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Grade 12 Math Lab: Analysis & Inference FREE

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The Chain Rule

70 min Grade 12 Math Lab: Analysis & Inference

Learning Objectives

  • Apply chain rule
  • Differentiate composite functions
  • Recognize nested functions

Key Concepts

Chain Rule

d/dx[f(g(x))] = f'(g(x)) · g'(x)

Composite Function

Function inside another function

Theory

**Chain Rule:** If y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx) **Steps:** 1. Identify outer and inner functions 2. Differentiate outer (keep inner unchanged) 3. Multiply by derivative of inner **Example:** d/dx[(3x + 1)⁵] - Outer: u⁵, Inner: 3x + 1 - 5(3x + 1)⁴ · 3 = 15(3x + 1)⁴

Examples

d/dx[sin(x²)]

Solution: cos(x²) · 2x = 2x·cos(x²)

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Knowledge Check

Test your understanding of Chain Rule Gears!