Lesson 1: Introduction to Differentiation

👨🏽‍🏫 Introduction

Have you ever wondered how fast your car is going at a specific moment? Or how steep a hill is at a particular point on a trail? 🏞️ These are questions of rate of change, and differentiation is the tool we use to answer them!

Differentiation is a core concept in calculus that helps us understand how things change. Whether it’s speed, slope, or growth rate, calculus gives us the math to model it. 🚗📈

• 🧠 Understand what a derivative is and how it’s related to slope
• 📐 Learn the first principles of differentiation
• 🛠 Solve real-life problems using derivatives

🧠 Key Concepts

📌 1. What is Differentiation?

• Differentiation is the process of finding the derivative of a function.
• The derivative measures the rate of change of one quantity with respect to another.
• In graphs, it tells us the slope of the tangent line at any given point.

📌 2. Derivative from First Principles

The derivative of a function f(x) at a point x is defined as:

This formula shows how the change in function values over a small interval becomes the instantaneous rate of change.

📌 3. Physical Meaning

• If f(x) represents distance, then f'(x) represents speed.
• If f(x) represents cost, then f'(x) could represent the rate at which cost is increasing or decreasing.

🔍 Worked Examples

1. 1. Example 1 – Using First Principles:
      Find the derivative of f(x) = x^2 using the first principle.

      Step 1: Apply the definition:
      
      Step 2: Expand the numerator:
      
      Step 3: Divide by h:
      
      Step 4: Take the limit as h → 0:
      
      🟢 Answer: The derivative of x^2 is 2x.

1. 1. Example 2 – Rate of Change in Real Life:
      A car’s position at time t is given by s(t) = 5t^2. Find its speed at t = 3 seconds.
      Step 1: Differentiate s(t):
      
      Step 2: Substitute t = 3:
      
      🟢 Answer: Speed is 30 m/s at 3 seconds.

1. Example 3 – Differentiating Polynomials:
   Find the derivative of f(x) = 3x^3 – 5x + 2.
   
   🟢 Answer: f'(x) = 9x^2 – 5

✍🏽 Practice Exercises

🅰️ Section A – First Principles

1. Find the derivative of f(x) = x^3 using first principles.

🅱️ Section B – Basic Differentiation

1. Differentiate: f(x) = 4x^2 + 6x – 1
2. Differentiate: y = 2x^4 – 3x^2 + x

🅾️ Section C – Real-Life Application

1. The height of a ball thrown into the air is given by h(t) = -5t^2 + 20t + 2. Find its velocity at t = 2 seconds.

✅ Answers and Explanations

• • A1:
    Use first principles:
    
    
    
    As h → 0:

• B1:
• B2:
• C1:
  Differentiate:
  At t = 2:
  🟢 Answer: Velocity = 0 m/s (The ball is at its highest point!)

📌 Recap

• Differentiation helps us calculate rates of change and slopes of curves
• The first principle defines the derivative as a limit of small changes
• Derivatives are applied in real-life contexts like speed, cost, and motion

💭 Reflection Prompt

Think of something around you that changes with time — your phone battery, bus arrival time, your height over years. Can you describe how fast it changes at a particular point? That’s the power of derivatives in action! 📱⏱️📉