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Grade 12 · Math · Calculus

12th grade Product and quotient rules worksheet

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Product and quotient rules

Grade 12 Math Worksheet

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  1. Ari uses the quotient rule on h(x) = (x² + 5)/(x + 1). What is h′(2)?

  2. Bo uses the quotient rule on h(x) = (x² − 3)/(x + 2). What is h′(4)?

  3. Esme uses the quotient rule on h(x) = (x² + 7)/(x − 1). What is h′(4)?

  4. Jun uses the quotient rule on h(x) = (x² + 1)/(x + 3). What is h′(-1)?

  5. Kira uses the product rule on h(x) = (2x + 1)√x. Which is h′(x)?

    • h′(x) = 2√x − (2x + 1)/(2√x)
    • h′(x) = 2√x + (2x + 1)/(2√x)
    • h′(x) = 1/√x
    • h′(x) = 2√x

  6. Luz uses the product rule on h(x) = (x² − 3)√x. Which is h′(x)?

    • h′(x) = 2x√x + (x² − 3)/(2√x)
    • h′(x) = 2x√x − (x² − 3)/(2√x)
    • h′(x) = x/√x
    • h′(x) = 2x√x

  7. Read this first

    Values now (example data)

    Price p = $2 per cup, p′ = $0.10 per week; sales q = 150 cups a week, q′ = −4 per week.

  8. Weekly revenue is R = pq, price times weekly sales, with the values shown. Pia finds how fast R changes. What is R′, in dollars per week?

  9. Values now (example data)

    Price p = $3 per bag, p′ = −$0.20 per week; sales q = 80 bags a week, q′ = 6 per week.

  10. Weekly revenue is R = pq, price times weekly sales, with the values shown. Ren finds how fast R changes. What is R′, in dollars per week?

  11. Values now (example data)

    Length L = 12 cm, L′ = 0.5 cm per s; width W = 8 cm, W′ = −0.25 cm per s.

  12. A rectangle's area is A = LW, with the values shown. Tala finds how fast the area changes. What is A′, in cm² per second?

  13. Values now (example data)

    Length L = 20 cm, L′ = −0.4 cm per s; width W = 15 cm, W′ = 0.6 cm per s.

  14. A rectangle's area is A = LW, with the values shown. Zee finds how fast the area changes. What is A′, in cm² per second?

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12th grade Product and quotient rules worksheet – printable
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About this worksheet

The product and quotient rules. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can use the product rule from a table of values or from formulas.
  • I can use the quotient rule in the right order, over g².
  • I can use these rules for rates in context and for three factors.

Standard: none (beyond CCSS)

Words to know

product rule: (fg)′ = f′g + fg′.

quotient rule: (f/g)′ = (f′g − fg′)/g².

revenue: Price times number sold.

Common mistakes

Multiplied the two derivatives. The product rule is f′g + fg′: one factor changes at a time.

Used only one part of the product rule. Each factor's change adds its own term.

Subtracted the quotient rule's top in the wrong order, so the sign flipped. The top is f′g − fg′.

Questions parents ask

What is the most common mistake with product and quotient rules?

Multiplied the two derivatives. The product rule is f′g + fg′: one factor changes at a time.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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