Related Rates

Finding the rate of change of one quantity given the rate of another

Golden Rule: Write an equation that relates the two quantities, differentiate both sides with respect to time t, then plug in known values last.

• Step 1: Draw and label the diagram — Assign variable names (x, y, r, h, etc.) to all quantities that change with time. Also note which quantities are constant. Do not plug in specific values yet — that comes last.
• Step 2: Write the geometric/physical equation — Use a known formula (Pythagorean theorem, similar triangles, volume/area formula, trig identities) to write one equation relating the variables. This equation must hold for all time t, not just at the instant of interest.
• Step 3: Differentiate with respect to t — Every variable that changes with time gets a d/dt applied to it via the chain rule. Constants stay constant. If you have a product of two changing quantities, use the product rule.
• Step 4: Substitute known values — Now substitute the values given in the problem: the known rates (dx/dt, dy/dt) and the known position values (x, y) at the instant asked about. Solve for the unknown rate.