Navigation: Wiki Home > Skills > Interpreting Rates of Change with Units
Velocity isn't the only rate of change we care about. In the real world, we constantly encounter questions like:
All of these are rates of change, and all of them are derivatives! The key is learning to interpret what the derivative means in context and to track units so your answers make physical sense.
| Property | Value |
|---|---|
| Section | Stewart 2.1 |
| Course | MATH161 |
| Difficulty | Intermediate |
| Time | ~15 minutes |
If $y = f(x)$, then the instantaneous rate of change of $y$ with respect to $x$ at $x = a$ is:
$$\boxed{f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}}$$
This is the same definition as before - but now we interpret it more broadly than just "slope" or "velocity."
The units of $f'(a)$ are always:
$$\boxed{\text{units of } f'(a) = \frac{\text{units of } y}{\text{units of } x}}$$
This is because the derivative is the limit of $\frac{\Delta y}{\Delta x}$.
| If $y$ is measured in... | And $x$ is measured in... | Then $f'(x)$ has units... |
|---|---|---|
| meters | seconds | meters/second (velocity) |
| dollars | items | dollars/item (marginal cost) |
| bacteria | hours | bacteria/hour (growth rate) |
| degrees | minutes | degrees/minute (cooling rate) |
The statement "$f'(a) = k$" means:
"When $x = a$, the quantity $y$ is changing at a rate of $k$ [units of y] per [unit of x]."
Important: A positive derivative means $y$ is increasing; a negative derivative means $y$ is decreasing.
If $C(x)$ = cost (in dollars) to produce $x$ items, then:
$$C'(x) = \text{marginal cost at production level } x$$
Interpretation: $C'(100) = 12$ means "when you're already producing 100 items, producing one more item costs approximately $12."
The marginal cost is the rate of change of cost with respect to quantity - it tells you how expensive it is to increase production right now.
| Quantity | Formula | Interpretation |
|---|---|---|
| Average rate of change | $\frac{f(b) - f(a)}{b - a}$ | Overall change divided by interval length |
| Instantaneous rate of change | $f'(a)$ | Rate at the exact moment $x = a$ |
The instantaneous rate is what the average rate approaches as the interval shrinks to zero.
The population $P$ of a city (in thousands) is a function of time $t$ (in years since 2020).
What are the units of $P'(t)$? What does $P'(5) = 3.2$ mean in context?
The amount of a radioactive substance (in grams) remaining after $t$ days is given by $A(t)$.
If $A'(10) = -0.5$, what does this tell you about the substance?
The temperature $T$ (in °C) of a cup of coffee $t$ minutes after being poured is modeled by $T(t) = 20 + 60e^{-0.1t}$.
The table shows some values:
| $t$ | 0 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| $T(t)$ | 80 | 56.5 | 42.1 | 33.4 | 28.1 |
(a) Estimate $T'(10)$ using the average rate of change over $[5, 15]$.
(b) Interpret your answer in context.
A company's cost function is $C(x) = 1000 + 8x + 0.01x^2$ dollars to produce $x$ units.
(a) Use the limit definition to find $C'(x)$.
(b) Find the marginal cost when $x = 200$ and interpret it.
(c) What is the actual cost of producing the 201st unit? Compare with your answer in (b).
The national debt $D(t)$ (in trillions of dollars) at the end of year $t$ is given in the table:
| Year $t$ | 2016 | 2018 | 2020 | 2022 | 2024 |
|---|---|---|---|---|---|
| $D(t)$ | 19.6 | 21.5 | 27.0 | 30.9 | 34.2 |
(a) Estimate $D'(2020)$ using the symmetric difference quotient (average of rates from $[2018, 2020]$ and $[2020, 2022]$).
(b) Estimate $D'(2020)$ using the average rate over $[2018, 2022]$.
(c) Which estimate do you think is more accurate? Explain.
(d) Interpret your estimate in context.
The graph shows the amount $A$ of water (gallons) in a tank over time $t$ (hours).
A (gallons)
│ ●───────────●
│ / \
│ / \
│ / \
│ ● ●
│
└─────────────────────── t (hours)
0 1 2 3 4 5At which time is the rate of change of water $A'(t)$ greatest (most positive)?
(A) $t = 0$ (B) $t = 1.5$ (C) $t = 3$ (D) $t = 4.5$
The "Per" Relationship: Whenever you see "per" in real life, you're looking at a rate of change:
The derivative gives you the instantaneous version of any "per" relationship.
Looking back:
Looking ahead:
| Previous | Up | Next |
|---|---|---|
| Instantaneous Velocity | Skills Index | The Derivative as a Function |
Last updated: 2026-01-22