Introduction to Related Rates

When two quantities are connected by an equation, they're forced to change together. A circle's area depends on its radius, so if the radius grows, the area must grow too, and the equation A = π r² controls exactly how fast. The job of related rates is to turn that controlling equation into a statement about speeds: how fast one thing changes compared to how fast the other does.

The move is to differentiate the whole equation with respect to time, treating every quantity as a function of t, even if t never appears explicitly. Differentiating A = π r² with respect to time uses the chain rule, since r itself is changing: dA/dt = 2 π r (dr/dt). This single line now connects the two rates, dA/dt and dr/dt, through the current value of r. Plug in r = 3 and dr/dt = 2, and dA/dt = 2 π (3)(2) = 12 π square cm per second. The radius has to stay a variable through the differentiation step; it only becomes a number afterward, once the derivative has captured how A responds to changes in r.

This same pattern, write an equation linking two quantities, differentiate with respect to time, substitute known values last, is the entire toolkit for every related rates problem that follows. A ladder sliding down a wall uses the Pythagorean theorem instead of a circle's area formula, and a draining cone uses the volume of a cone, but the mechanism never changes. Mastering this simple circle case is really mastering the whole method.