Composition Chains

Calculus 3 · Axiom Academy

When a quantity depends on things that themselves depend on time, its rate of change flows down every path at once. That is the multivariable chain rule. One thing, riding on two moving parts Picture a drone crossing a landscape. Its altitude z depends on where it is — on the two coordinates x and y . But the drone is moving, so its position is itself changing with time: x = x(t) and y = y(t) . So the altitude changes over time — not because time pushes on it directly, but because time moves the position, and the position sets the altitude. Watch the dependency chain. A nudge in time t flows down to both x(t) and y(t) , and each of those flows up into z . Time reaches z along two separate paths — one through x , one through y . Two arrows into z means two paths for time to travel — and the chain rule adds their effects together. Each path carries part of the rate Take a concrete example: z = x^2y , with and . Drag time and watch the two contributions. The x -path carries ; the y -path carries . Their sum is the total rate — sometimes they push the same way, sometimes they fight. The total rate is the two path terms added together — never just one of them. Position moves, so altitude has a rate Same example. On the left, drag time to send the position around its circular path. On the right, the altitude z it feels traces out a single curve z(t) — and the slope of that curve, right now, is exactly , the number the two paths added up to.

This is the written version of the interactive lesson above. See the full Calculus 3 course.