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Calculus 1 · Axiom Academy
When an equation ties two changing quantities together, it ties their rates of change together too — so one rate reveals the other. 1. One Constraint Locks Two Rates Pin a point to the circle x^2 + y^2 = 25 and let it travel. It can never leave the curve — so the instant x changes, y is forced to change to compensate. Watch the two velocity components: the horizontal rate (blue) and the vertical rate (orange) trade size as the point moves. They are not independent; the equation binds them. The constraint both variables must obey at all times Differentiate it in time and the rates become related 2. Differentiate the Equation in Time Take of both sides of x^2 + y^2 = 25 . Because x and y are functions of t , the Chain Rule attaches a rate to each term: , and likewise for y . The constant 25 has rate 0 . As the point moves, the two terms below stay perfectly equal and opposite — their sum is pinned to zero, which is exactly what keeps the point on the curve. — the x -side's contribution to the change. — the y -side's contribution to the change. — total change is nothing, so the point stays on the circle. — one rate now reveals the other. Freeze the motion at the moment the point reaches (3, 4) with its x -coordinate growing at 2 units per second. Substitute everything you know into and solve. The animation lands the point at (3,4) , draws the known blue vector, and resolves the orange — which comes out negative , pointing down. Substitute the moment: , which simplifies to .
This is the written version of the interactive lesson above. See the full Calculus 1 course.