What the Derivative Means
A derivative is more than a calculation. In a real-world problem, the derivative tells you how quickly one quantity is changing with respect to another quantity.
If a quantity depends on another quantity , then the derivative represents the instantaneous rate at which is changing with respect to .
The word instantaneous matters. A derivative gives the rate of change at one particular input value, rather than the average rate of change over an interval.
The Basic Formula
The derivative can be viewed as the limit of average rates of change over smaller and smaller intervals:
For a function , the derivative tells us the instantaneous rate of change of with respect to at .
Average Rate vs. Instantaneous Rate
This distinction is one of the most important ideas in AP Calculus.
Suppose the distance traveled by a car is modeled by , where is measured in hours and is measured in miles.
The average speed from to hours is:
This tells us how much the distance changed on average during the interval.
If we instead calculate , we are asking: "How fast is the distance changing at exactly 3 hours?"
That is an instantaneous rate, which is what a derivative gives us.
Units: The Secret to Interpreting Derivatives Correctly
On AP Calculus questions, the units of a derivative are often the easiest way to construct a correct interpretation.
If is measured in miles and is measured in hours, then:
So the derivative's units are always:
Think of this as "units of the output per unit of the input."
A Step-by-Step Method for Context Questions
- Identify what each variable represents. Determine the units and meaning of the input and output.
- Identify the derivative. Determine exactly what rate of change the derivative represents.
- Evaluate at the requested value. Substitute the given input into the derivative.
- Attach the units. The units are output units per input unit.
- Write a sentence. State what the numerical value means in the context of the problem.
Worked Examples
A city's population, , is modeled by , where is measured in years. Find and interpret .
Therefore, at year 5, the city's population is increasing at a rate of 2,500 people per year.
The depth of water in a tank is modeled by , where is measured in minutes and is measured in centimeters. Find and interpret .
The negative sign means the water level is decreasing. Therefore, at 4 minutes, the water level is decreasing at a rate of 2 cm per minute.
The temperature of a chemical reaction 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 .
Let represent the amount of medication in a patient's bloodstream, measured in milligrams, hours after a dose. Suppose . Interpret .
Therefore, 3 hours after the dose, the amount of medication in the bloodstream is decreasing at a rate of 4 mg per hour.
What the Sign of the Derivative Tells You
The sign of the derivative gives immediate information about the direction in which the quantity is changing.
- Positive derivative: the quantity is increasing.
- Negative derivative: the quantity is decreasing.
- Zero derivative: the quantity is momentarily not changing.
For example, if is measured in dollars per month, then means the revenue is increasing at that particular time.
Be Precise About What Is Changing
A common AP Calculus mistake is to say that the derivative is "the value of the function." It is not.
If represents the amount of water in a tank, then tells you the amount of water, while tells you the rate at which the amount of water is changing.
The first quantity might be measured in liters. The second might be measured in liters per minute. They answer completely different questions.
Common Mistakes
A derivative in context has units. If the question asks for an interpretation, include the units in your answer. "The rate is 12" is incomplete; "the population is increasing at 12 people per year" is a proper interpretation.
If represents distance, then represents the rate at which distance changes. Don't describe as a distance.
A negative derivative has important meaning. It tells you that the quantity is decreasing. Don't report only the magnitude and leave out the direction.
If is measured in kilograms and in hours, then has units of kilograms per hour, not hours per kilogram.
An average rate describes change over an interval. A derivative describes the instantaneous rate at a specific point.
Practice Problems
The position of a particle is given by , where is measured in seconds and position is measured in meters. Find and interpret .
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The amount of water in a reservoir is modeled by , where is measured in days and is measured in millions of liters. Find and interpret .
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A company's revenue is represented by , where is measured in thousands of dollars and is measured in months. Find the average rate of change from month 2 to month 6.
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The temperature of a cooling object is modeled by , where is measured in hours and temperature is measured in degrees Fahrenheit. Interpret the meaning of .
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