One of my favorite applications of derivatives is solving related rates problems. In this type of problem, two related parameters are changing at different rates over time. For example, throwing a stone in a pool of water creates ripples that expand outward. The rate of change of the area enclosed by the first ripple is related to the rate of change of the radius of the ripple. In this screencast, you can see how calculus helps us solve this type of problem.
FracTad's Fractopia 