Linked quantities change with the same time variable. Differentiate their relationship before substituting the instantaneous values.

Draw and label the changing quantities; identify what is fixed.

Write the geometric or physical relation.

Differentiate with respect to time, treating every changing quantity as a function of time.

Substitute the values at the specified instant and solve for the required rate.

A decreasing volume has \(\dfrac{\mathrm{d}V}{\mathrm{d}t}<0\). A negative derivative describes decrease; if asked for the rate of decrease, report its positive magnitude with the word “decreasing”. Check units: volume rate and radius rate have different units.

Connected Rates of Change

Differentiation

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Related Rates of Change

Related Rates of Change