Related rates: translate the geometry before differentiating
Connect changing quantities with one equation, differentiate with respect to time, and substitute only after the rates appear.
LaTeX article Updated July 11, 2026
Rates belong to a moment
A related-rates problem supplies values at a particular instant, not constants valid for all time. Substituting them before differentiating can erase the dependency that creates the requested rate.
Keep every changing quantity as a function of time until the derivative equation is formed.
Choose the connecting equation
The geometry or physical constraint is the bridge between known and unknown rates. For a circle use area or circumference; for a right triangle use the Pythagorean theorem; for a cone use similar triangles when dimensions co-vary.
Use the equation with the fewest unneeded variables.
Signs and units are part of the answer
A decreasing length has a negative rate. A positive computed rate may contradict a draining or shrinking description if signs were assigned carelessly.
Track units through every derivative: area changes in square units per time, while length changes in units per time.
Worked example
Common mistakes
- Substituting the snapshot values before differentiating.
- Forgetting a chain factor such as dr/dt.
- Reporting a magnitude without the sign or units.
Keep these ideas
- Model first, differentiate second, substitute last.
- Every changing variable contributes a time derivative.
- Check whether the sign matches the story.