Worksheets · Calculus

Instantaneous rate of change worksheet

A derivative at a point is a rate: how fast the output moves when the input moves by one. Outside a math class that is what it is always called, and marginal cost is exactly this.

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Name

Instantaneous Rate of Change

Date Period

Answer each question.

  1. Find rates of change in business

    If P(x)=x3+4x2+2x+258 is the profit in dollars from selling x units, find the marginal profit at x=8.
  2. If P(x)=x3+x2+17x+270 is the profit in dollars from selling x units, find the marginal profit at x=10.
  3. If C(x)=x3−4x2+17x+213 is the cost in dollars of making x units, find the marginal cost at x=3.
  4. If P(x)=x3−x2+2x+351 is the profit in dollars from selling x units, find the marginal profit at x=9.
  5. If C(x)=2x3−x2+20x+213 is the cost in dollars of making x units, find the marginal cost at x=10.

  1. Find and interpret a rate of change

    A population is P(t)=900e0.03t people after t years. Find P′(7) to the nearest hundredth and say what it means.
  2. The temperature is T(t)=72+14e−0.1t degrees after t minutes. Find T′(5) to the nearest hundredth and say what it means.
  3. A tank holds V(t)=316−3t2 gallons after t minutes. Find V′(6) and say what it means.
  4. The temperature is T(t)=64+36e−0.05t degrees after t minutes. Find T′(9) to the nearest hundredth and say what it means.
  5. A population is P(t)=600e0.05t people after t years. Find P′(6) to the nearest hundredth and say what it means.

  1. Compare average and instantaneous rates

    For f(x)=−x3−2x2−3x−2, find the average rate of change on [−1,2] and the instantaneous rate of change at x=−1.
  2. For f(x)=−x3+x2+2x−7, find the average rate of change on [−3,−1] and the instantaneous rate of change at x=−3.
  3. For f(x)=−x2−x−3, find the average rate of change on [1,3] and the instantaneous rate of change at x=1.
  4. For f(x)=2x3−x2+x+2, find the average rate of change on [1,4] and the instantaneous rate of change at x=1.
  5. For f(x)=x3+x2−2, find the average rate of change on [0,2] and the instantaneous rate of change at x=0.

  1. Find the secant and tangent lines

    For f(x)=2x3−x2−5x−6, find the secant line through x=−2 and x=−1, and the tangent line at x=−2.
  2. For f(x)=−x3+2x2+5x−5, find the secant line through x=1 and x=4, and the tangent line at x=1.
  3. For f(x)=x3−2x2, find the secant line through x=2 and x=5, and the tangent line at x=2.
  4. For f(x)=−3x2+2x+5, find the secant line through x=−1 and x=2, and the tangent line at x=−1.
  5. For f(x)=2x3−x2+4x+5, find the secant line through x=2 and x=5, and the tangent line at x=2.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Instantaneous Rate of ChangeAnswer keyVersion 1

Date Period

Answer each question.

  1. Find rates of change in business

    If P(x)=x3+4x2+2x+258 is the profit in dollars from selling x units, find the marginal profit at x=8.258 dollars per unit
  2. If P(x)=x3+x2+17x+270 is the profit in dollars from selling x units, find the marginal profit at x=10.337 dollars per unit
  3. If C(x)=x3−4x2+17x+213 is the cost in dollars of making x units, find the marginal cost at x=3.20 dollars per unit
  4. If P(x)=x3−x2+2x+351 is the profit in dollars from selling x units, find the marginal profit at x=9.227 dollars per unit
  5. If C(x)=2x3−x2+20x+213 is the cost in dollars of making x units, find the marginal cost at x=10.600 dollars per unit

  1. Find and interpret a rate of change

    A population is P(t)=900e0.03t people after t years. Find P′(7) to the nearest hundredth and say what it means.P′(7)≈33.31: at t=7 the population is growing by about 33.31 people per year
  2. The temperature is T(t)=72+14e−0.1t degrees after t minutes. Find T′(5) to the nearest hundredth and say what it means.T′(5)≈−0.85: at t=5 the temperature is falling about 0.85 degrees per minute
  3. A tank holds V(t)=316−3t2 gallons after t minutes. Find V′(6) and say what it means.V′(6)=−36: at t=6 the tank is losing 36 gallons per minute
  4. The temperature is T(t)=64+36e−0.05t degrees after t minutes. Find T′(9) to the nearest hundredth and say what it means.T′(9)≈−1.15: at t=9 the temperature is falling about 1.15 degrees per minute
  5. A population is P(t)=600e0.05t people after t years. Find P′(6) to the nearest hundredth and say what it means.P′(6)≈40.50: at t=6 the population is growing by about 40.50 people per year

  1. Compare average and instantaneous rates

    For f(x)=−x3−2x2−3x−2, find the average rate of change on [−1,2] and the instantaneous rate of change at x=−1.Average: −8Instantaneous: −2
  2. For f(x)=−x3+x2+2x−7, find the average rate of change on [−3,−1] and the instantaneous rate of change at x=−3.Average: −15Instantaneous: −31
  3. For f(x)=−x2−x−3, find the average rate of change on [1,3] and the instantaneous rate of change at x=1.Average: −5Instantaneous: −3
  4. For f(x)=2x3−x2+x+2, find the average rate of change on [1,4] and the instantaneous rate of change at x=1.Average: 38Instantaneous: 5
  5. For f(x)=x3+x2−2, find the average rate of change on [0,2] and the instantaneous rate of change at x=0.Average: 6Instantaneous: 0

  1. Find the secant and tangent lines

    For f(x)=2x3−x2−5x−6, find the secant line through x=−2 and x=−1, and the tangent line at x=−2.Secant: y=12x+8Tangent: y=23x+30
  2. For f(x)=−x3+2x2+5x−5, find the secant line through x=1 and x=4, and the tangent line at x=1.Secant: y=−6x+7Tangent: y=6x−5
  3. For f(x)=x3−2x2, find the secant line through x=2 and x=5, and the tangent line at x=2.Secant: y=25x−50Tangent: y=4x−8
  4. For f(x)=−3x2+2x+5, find the secant line through x=−1 and x=2, and the tangent line at x=−1.Secant: y=−x−1Tangent: y=8x+8
  5. For f(x)=2x3−x2+4x+5, find the secant line through x=2 and x=5, and the tangent line at x=2.Secant: y=75x−125Tangent: y=24x−23

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

If C(x)=2x2+5x+100 is the cost of making x units, find the marginal cost at x=10

  1. The derivative is 4x plus 5.
  2. At x equals 10 that is 45.
  3. So the next unit costs about 45 dollars.

The answer is 45 dollars per unit.

Where students go wrong

Working out the cost instead of the rate. Marginal cost is the derivative, not the value: C of 10 is what ten units cost altogether, which is a much bigger number and answers a different question.

Also called marginal cost, marginal revenue, marginal profit or rate of change at a point.

What you can put on this worksheet

Instantaneous rate of change
If R(x)=x2+18x+223 is the revenue in dollars from selling x units, find the marginal revenue at x=7. → 32 dollars per unit
In context, with units
A population is P(t)=600e0.05t people after t years. Find P′(3) to the nearest hundredth and say what it means. → P′(3)≈34.86: at t=3 the population is growing by about 34.86 people per year
Average rate against instantaneous rate
For f(x)=−2x2+x+1, find the average rate of change on [−2,1] and the instantaneous rate of change at x=−2. → Average: 3Instantaneous: 9
The secant line and the tangent line
For f(x)=−2x2+x+1, find the secant line through x=−2 and x=1, and the tangent line at x=−2. → Secant: y=3x−3Tangent: y=9x+9

Questions about these worksheets

Yes. Every instantaneous rate of change sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the instantaneous rate of change questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: instantaneous rate of change, in context, with units, average rate against instantaneous rate and the secant line and the tangent line. Tick as many as you want and set how many of each, or let it spread them evenly.

Working out the cost instead of the rate. Marginal cost is the derivative, not the value: C of 10 is what ten units cost altogether, which is a much bigger number and answers a different question.