What a Rate of Change Really Means
A derivative does not have to describe motion. In AP Calculus, a derivative is fundamentally a rate of change: it tells you how quickly one quantity changes as another quantity changes.
When the independent variable is time, the derivative often describes how something changes per unit of time. But the same idea applies when the independent variable is something else. A derivative can describe how revenue changes as the number of products sold changes, how area changes as a radius changes, or how concentration changes as the volume of a solution changes.
If represents a quantity that depends on , then the derivative represents the instantaneous rate at which changes with respect to .
The Basic Interpretation
The most important skill is translating the derivative into words.
For example, suppose represents the number of customers at a store and is measured in hours. Then might mean that at a particular time, the number of customers is increasing at a rate of 18 customers per hour.
Notice that the derivative's units matter. The units of a derivative are always:
That unit analysis is often the key to answering AP Calculus questions correctly.
Average Rate vs. Instantaneous Rate
Two ideas that look similar are often confused.
Average Rate of Change
The average rate of change over an interval measures the overall change from one point to another:
Geometrically, this is the slope of a secant line.
Instantaneous Rate of Change
The instantaneous rate of change at a particular input is the derivative:
Geometrically, this is the slope of the tangent line at that point.
So if a problem asks for the rate of change at a particular time or input, you are usually looking for a derivative evaluated at that input—not an average over an interval.
A Step-by-Step Method
- Identify the dependent and independent variables. Determine what quantity is changing and what it is changing with respect to.
- Find or use the derivative. If a function is given, differentiate it. If a table or graph gives derivative information, use that directly.
- Evaluate at the requested input. Substitute the given value into the derivative.
- Attach the correct units. State the answer as "units of the dependent variable per unit of the independent variable."
- Interpret the sign. A positive derivative means the quantity is increasing; a negative derivative means it is decreasing.
Worked Examples
A town's population is modeled by , where is measured in years. What is the population's instantaneous rate of change at ?
A company sells units and has revenue , measured in dollars. Find the instantaneous rate at which revenue changes when units have been sold.
The temperature of a chemical solution is modeled by , where is measured in minutes and temperature is measured in degrees Celsius. Find the rate at which the temperature is changing at .
The area of a circular region is , where is the radius in centimeters. Find the rate at which area changes with respect to radius when .
The number of customers entering a store is represented by . A table gives at . What does this information mean?
Positive and Negative Rates
The sign of a derivative carries important information.
A positive rate of change means the dependent quantity is increasing as the independent variable increases. A negative rate means it is decreasing.
For example, if a company's revenue satisfies , then revenue is increasing at that input. If instead , revenue is decreasing.
Rates of Change Do Not Have to Involve Time
This is one of the most important distinctions for this type of problem. Students often see the word "rate" and immediately assume that time must be involved.
That is not true. A rate of change simply describes one quantity changing with respect to another quantity.
For example:
- Revenue with respect to units sold: dollars per unit
- Area with respect to radius: square centimeters per centimeter
- Volume with respect to height: cubic meters per meter
- Population with respect to time: people per year
- Concentration with respect to volume: concentration units per liter
The derivative always answers the question: "How much does the output change for a small change in the input?"
Common Mistakes
If the question asks for the rate at a particular input, you usually need the derivative evaluated there. The average rate over an interval uses the secant-line formula instead.
A numerical derivative is often incomplete without units. If revenue is measured in dollars and quantity is measured in units, then the derivative is measured in dollars per unit.
Do not report only the magnitude. A negative derivative means the quantity is decreasing, while a positive derivative means it is increasing.
Velocity and acceleration are important applications of derivatives, but they are only special cases. Population, revenue, cost, temperature, area, volume, and concentration can all have meaningful rates of change.
Always identify what the derivative is "with respect to." The expressions and represent different rates and generally have different meanings and units.
Practice Problems
The population of a city is modeled by , where is measured in years. Find the instantaneous rate of population change at .
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A company's revenue is modeled by , where is the number of products sold. Find and interpret .
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The area of a circle is . Find the rate at which the area changes with respect to radius when .
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Suppose represents the concentration of a solution as a function of volume. If , interpret this value.
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