Why Compare Rates at Different Inputs?
A derivative describes an instantaneous rate of change at a particular input. Comparing derivative values lets us describe how that rate differs from one input to another. For example, a growing quantity might still be growing at both times, but at a slower rate at the later time.
The comparison is between rates, not necessarily between the values of the quantity itself. A quantity can be larger at one input even though its rate there is smaller. Keeping those two questions separate is essential when interpreting derivatives in context.
What Is Being Compared?
Suppose \(Q=f(x)\) represents a contextual quantity, and \(x\) is its input. At input \(a\), the instantaneous rate is \(f'(a)\); at input \(b\), it is \(f'(b)\). Both rates must refer to the same quantity and input, so they have the same units. For instance, if \(Q\) is measured in liters and \(x\) in hours, both derivative values are measured in liters per hour.
To compare them, first order the signed numbers. Then interpret the result in context. If \(f'(b)>f'(a)\), the rate at \(b\) is greater than the rate at \(a\). This does not by itself tell you whether the quantity is larger at \(b\), or what its rate did at every input between \(a\) and \(b\).
The units of this difference are still the units of rate. It is a difference between two rates, not the total change in the original quantity.
A Short Justification for the Comparison
The derivative \(f'(x)\) is the instantaneous rate of change of \(f\) at input \(x\). Applying that meaning separately at \(a\) and \(b\) gives the two rates \(f'(a)\) and \(f'(b)\). Subtracting the first from the second measures the change in the rate value. In particular, if that difference is positive, the later-listed rate is greater; if it is negative, it is smaller.
This conclusion compares rates at two inputs. To claim that the rate keeps increasing throughout an entire interval, you would need information about the rate at all inputs in that interval, not only its values at two selected points.
Worked Example: A Positive Rate That Gets Smaller
Worked Example: A Positive Rate That Gets Smaller
A community’s population \(P(t)\) is measured in thousands of people, with \(t\) in years. A model is \(P(t)=12+3t-0.25t^2\), for \(0\leq t\leq 8\). Compare the population’s instantaneous rates at \(t=2\) and \(t=5\).
Differentiate the model:
Evaluate at each input:
Both rates are positive, so the model indicates that the population is increasing at both inputs. The rate at \(t=5\) is smaller. The difference is:
Thus the rate value is \(1.5\) thousand people per year smaller at year \(5\) than at year \(2\). The units are thousands of people per year because that is the unit of \(P'(t)\). As a check, substituting the two rates into the difference gives \(0.5-2=-1.5\), consistent with the comparison. The population values themselves are \(P(2)=12+6-1=17\) and \(P(5)=12+15-6.25=20.75\) thousand people; these values answer a different question from the comparison of rates.
Negative Rates Require Careful Ordering
When both derivative values are negative, the quantity is decreasing at both inputs. But the greater signed rate is the one closer to zero. For example, \(-1\) is greater than \(-4\), even though \(4\) is greater than \(1\). The rate with the larger magnitude describes a faster decrease; the greater signed rate describes the larger number on the number line. Those are related but different comparisons.
Worked Example: A Decrease That Becomes Slower
The amount \(L(t)\) of liquid in a container is measured in liters, with \(t\) in hours. Suppose \(L(t)=20-3t+0.25t^2\) for \(0\leq t\leq 6\). Compare its rates at \(t=2\) and \(t=4\).
Differentiate:
Evaluate the derivative at each time:
Both rates are negative, so the amount is decreasing at both times. Since \(-1>-2\), the rate at hour \(4\) is greater as a signed number. The amount is decreasing more slowly at hour \(4\): its rate is \(-1\) liter per hour rather than \(-2\) liters per hour. The rate difference is:
The rate value has increased by \(1\) liter per hour between these two inputs, even though the amount is still decreasing at each one. Check the function values: \(L(2)=20-6+1=15\) liters and \(L(4)=20-12+4=12\) liters. These confirm that the modeled amount is positive at both inputs; the derivative values, not these function values, give the instantaneous rates being compared.
Comparing Rates Supplied in a Table
Sometimes a problem gives derivative values directly rather than asking you to calculate them from a formula. Read the input labels and rate units carefully, then compare the signed values. If the table gives rates only at selected inputs, do not assume that the rate followed a steady pattern between them.
Worked Example: Comparing Rates from a Table
A table lists the instantaneous rates of change of a town’s population \(N(t)\), measured in people per year. Compare the rates at years \(2\), \(5\), and \(8\).
| Input \(t\) (years) | Rate \(N'(t)\) (people per year) |
|---|---|
| 2 | 120 |
| 5 | 85 |
| 8 | -20 |
From year \(2\) to year \(5\), the difference in rate values is:
The rate at year \(5\) is \(35\) people per year smaller than the rate at year \(2\). Both rates are positive, so the population is increasing at both listed inputs.
From year \(5\) to year \(8\):
The rate at year \(8\) is \(105\) people per year smaller than the rate at year \(5\). It is negative, so the population is decreasing at year \(8\). Overall, the difference between the rates at years \(2\) and \(8\) is:
The rate at year \(8\) is \(140\) people per year smaller than the rate at year \(2\). These comparisons follow directly from the table values. They do not establish exactly when the population’s rate changed sign, or how the rate behaved between the listed years.
Common Mistake: Comparing Magnitudes Instead of Rates
Suppose a quantity has rates \(-4\) units per hour at one input and \(-1\) unit per hour at another. A mistaken comparison might say, “\(-4\) is greater because its magnitude is greater.” That confuses the size of the rate with its signed value.
So the rate \(-1\) is greater as a signed number, while the rate \(-4\) has greater magnitude and describes the faster decrease. State which comparison the context calls for. If the question asks which derivative value is greater, compare signed values. If it asks which decrease is faster, compare magnitudes after confirming both rates are negative.
What Two-Input Comparisons Do—and Do Not—Show
Comparing rates at two inputs gives a precise conclusion about those inputs. It can show that the rate is greater or smaller at one than at the other, and by how much. It does not automatically describe every input between them. A function’s rate could vary in between and still have the same two endpoint derivative values.
For a careful interpretation, keep three questions separate:
- What is the derivative value at each input?
- Which signed derivative value is greater, and what is their difference?
- What does the sign of each derivative say about whether the quantity is increasing or decreasing at that input?
Answering these in order helps prevent a common mix-up: a larger derivative value does not necessarily mean a larger quantity, and two rate values alone do not describe the entire interval between their inputs.
Check Your Understanding
For each question, compare the derivative values as signed rates before interpreting them in context.
- A quantity has derivative values \(6\) and \(2\) units per hour at two inputs. Which rate is greater, and by how much?
- A quantity has rates \(-3\) and \(-7\) liters per minute at two times. Which signed rate is greater? At which time is the quantity decreasing faster?
- A model gives \(f'(1)=4\) and \(f'(6)=-2\). What does each sign say about the quantity at those inputs, and what is \(f'(6)-f'(1)\)?
- A table gives rates at inputs \(2\) and \(9\), with the later-input rate greater. Can you conclude that the rate increased at every input between \(2\) and \(9\)? Explain.
- Why does comparing \(f'(a)\) and \(f'(b)\) not by itself determine which of \(f(a)\) and \(f(b)\) is larger?
