4.1 Interpreting the Meaning of the Derivative in Context

In applied problems, the derivative represents an instantaneous rate of change, so its meaning depends on the situation and on the units involved. If y=f(x), then f'(x) tells how fast y is changing with respect to x at a specific value of x. For example, if s(t) is position in meters and t is time in seconds, then s'(t) is velocity in meters per second. If P(t) is population in thousands of people and t is measured in years, then P'(t) is the rate at which the population is changing in thousands of people per year. So interpreting a derivative is not just about computing a number; it is about explaining what quantity is changing, with respect to what input, and in what units.