- Show that the rate at which the area of a circle changes is equal to the circumference of the circle.
- Show that the rate at which the volume of a sphere changes is equal to the surface area of the sphere.

- The area of a circle of radius is . Therefore, the rate of change of the area with respect to is the derivative,
- The volume of a sphere of radius is . Therefore, the rate of change of the volume with respect to 4r$ is the derivative,