Related Rates: Step-by-Step Guide
Related rates problems link the rates of change of two or more quantities through an equation. By implicitly differentiating that equation with respect to time, you can solve for an unknown rate given the others. The key is to set up a correct geometric or algebraic relationship before differentiating.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Identify all variables and given rates. Read the problem and label every changing quantity. Note which rates are given and which is unknown. All variables are functions of time t.\[\frac{dx}{dt}, \;\; \frac{dy}{dt}, \;\; \frac{dV}{dt}, \;\dots\]
- Write an equation relating the variables. Use geometry, the Pythagorean theorem, similar triangles, area/volume formulas, or other relationships to link the variables in a single equation.\[F(x, y, \dots) = 0\]
- Differentiate both sides with respect to t. Apply implicit differentiation, treating each variable as a function of t. The chain rule introduces dx/dt, dy/dt, etc.\[\frac{d}{dt}[F(x, y, \dots)] = 0\]
- Substitute the known values and solve. Plug in the known instantaneous values of variables and rates, then solve for the unknown rate.\[\text{solve for unknown } \frac{d?}{dt}\]
Worked example
A 10-ft ladder leans against a wall. The bottom is pulled away from the wall at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?
Step 1: Let x = distance from the wall to the bottom and y = height of the top. Given dx/dt = 2 ft/s; find dy/dt when x = 6.
Step 2: The ladder forms a right triangle: x^2 + y^2 = 10^2 = 100.
Step 3: Differentiate implicitly with respect to t: 2x*(dx/dt) + 2y*(dy/dt) = 0, so x*(dx/dt) + y*(dy/dt) = 0.
Step 4: When x = 6, y = sqrt(100 - 36) = sqrt(64) = 8.
Step 5: Substitute: 6*(2) + 8*(dy/dt) = 0, so 12 + 8*(dy/dt) = 0, giving dy/dt = -12/8 = -1.5 ft/s.
Step 6: The top is sliding down at 1.5 ft/s (negative sign indicates decreasing height).
Verification: Units check (ft and ft/s). The answer is reasonable: as x grows, y must shrink, so dy/dt should be negative.
Tips and common mistakes
- Draw a clear diagram and label every variable before writing equations.
- Distinguish quantities that change with time (variables) from quantities that are fixed (constants like ladder length).
- Differentiate WITH RESPECT TO t — every variable becomes a chain-rule term.
- Plug in numerical values ONLY AFTER differentiating — substituting too early kills the rate information.
- Negative rates indicate decrease; positive rates indicate increase. The sign matters for the answer.
- Common geometric formulas to know: Pythagorean theorem, area of circle/triangle, volume of sphere/cone, similar-triangle ratios.
- For cone problems, similar triangles often let you eliminate one variable from the volume formula.
- Always answer the question in the right units (e.g., ft/s) and with a complete sentence interpreting the rate.