Differentiate a Product with Both Contributions
The derivative of uv is u′v+uv′.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: For f(x)=x²(x+1), find f′(2). Candidate answers: (A) 15 (B) 16 (C) 17 (D) 32.
- Independent check inputs: For f(x)=x²(x+1), find f′(3). Candidate answers: (A) 32 (B) 33 (C) 34 (D) 66.
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. The derivative of uv is u′v+uv′. Each factor’s change contributes while the other factor is retained, which explains why the two terms must be added.
Here is the method on the supplied teaching case. The product rule gives f′(x)=2x(x+1)+x²=3x²+2x. At x=2, this is 16. Differentiating each factor and multiplying would omit essential terms.
Multiplying u′ and v′ is not the product rule. Expand first as an independent check when possible, then confirm both approaches give the same derivative polynomial and evaluation value.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: For f(x)=x²(x+1), find f′(2). Candidate answers: (A) 15 (B) 16 (C) 17 (D) 32. Solution: The product rule gives f′(x)=2x(x+1)+x²=3x²+2x. At x=2, this is 16. Differentiating each factor and multiplying would omit essential terms. The final answer is 16. Multiplying u′ and v′ is not the product rule. Expand first as an independent check when possible, then confirm both approaches give the same derivative polynomial and evaluation value.
Check your reasoning
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Helpful tips
- The derivative of uv is u′v+uv′.
- Multiplying u′ and v′ is not the product rule.
See the answer and explanation
33
The product rule gives f′(x)=2x(x+1)+x²=3x²+2x. At x=3, this is 33. Differentiating each factor and multiplying would omit essential terms.
Take it further
Changed-input task: For f(x)=x²(x+1), find f′(9). Candidate answers: (A) 260 (B) 261 (C) 262 (D) 522. Show the mathematical reason, not only the selected answer. Expected result: 261. The product rule gives f′(x)=2x(x+1)+x²=3x²+2x. At x=9, this is 261. Differentiating each factor and multiplying would omit essential terms.
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