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.
How to recognize the method, run it, and know when it is the wrong choice.
Reviewed July 11, 2026Draw the situation, name time-dependent quantities, write one equation, differentiate implicitly with respect to time, then substitute the instant’s values.
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.
A circle’s radius grows at 3 cm/s. How fast is its area growing when r = 4 cm?
Connect area and radius.
Differentiate with respect to time.
Substitute the instant’s radius and known rate.
Common mistakes
- Substituting the snapshot values before differentiating.
- Forgetting a chain factor such as dr/dt.
- Reporting a magnitude without the sign or units.
Three takeaways
- Model first, differentiate second, substitute last.
- Every changing variable contributes a time derivative.
- Check whether the sign matches the story.