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 maths lesson that is what it is always called, and marginal cost is exactly this.

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Sigma Prep · sigmaprep.io/worksheets

Name

Instantaneous Rate of Change

Date Period

Answer each question.

  1. 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 C(x)=−x3−2x2+30x+182 is the cost in dollars of making x units, find the marginal cost at x=4
  3. If R(x)=−x3−4x2+3x+206 is the revenue in dollars from selling x units, find the marginal revenue at x=2
  4. If C(x)=2x3−x2+25x+244 is the cost in dollars of making x units, find the marginal cost at x=10
  5. If C(x)=−2x3+4x2+15x+294 is the cost in dollars of making x units, find the marginal cost at x=3
  6. If C(x)=x3−3x2+9x+177 is the cost in dollars of making x units, find the marginal cost at x=4
  7. If P(x)=x3−4x2+27x+257 is the profit in dollars from selling x units, find the marginal profit at x=5
  8. If C(x)=2x3+4x2+28x+382 is the cost in dollars of making x units, find the marginal cost at x=3
  9. If P(x)=2x3+3x2+10x+350 is the profit in dollars from selling x units, find the marginal profit at x=3
  10. If C(x)=−x3+x2+5x+381 is the cost in dollars of making x units, find the marginal cost at x=6
  11. If C(x)=2x3+x2+16x+270 is the cost in dollars of making x units, find the marginal cost at x=5
  12. If C(x)=2x3−x2+27x+135 is the cost in dollars of making x units, find the marginal cost at x=5
  13. If R(x)=x3−3x2+3x+73 is the revenue in dollars from selling x units, find the marginal revenue at x=7
  14. If C(x)=2x3−4x2+24x+100 is the cost in dollars of making x units, find the marginal cost at x=2
  15. If P(x)=x3−3x2+30x+30 is the profit in dollars from selling x units, find the marginal profit at x=4
  16. If C(x)=−x3−x2+24x+322 is the cost in dollars of making x units, find the marginal cost at x=7
  17. If R(x)=2x3+4x2+11x+357 is the revenue in dollars from selling x units, find the marginal revenue at x=4
  18. If C(x)=x3−4x2+7x+137 is the cost in dollars of making x units, find the marginal cost at x=5
  19. If P(x)=−2x3−x2+24x+33 is the profit in dollars from selling x units, find the marginal profit at x=5
  20. If P(x)=−2x3+2x2+14x+220 is the profit in dollars from selling x units, find the marginal profit at x=10

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. If P(x)=x3+4x2+2x+258 is the profit in dollars from selling x units, find the marginal profit at x=8258 dollars per unit
  2. If C(x)=−x3−2x2+30x+182 is the cost in dollars of making x units, find the marginal cost at x=4−34 dollars per unit
  3. If R(x)=−x3−4x2+3x+206 is the revenue in dollars from selling x units, find the marginal revenue at x=2−25 dollars per unit
  4. If C(x)=2x3−x2+25x+244 is the cost in dollars of making x units, find the marginal cost at x=10605 dollars per unit
  5. If C(x)=−2x3+4x2+15x+294 is the cost in dollars of making x units, find the marginal cost at x=3−15 dollars per unit
  6. If C(x)=x3−3x2+9x+177 is the cost in dollars of making x units, find the marginal cost at x=433 dollars per unit
  7. If P(x)=x3−4x2+27x+257 is the profit in dollars from selling x units, find the marginal profit at x=562 dollars per unit
  8. If C(x)=2x3+4x2+28x+382 is the cost in dollars of making x units, find the marginal cost at x=3106 dollars per unit
  9. If P(x)=2x3+3x2+10x+350 is the profit in dollars from selling x units, find the marginal profit at x=382 dollars per unit
  10. If C(x)=−x3+x2+5x+381 is the cost in dollars of making x units, find the marginal cost at x=6−91 dollars per unit
  11. If C(x)=2x3+x2+16x+270 is the cost in dollars of making x units, find the marginal cost at x=5176 dollars per unit
  12. If C(x)=2x3−x2+27x+135 is the cost in dollars of making x units, find the marginal cost at x=5167 dollars per unit
  13. If R(x)=x3−3x2+3x+73 is the revenue in dollars from selling x units, find the marginal revenue at x=7108 dollars per unit
  14. If C(x)=2x3−4x2+24x+100 is the cost in dollars of making x units, find the marginal cost at x=232 dollars per unit
  15. If P(x)=x3−3x2+30x+30 is the profit in dollars from selling x units, find the marginal profit at x=454 dollars per unit
  16. If C(x)=−x3−x2+24x+322 is the cost in dollars of making x units, find the marginal cost at x=7−137 dollars per unit
  17. If R(x)=2x3+4x2+11x+357 is the revenue in dollars from selling x units, find the marginal revenue at x=4139 dollars per unit
  18. If C(x)=x3−4x2+7x+137 is the cost in dollars of making x units, find the marginal cost at x=542 dollars per unit
  19. If P(x)=−2x3−x2+24x+33 is the profit in dollars from selling x units, find the marginal profit at x=5−136 dollars per unit
  20. If P(x)=−2x3+2x2+14x+220 is the profit in dollars from selling x units, find the marginal profit at x=10−546 dollars per unit

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.

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.

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.