related rates
When you inflate a balloon, several things change together: the radius grows, the surface stretches, the volume swells — all at once, all linked. Related rates is the technique for answering a question like, if I pump air in at a steady rate, how fast is the radius growing right now? You know one rate of change and you want another, and the two are tied together because the quantities themselves are tied together by a formula.
The method hinges on differentiating with respect to time. Start from the equation linking the quantities — say V = (4/3) pi r^3 for a sphere — then differentiate both sides with respect to t, treating each variable as a function of time. The chain rule turns r^3 into 3 r^2 (dr/dt), giving dV/dt = 4 pi r^2 (dr/dt). Now plug in the known rate and the known instant to solve for the unknown rate. The classic ladder problem works the same way: a ladder of fixed length L against a wall has x^2 + y^2 = L^2, so differentiating gives 2x(dx/dt) + 2y(dy/dt) = 0, linking how fast the foot slides out to how fast the top slides down.
The order of operations is what trips people up. Differentiate first, treating the changing quantities as functions of time, and only then substitute the specific numbers for that instant. If you plug the numbers in too early — fixing r to a value before differentiating — you freeze a variable that was supposed to be moving, and its rate vanishes from the equation, giving nonsense.
Differentiating the volume formula with respect to time links the rate of inflation dV/dt to the rate the radius grows dr/dt.
Quantities that stay constant throughout the motion (like a ladder's length) may be substituted before differentiating, but any quantity that is changing must be left as a variable until after you differentiate.