Derivative of \( \displaystyle 3 x \left(3 x^{2} + 6 x + 4\right) \)
Problem 2.1004 · hard Video
Differentiate \( \displaystyle f(x) = 3 x \left(3 x^{2} + 6 x + 4\right) \).
- \[ \frac{d}{d x} 3 x \left(3 x^{2} + 6 x + 4\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 3 \frac{d}{d x} x \left(3 x^{2} + 6 x + 4\right) \]constant-multiplePull out the constant factor 3.✓ Proved
- \[ = 3 x \frac{d}{d x} \left(3 x^{2} + 6 x + 4\right) + 3 \left(3 x^{2} + 6 x + 4\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 9 x^{2} + 3 x \left(\frac{d}{d x} 4 + \frac{d}{d x} 6 x + \frac{d}{d x} 3 x^{2}\right) + 18 x + 12 \]derivativeDifferentiate each term in the sum.✓ Proved
- \[ = 9 x^{2} + 3 x \left(6 x + 6\right) + 18 x + 12 \]derivative algebraEvaluate the derivatives of the individual terms. Simplify the expression inside the parentheses.✓ Proved
- \[ = 27 x^{2} + 36 x + 12 \]algebra algebra algebraDistribute x into the parentheses. Combine like terms. Distribute the 3 to get the final answer.✓ Proved
Answer \( 3 \left(3 x + 2\right)^{2} \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The final answer provided (27*x**2 + 36*x + 12) does not match the stated answer (3*(3*x + 2)**2 = 27*x**2 + 36*x + 12). Wait, 3*(3x+2)^2 = 3*(9x^2 + 12x + 4) = 27x^2 + 36x + 12. They are equal. Let me re-read the prompt. 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. So the math is correct. Is there a rule violation? Step 4 applies 'derivative' to a sum. The rule 'sum' is in the vocabulary. Step 4 says 'rule: derivative' but the note says 'Differentiate each term in the sum'. This is applying the linearity of the derivative (sum rule) AND evaluating the derivatives of the terms inside? No, Step 4 expands Derivative(3*x**2 + 6*x + 4, x) into Derivative(3*x**2, x) + Derivative(6*x, x) + Derivative(4, x). This is the sum rule. The label is 'derivative'. The vocabulary has 'sum'. Is 'derivative' acceptable for applying the sum rule? The prompt says: 'derivative' means unfolding d/dx on a known form (the derivative of x, of a constant, of sin, of exp). It does NOT mean applying the sum rule. Applying the sum rule should be labeled 'sum'. Step 4 applies the sum rule to split the derivative of a sum into a sum of derivatives. It labels this 'derivative'. This is a label defect. Step 5 evaluates the derivatives. It labels this 'derivative'. This is correct. So Step 4 is the issue. It applies the sum rule but labels it 'derivative'. The prompt says: 'Each step names its rule from this fixed vocabulary... A label is correct if it names a rule the step legitimately applied.' Did Step 4 apply 'derivative'? No, it applied the sum rule (linearity). It did not evaluate any derivatives yet. It just split the operator. So 'derivative' is the wrong label. It should be 'sum'. Therefore, it is a defect.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The final answer provided (27*x**2 + 36*x + 12) does not match the stated answer (3*(3*x + 2)**2 = 27*x**2 + 36*x + 12). Wait, 3*(3x+2)^2 = 3*(9x^2 + 12x + 4) = 27x^2 + 36x + 12. They are equal. Let me re-read the prompt. 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. So the math is correct. Is there a rule violation? Step 4 applies 'derivative' to a sum. The rule 'sum' is in the vocabulary. Step 4 says 'rule: derivative' but the note says 'Differentiate each term in the sum'. This is applying the linearity of the derivative (sum rule) AND evaluating the derivatives of the terms inside? No, Step 4 expands Derivative(3*x**2 + 6*x + 4, x) into Derivative(3*x**2, x) + Derivative(6*x, x) + Derivative(4, x). This is the sum rule. The label is 'derivative'. The vocabulary has 'sum'. Is 'derivative' acceptable for applying the sum rule? The prompt says: 'derivative' means unfolding d/dx on a known form (the derivative of x, of a constant, of sin, of exp). It does NOT mean applying the sum rule. Applying the sum rule should be labeled 'sum'. Step 4 applies the sum rule to split the derivative of a sum into a sum of derivatives. It labels this 'derivative'. This is a label defect. Step 5 evaluates the derivatives. It labels this 'derivative'. This is correct. So Step 4 is the issue. It applies the sum rule but labels it 'derivative'. The prompt says: 'Each step names its rule from this fixed vocabulary... A label is correct if it names a rule the step legitimately applied.' Did Step 4 apply 'derivative'? No, it applied the sum rule (linearity). It did not evaluate any derivatives yet. It just split the operator. So 'derivative' is the wrong label. It should be 'sum'. Therefore, it is a defect.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The final answer provided in the task description, 3*(3*x + 2)**2, is mathematically incorrect (it expands to 27*x**2 + 36*x + 12, but the derivative of the original function is 27*x**2 + 36*x + 12? Wait. Let's check. f(x) = 9x^3 + 18x^2 + 12x. f'(x) = 27x^2 + 36x + 12. The stated answer is 3*(9x^2 + 12x + 4) = 27x^2 + 36x + 12. Okay, the stated answer IS correct. The steps arrive at 27*x**2 + 36*x + 12. So the math is correct. Why did I think it was wrong? Let me re-read the contract. 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. So I don't need to check the algebra. I need to check the rules. Step 4 applies 'derivative' to a sum. The rule 'sum' is in the vocabulary. Step 4 does Derivative(A+B+C) -> Derivative(A)+Derivative(B)+Derivative(C). This is the sum rule. The label is 'derivative'. Is 'derivative' a valid label for applying the sum rule? The contract says: 'derivative' means unfolding d/dx on a known form (the derivative of x, of a constant, of sin, of exp). It does NOT say 'derivative' covers the sum rule. The sum rule is 'sum'. Therefore, labeling the application of the sum rule as 'derivative' is a defect because 'derivative' is reserved for basic derivatives, and 'sum' is the specific label for linearity over addition. Also, Step 4 applies the sum rule AND evaluates Derivative(x,x)=1. That is two rules: sum and derivative. This violates the 'one thing per step' rule. Step 4 is a multi-rule step labeled incorrectly.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-09-28 with SymPy 1.14.0.