Description

DIFFERENTIATION — Complete Study Guide (008)
Advanced Level Mathematics — Form Five

A comprehensive, exam-ready resource covering everything from first principles to stationary points and curve sketching — all in one clearly organized document.

📘 What’s Inside
1. Introduction to Differentiation

Definition and real-life applications (physics, economics, engineering, biology, medicine, business)

Average vs. instantaneous rate of change explained

Visual: secant vs. tangent line

2. Differentiation from First Principles

Definition of the derivative and alternative notation (dy/dx)

8 fully worked examples from first principles:

Linear function

Quadratic function

Cubic function

Reciprocal function

Square root function

Exponential function

Sine function (using compound angle formula)

Cosine function (using compound angle formula)

3. Differentiation Rules

Power Rule — with worked examples including negative and fractional powers

Sum and Difference Rules

Product Rule — two fully worked examples

Quotient Rule — two fully worked examples

Chain Rule — four fully worked examples (polynomial, sine, exponential, square root)

4. Implicit Differentiation

3-step method

Three fully worked examples (circle, x² + xy + y² = 4, and trigonometric implicit)

5. Derivatives of Trigonometric Functions

Complete formula table (sin, cos, tan, cot, sec, cosec)

Proofs of the derivatives of tan x and sec x using the quotient rule

Worked examples with composite trigonometric functions

6. Derivatives of Inverse Trigonometric Functions

All six inverse derivative formulas

Full proof that d/dx(sin⁻¹x) = 1/√(1−x²)

Worked examples with composite inverse functions

7. Higher Order Derivatives

Notation table (first through nth derivative)

Worked example finding derivatives up to fourth order

8. Applications: Tangents, Normals, and Stationary Points

Tangent and normal line equations with worked examples

Stationary points table: maximum, minimum, point of inflection

Worked examples:

Finding and classifying stationary points

Full curve sketching (intercepts, extrema, end behaviour) with graph

9. Chapter Summary

Key derivative rules table (power, sum, product, quotient, chain)

Complete derivative table (constant through inverse functions)

10. Quick Reference Card

First principles, all rules, implicit differentiation, tangent/normal formulas, and stationary point tests on one card

11. Practice Questions

Eight sections: First Principles, Power Rule, Product & Quotient Rules, Chain Rule, Implicit Differentiation, Trig Derivatives, Inverse Trig, and Applications

✅ Why This Guide Works
Every rule proven where it matters — first principles for 8 functions and full proofs of tan x, sec x, and sin⁻¹x

Compound angle formulas applied — showing exactly how sin and cos derivatives are derived

Curve sketching integrated — connecting derivatives to graph features

Exam-focused — quick reference card, key rules table, and practice questions by topic

Complete coverage — from the definition of a derivative to stationary point classification in a single document

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