Section 3.10
The Chain Rule
The most powerful tool in your arsenal. It allows you to differentiate composite functions by working from the outside in.
1
The Onion Analogy
Peel the Layers
Composite functions equate to layers. To differentiate , you take the derivative of the outer layer, leave the inside alone, then multiply by the derivative of the inside.
Outer Derivative×Inner Derivative
2
Worked Example
Differentiate .
Step 1: Identify Layers
- Outside: (Power function)
- Inside: (Cubic function)
Step 2: Differentiate Outside
Power rule: bring down 100, subtract 1. Keep the inside!
Step 3: Multiply by Inner Derivative
Derivative of is .
3
More Examples
Trig Function Composition
Differentiate .
Identify Layers:
Chain Rule:
Exponential Chain
Differentiate .
Identify Layers:
Chain Rule:
Double Chain (Nested)
Differentiate .
Identify 3 Layers:
Chain Rule Twist:
5
Practice Quiz
Loading...