Composition Gears

Calculus 1 · Axiom Academy

Composition Is Gears in Series Stack one function inside another and you build a gear train: turn the input, the inner gear drives the outer gear, and the overall rate is the product of the two. Two gears, one turn — and the rates multiply When you write f(g(x)) you nest one function inside another: the input feeds g , and whatever g puts out feeds f . That is exactly a gear train — turn the crank, the inner gear spins, and it drives the outer gear. The question calculus asks is simple: if you nudge the input a little, how fast does the final output move? Watch the input gear drive the inner gear g , which meshes into and drives the outer gear f . The inner gear turns at one ratio; the outer turns at another. The rate the final gear spins, compared to your input, is neither of those alone — it is the two ratios multiplied . The final gear runs at the inner ratio times the outer ratio — that product is the whole idea behind the chain rule. Crank each ratio — the combined rate is their product Set how fast the inner gear g turns the input, and how fast the outer gear f turns its input. The two gears spin live, and the final gear runs at the product of the two ratios. Push one ratio to zero and the whole train stalls — a slow link anywhere slows everything. Combined rate = inner ratio × outer ratio. Multiply the links, and you have the rate of the whole chain. The same product is the chain rule

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