calculus

Related Rates

Related rates problems link the rates of change of two or more variables connected by an equation. Use implicit differentiation with respect to time.

Related rates problems involve multiple changing quantities linked by an equation, where you know one rate of change and need to find another.

Standard procedure:

  1. Identify the quantities and the relationship (geometric formula, physical law).
  2. Differentiate the relationship with respect to time tt — implicit differentiation, treating each variable as a function of tt.
  3. Plug in the known values of variables and rates.
  4. Solve for the unknown rate.

Classic problems: a ladder slides down a wall (how fast does the bottom move?), water fills a conical tank (how fast does the water level rise?), two cars approach an intersection (how fast does the distance between them change?).

Critical setup tip: never substitute numbers before differentiating. Differentiate first while everything is still a variable, then substitute the snapshot values.