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Math Practice

Chain Rule Practice Problems

Differentiate the outside function, keep the inside as it is, then multiply by the derivative of the inside. With more layers, multiply one derivative per layer.

  1. 1.

    Differentiate .

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    . The inner derivative is 1, so the chain rule changes nothing here.

  2. 2.

    Differentiate .

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    .

  3. 3.

    Differentiate .

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    .

  4. 4.

    Differentiate . Type square roots as sqrt( ).

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    .

  5. 5.

    Differentiate using the chain rule.

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    Write : .

  6. 6.

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

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    .

  7. 7.

    Differentiate . Type e^x as e^( ) or exp( ).

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    .

  8. 8.

    Differentiate . Type the natural log as ln( ).

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    .

  9. 9.

    If , , and , find the derivative of at .

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    . The value is a distractor: is evaluated at .

  10. 10.

    Differentiate .

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    Three layers: the square gives , the cosine gives , and gives . Multiplying: .

  11. 11.

    Differentiate and factor your answer.

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    Product rule with the chain rule inside: .

  12. 12.

    Find the slope of the tangent line to at .

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    . At : .
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MPhil in Mathematics, Founder of Mathovia
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Chain Rule Practice Problems

These problems follow the chain rule lesson. They start with powers of simple expressions and build up to three layers, tables of values and products.

The method

  1. Split the function into an outer layer and the inside.
  2. Differentiate the outer layer, keeping the inside unchanged.
  3. Multiply by the derivative of the inside.
  4. Repeat for each extra layer, working from the outside in.

Common mistakes to watch for

  • Forgetting the inner derivative, the most common chain rule error.
  • Changing the inside when differentiating the outer function.
  • Evaluating at instead of with a table of values.
  • Confusing with : which one is the outer layer?

Frequently Asked Questions

How do I type e, ln and trig functions?+

Use e^(-2x) or exp(-2x), ln(3x + 1), sin(3x) and cos(3x). Put * between factors if a term could be ambiguous, such as -4x*sin(x^2)*cos(x^2).

Do I need to factor or simplify?+

Only where a problem asks. Any expression equal to the correct derivative is accepted, expanded or factored.

More practice problems

Quotient Rule Practice Problems Product Rule Practice Problems Power Rule Practice Problems