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Math Practice

Product Rule Practice Problems

For a product f times g, the derivative is f′g + fg′. Name the two factors, differentiate each, and combine. Every problem has a full worked solution.

  1. 1.

    Use the product rule to differentiate .

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    , which matches the power rule for .

  2. 2.

    Differentiate .

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    . (Check: the product is .)

  3. 3.

    Differentiate .

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    .

  4. 4.

    Differentiate and simplify.

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    . The terms cancel, because the product is just .

  5. 5.

    For , find the slope at .

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    . At that is .

  6. 6.

    Differentiate . Type square roots as sqrt( ).

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    .

  7. 7.

    Differentiate . Type sin and cos as sin(x) and cos(x).

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    With : .

  8. 8.

    If , , and , find .

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    . Match each derivative with the OTHER function's value.

  9. 9.

    If , , and , find .

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    .

  10. 10.

    Differentiate using the three-factor product rule.

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    .

  11. 11.

    Find the slope of the tangent line to at .

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    . At : .

  12. 12.

    At which x-values does have a horizontal tangent line? List them separated by commas.

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    Write as if you haven't met the chain rule yet; its derivative is . Then , which is zero at and .
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MPhil in Mathematics, Founder of Mathovia
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Product Rule Practice Problems

These problems follow the product rule for derivatives lesson. They build from simple products to tables of values, three factors and tangent lines.

The rule

  1. Name the factors and .
  2. Differentiate each: and .
  3. Combine: . For three factors, .
  4. Simplify if it helps, or factor to find where the slope is zero.

Common mistakes to watch for

  • Multiplying the derivatives. .
  • Dropping a term. Each factor takes its turn being differentiated.
  • Missing brackets around a factor with more than one term.
  • Using a minus sign. The product rule adds; only the quotient rule subtracts.

Frequently Asked Questions

Do I have to simplify my answer?+

No. Any form that equals the correct derivative is accepted, so 3x^2 + 2x^2 and 5x^2 are both marked right. Simplifying is still good practice.

How do I type sin, e^x and square roots?+

Use sin(x), cos(x), e^x (or exp(x)) and sqrt(x). Use * for multiplication if a term could be ambiguous, such as 3x^2*sin(x).

More practice problems

Power Rule Practice Problems Limit Definition of the Derivative Practice Problems Differentiability Practice Problems