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Limit Definition of the Derivative Practice Problems

Each problem asks you to use the definition of the derivative, not the shortcut rules. Write f(x + h), subtract f(x), divide by h, and only then let h go to 0.

  1. 1.

    Use the limit definition to find for .

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    . A line's derivative is its slope.

  2. 2.

    Use the limit definition to find for .

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    .

  3. 3.

    Use the limit definition to find for .

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    . Divide by to get , which tends to . The +1 cancels, so shifting a graph up doesn't change its slopes.

  4. 4.

    Use the limit definition to find for the constant function .

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    for every , so the limit is 0. A horizontal line has slope 0.

  5. 5.

    Use the limit definition to find for .

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    . Subtracting leaves . Divide by : .

  6. 6.

    Use the limit definition to find for .

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    . Divide by : .

  7. 7.

    Use the limit definition to find for .

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    . Divide by : .

  8. 8.

    Does exist for ? Give its value, or type DNE.

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    The difference quotient is : it equals for and for . The one-sided limits disagree, so the derivative at the corner does not exist.

  9. 9.

    Use the alternate definition to find for .

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    , so the quotient is . As , this tends to .

  10. 10.

    Use the limit definition to find for .

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    Multiply by the conjugate: .

  11. 11.

    Use the limit definition to find for . Type square roots as sqrt( ).

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    Multiply by the conjugate : the top becomes , which cancels, leaving .

  12. 12.

    A stone dropped from a bridge falls feet in seconds. Use the limit definition to find its velocity, in feet per second, at .

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    . After 3 seconds the stone is falling at 96 feet per second.
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Limit Definition of the Derivative Practice Problems

These problems follow the limit definition of the derivative lesson. Some ask for a number, the derivative at a point, and some ask for the whole derivative function.

The four steps

  1. Write , replacing every with .
  2. Subtract . Every term without an should cancel.
  3. Divide by . Combine fractions first; multiply square roots by the conjugate.
  4. Let once the in the denominator is gone.

Common mistakes to watch for

  • Writing for . For it’s , not .
  • Dropping brackets when subtracting . Subtract the whole function, every term.
  • Substituting too soon. That gives ; cancel the first.
  • Answering with a function when the question asks for . Plug in the number at the end.

Frequently Asked Questions

Can I use the power rule on these?+

You can use it to check an answer, but each problem asks for the limit definition, which is what exams test with the wording "use the definition of the derivative". Work the limit, then compare with the shortcut.

How do I type square roots and fractions?+

Type sqrt(x + 3) for a square root and use a slash for fractions, such as 1/4 or 1/(2*sqrt(x + 3)). Equivalent forms of an expression are accepted.

More practice problems

Evaluating Limits Practice Problems One-Sided Limits Practice Problems L'Hôpital's Rule Practice Problems