Derivatives in Context — Units & Meaning

When a problem gives a real-world function, the derivative has real-world meaning.

If s(t) = position in miles, t in hours:

s'(t) = velocity in miles per hour

s''(t) = acceleration in miles per hour per hour

If C(x) = cost in dollars, x = units produced:

C'(x) = marginal cost in dollars per unit

C'(100) = 3.50 means the 101st unit costs approximately $3.50

If P(t) = population, t in years:

P'(t) = rate of population change (people per year)

P'(5) = -200 means the population is decreasing by 200 people per year at t = 5

AP rule: When interpreting, always include:

1. What is changing

2. The rate/units

3. When/where (the specific value of the independent variable)

4. Whether it's increasing or decreasing

Reference:

Wikipedia: Derivative

https://en.wikipedia.org/wiki/Derivative