The Chain Rule: Rates of Change Multiply

1. Choose the two stages: u = ax + b (a ≠ 0),  y = u2

The Chain Rule

2. The composition machine — three graphs, three slopes

Stage 1: u = ax + b

Stage 2: y = f(u) = uc

Composite: y = (ax + b)c

3. Why multiply? — the telescoping ratio

For an actual step h = Δx ≠ 0, the difference quotients obey the identity

ΔyΔx  =  ΔyΔu  ·  ΔuΔx

ratio at step hvaluelimit as h → 0value
Δu / Δxg′(x₀) = a
Δy / Δuf′(u₀) = c·u₀c−1
Δy / Δx  =  product of the twof′(u₀) · g′(x₀)

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