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Higher-Order Derivatives Practice Problems

Differentiate once, simplify, then differentiate again. Keep going for third and higher derivatives, and give your answer in terms of x.

  1. 1.

    Find for .

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    , so .

  2. 2.

    Find for .

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    , so .

  3. 3.

    Find the third derivative of .

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    , , .

  4. 4.

    Find for . Type sin(x) and cos(x).

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    , so .

  5. 5.

    Find for . Put a longer exponent in brackets: e^(5x).

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    Each derivative brings out a factor of 5: and .

  6. 6.

    Find for .

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    Write . Then and .

  7. 7.

    Find for .

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    The chain rule gives , then .

  8. 8.

    Find for .

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    Product rule: . Again: .

  9. 9.

    Find for . Type sec(x)^2 for sec squared.

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    . The chain rule gives .

  10. 10.

    Find for .

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    . The quotient rule gives .

  11. 11.

    A particle has position for . Find its acceleration at the moment its velocity is zero.

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    , so at (the root is outside ). , so .

  12. 12.

    Let . Find , the 27th derivative at .

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    The derivatives of repeat every 4. Since , , and .
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Higher-Order Derivatives Practice Problems

These problems follow the higher-order derivatives lesson. They start with second derivatives of polynomials and build up to products, chain-rule functions, acceleration and a high-order trig derivative.

The method

  1. Find the first derivative with the usual rules.
  2. Simplify it: combine terms and factor out common pieces like .
  3. Differentiate the simplified result to get the second derivative.
  4. Repeat for the third derivative and beyond.

Common mistakes to watch for

  • Squaring the first derivative instead of differentiating it again.
  • Dropping the chain-rule factor on the second pass, for example writing instead of .
  • Forgetting the product rule when the first derivative is still a product.

Frequently Asked Questions

How should I type my answer?+

Write it in terms of x. Use ^ for powers, put longer exponents in brackets like e^(2x), and type functions with brackets: sin(x), ln(x), sec(x)^2.

Does my answer have to be simplified?+

No. Any form that is equal to the correct derivative is accepted, so 4(x + 1)e^(2x) and (4x + 4)e^(2x) both count.

More practice problems

Power Rule Practice Problems Chain Rule Practice Problems Product Rule Practice Problems