Derivatives from First Principles

Introduction

• Finding derivatives using first principles involves a specific formula.
• Understanding the limit as h approaches zero is crucial.

Example 1: Linear Function

Function: f(x) = 4x - 3

1. Plug in x + h into the function:
   • f(x + h) = 4(x + h) - 3
   • = 4x + 4h - 3
2. Set up the limit:
   • Limit as h approaches 0: [ rac{f(x + h) - f(x)}{h} = \frac{(4x + 4h - 3) - (4x - 3)}{h} ]
3. Simplify the equation:
   • = [ \frac{4h}{h} = 4 ]
4. Conclusion: The derivative is 4.

Example 2: Quadratic Function

Function: f(x) = x² + 4x

1. Plug in x + h:
   • f(x + h) = (x + h)² + 4(x + h)
   • = x² + 2xh + h² + 4x + 4h
2. Set up the limit:
   • Limit as h approaches 0: [ \frac{(x² + 2xh + h² + 4x + 4h) - (x² + 4x)}{h} ]
3. Simplify the equation:
   • = [ \frac{(2xh + h² + 4h)}{h} ]
   • Cancel terms: [(2x + h + 4) ]
4. Apply the limit as h approaches 0:
   • Result: 2x + 4

Summary

• Derivatives can be found using first principles by evaluating limits.
• Key steps involve plugging in x + h, simplifying, and applying the limit as h approaches zero.