is a non-negative integer.

Basic Rules of Differentiation for Polynomials:

Power Rule: If y = ax^n, then dy/dx = anx^(n-1).
Constant Rule: If y = c (where c is a constant), then dy/dx = 0.
Sum/Difference Rule: If y = f(x) ± g(x), then dy/dx = df/dx ± dg/dx.

Example 1: Differentiate y = 4x^3 – 2x^2 + 7x – 10.

Step 1: Apply the power rule to each term.

dy/dx = d/dx (4x^3) – d/dx (2x^2) + d/dx (7x) – d/dx (10)

Step 2: Calculate each derivative.

d/dx (4x^3) = 4 * 3 * x^(3-1) = 12x^2

d/dx (2x^2) = 2 * 2 * x^(2-1) = 4x

d/dx (7x) = 7 * 1 * x^(1-1) = 7x^0 = 7 * 1 = 7

d/dx (10) = 0 (derivative of a constant)

Step 3: Combine the results.

dy/dx = 12x^2 – 4x + 7

Example 2: Find the derivative of f(x) = 5x^4 + 3x – 8.

Step 1: Differentiate each term.

f'(x) = d/dx (5x^4) + d/dx (3x) – d/dx (8)

Step 2: Apply the rules.

d/dx (5x^4) = 5 * 4 * x^(4-1) = 20x^3

d/dx (3x) = 3 * 1 * x^(1-1) = 3

d/dx (8) = 0

Step 3: Combine.

f'(x) = 20x^3 + 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 introduces the topic by asking pupils what they understand by “rate of change” in everyday life, for example, speed as the rate of change of distance.
Pupils’ Activity: Pupils respond by giving examples of rates of change and relate them to previous knowledge.
Learning Point: Pupils are prepared for the lesson and connect new concepts to familiar ideas.

Step 2: Meaning of Differentiation

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher explains the meaning of differentiation as the instantaneous rate of change of a function and its geometric interpretation as the slope of the tangent to a curve.
Pupils’ Activity: Pupils listen attentively and take notes on the definition and concept.
Learning Point: Pupils understand the fundamental concept and purpose of differentiation.

Step 3: Limits of a Function

Time: 10 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher explains what a limit is and demonstrates how to evaluate limits using direct substitution and factorization with worked examples on the board.
Pupils’ Activity: Pupils observe the examples, ask questions for clarification, and attempt similar problems provided by the teacher.
Learning Point: Pupils learn how to evaluate limits of various functions.

Step 4: Differentiation from First Principle

Time: 10 minutes
Teaching Skill: Demonstration/Guided Practice
Teacher’s Activity: The teacher introduces the formula for differentiation from first principle and demonstrates its application to simple polynomial functions with step-by-step examples.
Pupils’ Activity: Pupils copy the formula, follow the steps of the examples, and attempt to differentiate a given function from first principle.
Learning Point: Pupils understand and can apply the first principle to find derivatives.

Step 5: Differentiation of Polynomials

Time: 8 minutes
Teaching Skill: Explanation/Application
Teacher’s Activity: The teacher explains the basic rules for differentiating polynomials (power rule, constant rule, sum/difference rule) and provides examples.
Pupils’ Activity: Pupils listen, note down the rules, and apply them to solve simple polynomial differentiation problems.
Learning Point: Pupils can differentiate polynomial functions efficiently using the established rules.

Step 6: Further Practice/Application

Time: 5 minutes
Teaching Skill: Problem-Solving
Teacher’s Activity: The teacher provides additional problems combining limits, first principle, and polynomial differentiation for pupils to solve individually or in pairs.
Pupils’ Activity: Pupils work on the problems, applying the concepts learned, and seek assistance where needed.
Learning Point: Pupils reinforce their understanding and improve their problem-solving skills.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define differentiation.
Evaluate lim (x→1) (x^2 + 2x – 3) / (x – 1).
Differentiate y = x^3 from first principle.
Find dy/dx for y = 6x^4 – 5x + 9.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points of the lesson, emphasizing the definitions of differentiation and limits, and the methods for finding derivatives. The teacher then assigns relevant homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Pupils consolidate their understanding and are prepared for further practice.

Lesson Keywords

Differentiation – The process of finding the derivative of a function.
Rate of Change – How one quantity changes in relation to another.
Limit – The value a function approaches as the input approaches some value.
First Principle – The method of finding a derivative using the limit definition.
Polynomial – An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more complex polynomial functions or introduce differentiation of functions with negative or fractional exponents. For pupils who need more support, the teacher will provide simpler functions and offer one-on-one guidance or peer tutoring opportunities, focusing on understanding the step-by-step process of evaluating limits and applying the first principle.

Note for teachers using this lesson plan

Ensure that pupils have a strong foundation in algebraic manipulation, especially factorization, before proceeding to limits and differentiation from first principle. Emphasize the connection between the algebraic definition of the derivative and its graphical interpretation as the slope of a tangent. Encourage pupils to practice numerous examples to build proficiency in applying the rules of differentiation.