Solving Related Rates Problems

Every related rates problem hides a static equation connecting two or more quantities, like x²+y²=L² for a sliding ladder. The quantities change over time, so you differentiate the whole equation with respect to time t, using the chain rule on each variable. That turns x and y into dx/dt and dy/dt, linking the rates the same way the original equation linked the quantities. The algebra never changes; only the geometry does.

The setup does the real work. Draw the picture, label every quantity that changes, and write the one equation that is always true, not just true at the instant you care about. Plug in the known rate and the known value of the variable at that instant only after differentiating, never before. This is why the ladder's top accelerates without limit as the base approaches the wall: dy/dt depends on x and y, and as y shrinks toward zero while x stays positive, dy/dt blows up, even though dx/dt stays constant. The base moves at a steady rate; the top's rate is a function of position, not a constant.

Related rates sits right after implicit differentiation for a reason: every setup here is an implicit differentiation problem wearing a word problem's clothes. It also previews optimization in the next unit, where instead of rates you'll track changing quantities to find a maximum or minimum, using the same habit of writing one clean equation before touching a derivative.