The Chain Rule can be in many different ways. One of the most practical is to find the rate of change of some dimension of a solid when the volume of the solid is changing. For example, a spherical ballon is being blown up and we might want to find the rate of change of the radius, treating the balloon as a sphere, or oil might be poured onto a puddle of oil, and we might want to find the rate of increase of the radius of the oil puddle.

Suppose that a spherically shaped raindrop is falling. As it falls, water condenses on it. The rate of increase of the radius of the raindrop is 0.01 mm/s at a tuime when the radius of the raindrop is 3 mm. What is the rate at which the volume is increasing?

If the raindrop can be treated as a sphere, then the volume of the raindrop is The chain rule may be used in the form

so whenmm,

We are told in the question that the rate of increase ofmm/s, so

mm ^{3 }/s.

Suppose on the other hand that air is being blown into a balloon at the rate of 5 cm ^{3 }/s at a point when the radius of the balloon is 6 cm. Find the rate at which the radius of the ballon is increasing.

Again using the relationshipwithso when r=6 cm,

is the rate at which air is being blown into the balloon, so

cm/s.