Rates of Change
When quantities depend on other quantities that are changing, for example the volume of a sphere depends on the radius which is increasing at 1 cm per second, we have to be very methodical in our approach if we want to find the rate of change of volume of the sphere. We use the chain rule, which in this case can be made to relate the rate of change of volume to the rate of change of volume with radius and rate of change of radius:
Suppose then that the radius is increasing at 1 cm per second, soand suppose at some instant the radius is 5. Sinceso at that instant when
Substitute these values into (1) to give
The diagram is of a garden pond. The volume of the pond isWater is poured in at the rate of 0.03m^3 per minute.
From the chain rule,
Into this expression substituteand