Worksheets · Calculus

Product rule worksheet

The derivative of a product is not the product of the derivatives. Differentiate the first and keep the second, then keep the first and differentiate the second, and add the two.

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Name

The Product Rule

Date Period

Differentiate each function.

  1. f(x)=(x2+9)cos(4x)
  2. f(x)=(x3)sin(3x)
  3. f(x)=(3x+8)e3x
  4. f(x)=(4x2+3)e3x
  5. f(x)=(x21)cos(4x)
  6. f(x)=(5x2+7)cos(2x)
  7. f(x)=(3x+7)cos(2x)
  8. f(x)=(3x26)sin(4x)
  9. f(x)=(5x22)cos(2x)
  10. f(x)=(4x25)ln(2x+9)
  11. f(x)=(x9)ln(2x8)
  12. f(x)=(2x2)ln(4x9)
  13. f(x)=(4x4)cos(4x)
  14. f(x)=(x+4)cos(2x)
  15. f(x)=(2x25)sin(2x)
  16. f(x)=(3x+6)e2x
  17. f(x)=(x8)ln(3x5)
  18. f(x)=(x7)ln(4x1)
  19. f(x)=(x+2)cos(3x)
  20. f(x)=(4x+1)sin(2x)

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

Sigma Prep · sigmaprep.io/worksheets

Name

The Product RuleAnswer keyVersion 1

Date Period

Differentiate each function.

  1. f(x)=(x2+9)cos(4x)2xcos(4x)+(x2+9)(4sin(4x))
  2. f(x)=(x3)sin(3x)sin(3x)+(x3)(3cos(3x))
  3. f(x)=(3x+8)e3x3e3x+(3x+8)3e3x
  4. f(x)=(4x2+3)e3x8xe3x+(4x2+3)(3e3x)
  5. f(x)=(x21)cos(4x)2xcos(4x)+(x21)4sin(4x)
  6. f(x)=(5x2+7)cos(2x)10xcos(2x)+(5x2+7)(2sin(2x))
  7. f(x)=(3x+7)cos(2x)3cos(2x)+(3x+7)(2sin(2x))
  8. f(x)=(3x26)sin(4x)6xsin(4x)+(3x26)(4cos(4x))
  9. f(x)=(5x22)cos(2x)10xcos(2x)+(5x22)2sin(2x)
  10. f(x)=(4x25)ln(2x+9)8xln(2x+9)+(4x25)22x+9
  11. f(x)=(x9)ln(2x8)ln(2x8)+(x9)22x8
  12. f(x)=(2x2)ln(4x9)2ln(4x9)+(2x2)44x9
  13. f(x)=(4x4)cos(4x)4cos(4x)+(4x4)4sin(4x)
  14. f(x)=(x+4)cos(2x)cos(2x)+(x+4)(2sin(2x))
  15. f(x)=(2x25)sin(2x)4xsin(2x)+(2x25)(2cos(2x))
  16. f(x)=(3x+6)e2x3e2x+(3x+6)2e2x
  17. f(x)=(x8)ln(3x5)ln(3x5)+(x8)33x5
  18. f(x)=(x7)ln(4x1)ln(4x1)+(x7)44x1
  19. f(x)=(x+2)cos(3x)cos(3x)+(x+2)(3sin(3x))
  20. f(x)=(4x+1)sin(2x)4sin(2x)+(4x+1)(2cos(2x))

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

How to do these

f(x)=x2sinx

  1. Differentiate the first: 2x.
  2. Keep the second: sin x.
  3. Then the other way round: x squared times cos x.

The answer is 2xsinx+x2cosx.

Where students go wrong

Multiplying the two derivatives together. That is the one thing the rule exists to stop you doing, and it gives the wrong answer on almost every problem.

Also called derivative of a product or differentiation product rule.

Questions about these worksheets

Yes. Every product rule 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 product rule 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 two 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.

Multiplying the two derivatives together. That is the one thing the rule exists to stop you doing, and it gives the wrong answer on almost every problem.