Skip to main content
Mathovia

Math Practice

Quotient Rule Practice Problems

For f over g, the derivative is (f′g − fg′) / g²: low d-high minus high d-low, over low squared. Start with the derivative of the top, and keep the denominator squared.

  1. 1.

    Differentiate with the quotient rule, then check with the power rule.

    Show full solution

    . The power rule on gives , the same answer.

  2. 2.

    Differentiate .

    Show full solution

    .

  3. 3.

    Differentiate .

    Show full solution

    .

  4. 4.

    Differentiate .

    Show full solution

    .

  5. 5.

    Differentiate .

    Show full solution

    .

  6. 6.

    For , find the slope at .

    Show full solution

    , which is at . The graph has a horizontal tangent there.

  7. 7.

    Differentiate (for ). Type square roots as sqrt( ).

    Show full solution

    Quickest is to simplify first: , so . The quotient rule gives the same: .

  8. 8.

    If , , and , find .

    Show full solution

    .

  9. 9.

    Use the quotient rule to differentiate . Type sin and cos as sin(x) and cos(x).

    Show full solution

    .

  10. 10.

    Find the slope of the tangent line to at .

    Show full solution

    . At : .

  11. 11.

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

    Show full solution

    . The numerator is zero at . (Splitting first, , gives , the same thing.)

  12. 12.

    Differentiate .

    Show full solution

    . The derivative of the bottom is , and the two minus signs make a plus.
Written by
MPhil in Mathematics, Founder of Mathovia
Published

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Quotient Rule Practice Problems

These problems follow the quotient rule for derivatives lesson. They range from simple rational functions to tables of values, trig quotients and turning points.

The rule

  1. Name the top and the bottom , and differentiate each.
  2. Assemble : low d-high minus high d-low, over low squared.
  3. Simplify the numerator and leave the denominator squared.
  4. Check for a shortcut first: a constant top or bottom, or a single-term bottom, is quicker with the power rule.

Common mistakes to watch for

  • Swapping the order in the numerator, which flips the sign. Test on .
  • Forgetting to square the denominator.
  • Losing a minus sign when the derivative of the bottom is negative.
  • Differentiating the top and bottom separately. That’s L’Hôpital’s rule, not the quotient rule.

Frequently Asked Questions

Can I give my answer in a different form?+

Yes. Any expression equal to the correct derivative is accepted, so -2/(x-2)^2 and -2(x-2)^(-2) are both marked right. You don't need to expand the denominator.

How do I type fractions, roots and trig functions?+

Use / for fractions with brackets around the top and bottom, for example (x^2 + 6x)/(x + 3)^2. Use sqrt(x), sin(x) and cos(x), and * between factors if a term could be ambiguous.

More practice problems

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