Four steps, always in this order. The one that ends careers is doing step 4 too early.
Label the changing quantities. Mark which rate is given and which you want. Note what's constant.
Write the one equation linking the quantities — geometry, a volume formula, or similar triangles. Eliminate extra variables now.
Differentiate both sides with respect to t. Chain rule on every variable — each gets its own rate.
Only now plug in the instant's numbers and solve for the wanted rate. Substitute earlier and you freeze a moving quantity.
Pick a scene. Play it — the quantity you control changes at a steady rate you set, and the linked rate is computed live from the equation, so you watch them lock. The four setup steps fill in beside it. Then build the setup yourself in the challenge below.
This isn't a textbook trick. Anywhere two changing quantities obey one law, related rates reads a hidden rate off a measured one — often in real time, often when it matters.
A controller can't measure two jets' closing speed directly — she computes it. Their positions are tied by the distance formula, so one differentiation turns each plane's known speed into how fast the gap is shrinking. Safe separation versus near-miss.
A slick spreads as a growing circle. Responders need dA/dt — how fast the contaminated area expands — to size booms and crews. It falls straight out of A = πr² and a measured dr/dt: the ripple you dragged, at industrial stakes.
Operators release water at a controlled dV/dt, but what they watch is the level. The basin's shape links the two — and because it widens with height, the same outflow drops the level far faster when the reservoir runs low.
A ground camera holds a rising rocket in frame by matching its angle of elevation to the rocket's height: tan θ = h/d. Differentiate and the climb rate dh/dt sets exactly how fast the camera must pivot — quick near the pad, easing as it climbs away.
Three more, in words. For each, write the relation and its t-derivative before revealing — the numbers are the easy part.
Recognition drills. For each, decide the equation and its t-derivative in your head, then reveal to check — this is the fast half of every problem.
Everything at once. A balloon rises straight up while you watch from 500 ft away — and two different rates hide in the same scene: how fast it pulls away from you, and how fast you must tilt your gaze to follow it. Reveal each part step by step, then let the scene lock to the instant and confirm both answers.
The cardinal sin. If you plug in x = 3 before taking d/dt, you've frozen x — its rate vanishes and the whole problem collapses. Differentiate first, substitute last.
Every changing variable gets a rate when you differentiate. In x²+y²=L², both x and y change, so both dx/dt and dy/dt appear. Dropping one is the most common slip.
In the cone, V = ⅓πr²h has two changers. Use similar triangles (r = h/2) to get V in terms of h alone before differentiating — otherwise you'd need dr/dt too.
d/dt of r² is 2r·(dr/dt), not 2r. Every derivative is with respect to t, so a factor of the variable's rate rides along. Missing it is a silent, fatal error.
A shrinking quantity has a negative rate. The ladder's height falls, so dy/dt < 0. Let the equation produce the sign; don't force it.
"How fast is the shadow's tip moving" is not "how fast is the shadow lengthening." Re-read what's asked — tip speed is dx/dt + ds/dt, not ds/dt.