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Calculus practice

Product Rule worksheets

Differentiate products of two nonzero polynomials with selectable factor size and degree ceilings, then combine like terms exactly.

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Differentiate each product and combine like terms.

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Paper sizes and printing help · More calculus activities

Practice with exact answers

Keep fractions and rational powers exact. Differentiate the complete original function, then simplify without dropping its domain restrictions.

Learning objective

What this activity builds

Apply the product rule and normalize the resulting polynomial derivative.

Worked example

Differentiate y = (x^2 + 1)(3x - 2).

dy/dx = 9x^2 - 4x + 3

Use u'v + uv', expand, and combine like powers.

Teaching notes

Make the practice count

  • Differentiate one factor at a time.
  • Keep the unchanged factor beside each derivative.
  • Expand once and combine like powers for the final answer.

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Math SheetProduct RuleDifferentiate each product and combine like terms.1.y =8·x·6·x2+42.y =-3·x2-3·x·7·x4+8·x3-9·x3.y =x3-9·x2+1·-2·x4.y =-1·x4·x3+5·x25.y =8·x5+8·x·7·x5-2·x3+5·x26.y =-3·x-7·2·x27.y =4·x2·6·x4-2·x28.y =3·x3+6·x·4·x+49.y =-5·x4+x3+5·x2·-7·x310.y =-1·x5·5·x5+4·x3math-sheet.comPage 1 of 2
Math SheetProduct RuleDifferentiate each product and combine like terms.11.y =6·x+1·x2-2·x+712.y =-3·x2+4·x-1·-3·x413.y =-3·x3·6·x-714.y =7·x4-2·-5·x3-5·x2+3·x15.y =8·x5-7·x4+4·x2·2·x516.y =x·8·x2-8·x17.y =-1·x2-3·-8·x4-4·x+218.y =7·x3-8·x+5·4·x19.y =7·x4·-6·x3-9·x220.y =-3·x5-2·x·6·x5-2·x4+7·xmath-sheet.comPage 2 of 2

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