Lesson 2: Standard Derivatives and Their Applications

👨🏽‍🏫 Introduction

Imagine you’re riding a bike 🚲 uphill and want to know how steep the path is at different points. The steepness at each point comes from the slope, and you can find it using derivatives!

Now that you understand what a derivative is, it’s time to learn how to find them quickly using rules—no limits required! In this lesson, you’ll master the standard derivatives of common functions and apply them to solve real problems involving motion, cost, and more.

• 🧠 Learn the basic derivative rules
• 🧮 Apply rules to polynomials, exponentials, and trigonometric functions
• 🔧 Solve real-life problems involving rates and slopes

🧠 Key Concepts

📌 1. Derivative Rules to Know

• Power Rule:
• Constant Rule: where c is a constant
• Constant Multiple Rule:
• Sum Rule:

📌 2. Standard Derivatives of Common Functions

📌 3. Applications of Derivatives

• Motion: Velocity and acceleration from position functions
• Economics: Marginal cost and marginal revenue
• Geometry: Slopes of curves and tangents

🔍 Worked Examples

1. 1. Example 1 – Basic Polynomial:
      Find
      
      if
      .
      

1. 1. Example 2 – Exponential and Logarithmic Functions:
      Differentiate
      .
      

1. 1. Example 3 – Trigonometric Function:
      Differentiate
      .
      

1. Example 4 – Application in Motion:
   A particle moves with position function
   .
   Find its velocity and acceleration.

   Velocity:
   
   Acceleration:
   

✍🏽 Practice Exercises

🅰️ Section A – Basic Differentiation

1. Differentiate:
   
2. Find
   
   if
   

🅱️ Section B – Trig and Exponential Functions

1. Differentiate:
   
2. Find
   
   if
   

🅾️ Section C – Real-World Application

1. The cost function for producing
   
   items is
   .
   Find the marginal cost function
   
   and compute it at
   .

✅ Answers and Explanations

• A1:
• A2:
• B1:
• B2:
• C1:
  Differentiate:
  At
  :
  
  🟢 Answer: Marginal cost at 10 items is 5 units of currency per item

📌 Recap

• 📚 You now know the most important rules for differentiating functions
• 🧠 You can handle powers, trigonometric, exponential, and log functions with ease
• 💼 You applied derivatives to economics and motion—real-life math in action!

💭 Reflection Prompt

Try to think about your daily routines—like cooking time, walking pace, or fuel consumption. Can you imagine a function that models it and how quickly it’s changing? What would its derivative look like? 🔍📊