Rates of Change in Applied Contexts Other Than Motion

A derivative is just a rate of change, and it does not care what is changing. Position gives velocity, but volume, population, and money all work the same way: whatever quantity Q(t) you are tracking, dQ/dt tells you how fast it is growing or shrinking at a given instant. The unit after the word per is the only thing that changes from problem to problem.

The mechanism behind almost every one of these problems is rate-in minus rate-out. A tank's volume rises because water flows in and falls because water leaks out, so dV/dt equals the inflow rate minus the outflow rate at that moment. A population grows because births add people and shrinks because deaths remove them, so dP/dt equals births per day minus deaths per day. Whenever the problem hands you two competing rates pushing in opposite directions, subtract them to get the net rate of change. If the rates are constant, the net rate is constant too; if they are given as functions of time, you plug in the specific time you care about.

This topic sets up everything else in the unit. Once you can read a situation as rate-in minus rate-out, you are ready for related rates problems where several quantities change together, and for the idea that a derivative's sign tells you whether something is increasing or decreasing at an instant. The skill here is reading a word problem and translating it into a single subtraction of rates, which is the foundation for the more geometric setups that follow.