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Grade 12 · lesson 2 of 4

Apply the Power Rule to a Cubic

For a power x^n, the derivative is n x^(n−1) where the rule applies.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: For f(x)=x³, find f′(2). Candidate answers: (A) 11 (B) 12 (C) 13 (D) 24.
  • Independent check inputs: For f(x)=x³, find f′(3). Candidate answers: (A) 26 (B) 27 (C) 28 (D) 54.

Learn the idea

A derivative measures instantaneous rate through a limiting difference quotient. Differentiation rules summarize that process for suitable functions, while a tangent line uses the derivative slope and the function’s actual point to model local behavior. For a power x^n, the derivative is n x^(n−1) where the rule applies. Multiplying by the exponent and lowering the power produces the cubic’s instantaneous slope at each input.

Here is the method on the supplied teaching case. The power rule gives f′(x)=3x². Substitution x=2 yields 3(2)²=12. The exponent decreases by one; it is not divided by three.

Differentiate before substituting the evaluation input. The power rule changes both coefficient and exponent; simply dividing by the exponent or retaining the original exponent would produce a different expression.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: For f(x)=x³, find f′(2). Candidate answers: (A) 11 (B) 12 (C) 13 (D) 24. Solution: The power rule gives f′(x)=3x². Substitution x=2 yields 3(2)²=12. The exponent decreases by one; it is not divided by three. The final answer is 12. Differentiate before substituting the evaluation input. The power rule changes both coefficient and exponent; simply dividing by the exponent or retaining the original exponent would produce a different expression.

Your turn

Check your reasoning

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Helpful tips
  • For a power x^n, the derivative is n x^(n−1) where the rule applies.
  • Differentiate before substituting the evaluation input.
For f(x)=x³, find f′(3).

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See the answer and explanation

27

The power rule gives f′(x)=3x². Substitution x=3 yields 3(3)²=27. The exponent decreases by one; it is not divided by three.

Take it further

Changed-input task: For f(x)=x³, find f′(9). Candidate answers: (A) 242 (B) 243 (C) 244 (D) 486. Show the mathematical reason, not only the selected answer. Expected result: 243. The power rule gives f′(x)=3x². Substitution x=9 yields 3(9)²=243. The exponent decreases by one; it is not divided by three.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. For f(x)=x³, find f′(4).
Need a hint?
  • For a power x^n, the derivative is n x^(n−1) where the rule applies.
  • Differentiate before substituting the evaluation input.
2. For f(x)=x³, find f′(5).
Need a hint?
  • For a power x^n, the derivative is n x^(n−1) where the rule applies.
  • Differentiate before substituting the evaluation input.
3. For f(x)=x³, find f′(6).
Need a hint?
  • For a power x^n, the derivative is n x^(n−1) where the rule applies.
  • Differentiate before substituting the evaluation input.

Your choices are temporary. Feedback appears after you check all the answers.

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