Related Rates Guide
Classic related-rates scenarios solved step by step: differentiate the relation, substitute, solve.
Tip: write the geometric relation first, differentiate with respect to time t (chain rule!), then plug in the instant’s values.
① Answer
② Steps
③ Why it works
In related rates, two quantities change together because geometry ties them: A = πr² ties area to radius. Differentiating with respect to time — not x — uses the chain rule: dA/dt = dA/dr · dr/dt. The known rate flows through the derivative of the relation to reveal the unknown rate. Always substitute the values after differentiating.