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Differentiability Practice Problems

Check continuity first, then compare the slopes from the left and the right. A function is differentiable at a point only when both one-sided derivatives exist and agree.

  1. 1.

    For , find the right-hand derivative at .

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    For , , so the right-hand derivative is 1.

  2. 2.

    For , find the left-hand derivative at .

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    For , , so the left-hand derivative is .

  3. 3.

    Is differentiable at ? Type yes or no.

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    The one-sided derivatives are and . They differ, so the graph has a corner and there is no derivative at 4.

  4. 4.

    Is differentiable at every real number? Type yes or no.

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    Every polynomial is differentiable everywhere. Its graph is smooth and unbroken, with no corners or vertical tangents.

  5. 5.

    Let for and for . Find .

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    Continuous at 1 (both pieces give 1). Left slope , right slope 2. They agree, so .

  6. 6.

    Let for and for . Is differentiable at ? Type yes or no.

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    It is continuous: both pieces give 2 at . But the left slope is and the right slope is 3, so there is a corner.

  7. 7.

    For , evaluate . Type infinity or -infinity if it grows without bound.

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    , and is positive on both sides, so the quotient grows to . That is a vertical tangent, so does not exist.

  8. 8.

    At which values of is not differentiable? List them separated by commas.

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    The inside is zero at , where it changes sign with a nonzero slope, so the absolute value folds the graph into a corner at each. Everywhere else is a polynomial (or its negative).

  9. 9.

    Find so that for and for is differentiable at .

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    Heights: . Slopes: , so . Both conditions agree on .

  10. 10.

    Let for and for . If is differentiable everywhere, find .

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    The slopes must match at : the left slope is , so .

  11. 11.

    For the same function as the previous problem, find .

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    The heights must match at : , so . The line is the tangent to at .

  12. 12.

    Find for , or type DNE if it does not exist.

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    from both sides. The limit exists, so . The extra factor of smooths out the corner of .
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Differentiability Practice Problems

These problems follow the differentiability lesson. They mix one-sided derivatives, graph features and piecewise functions.

The three checks

  1. Continuous at ? If not, the function is not differentiable there.
  2. Left and right slopes equal? Compute the one-sided derivatives.
  3. Finite? A slope of is a vertical tangent, not a derivative.

Common mistakes to watch for

  • Thinking continuous means differentiable. at 0 is continuous with no derivative.
  • Matching slopes but not heights in piecewise problems. You need both equations.
  • Treating a vertical tangent as differentiable. Its slope is infinite.
  • Assuming every absolute value makes a corner. is differentiable at 0.

Frequently Asked Questions

Do I need the limit definition for every problem?+

No. For piecewise functions made of polynomials, compare the derivatives of the two pieces at the break point, after checking that the function is continuous there. Use the limit definition for absolute values and roots at zero.

What is the difference between continuous and differentiable?+

Continuous means the heights match up at a point; differentiable means the heights and the slopes both match. Every differentiable function is continuous, but corners, cusps and vertical tangents are continuous without being differentiable.

More practice problems

Limit Definition of the Derivative Practice Problems Continuity Practice Problems One-Sided Limits Practice Problems