Derivative of \( \displaystyle 2 x \left(x^{2} + 6 x + 12\right) \)
Problem 2.829 · hard Beautiful Video
- \[ \frac{d}{d x} 2 x \left(x^{2} + 6 x + 12\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(2 x^{3} + 12 x^{2} + 24 x\right) \]algebraExpand the expression using distributive property.✓ Proved
- \[ = \frac{d}{d x} 24 x + \frac{d}{d x} 12 x^{2} + \frac{d}{d x} 2 x^{3} \]sumApply the sum rule for derivatives.✓ Proved
- \[ = 6 x^{2} + \frac{d}{d x} 24 x + \frac{d}{d x} 12 x^{2} \]powerDifferentiate the first term using the power rule.✓ Proved
- \[ = 6 x^{2} + 24 x + \frac{d}{d x} 24 x \]powerDifferentiate the second term using the power rule.✓ Proved
- \[ = 6 x^{2} + 24 x + 24 \]power simplifyDifferentiate the third term using the power rule. Factor the expression to show the simplified form.✓ Proved
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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 |
| 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: pass — The solution correctly applies the rules in sequence, changing only one aspect of the expression per step. The labels accurately reflect the operations performed.
Senior review claude-sonnet-5-5, 2026-10-13: pass — Each step changes one thing and the labels are legitimate. Expanding is algebra, splitting is sum, each term is differentiated by the power rule, and factoring out 6 is simplify. The last line 6(x^2+4x+4) equals the stated 6(x+2)^2.
panel (pass and objection under the same prompt): dismiss — No concrete defect is stated, and every line is SymPy-proved equal to the previous one, with sound rule labels (the 12x^2 and 24x terms use the power rule, with the constant multiple folded in).agent:pipeline: uphold — Beautiful nomination: f = 2x(x^2+6x+12) is 2((x+2)^3 - 8), a shifted cube. A messy cubic therefore collapses to the compact derivative 6(x+2)^2.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-13 — Each step changes one thing and the labels are legitimate. Expanding is algebra, splitting is sum, each term is differentiated by the power rule, and factoring out 6 is simplify. The last line 6(x^2+4x+4) equals the stated 6(x+2)^2.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the rules in sequence, changing only one aspect of the expression per step. The labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (error) 2026-09-26 — Step 1 applies the derivative rule directly to the product 2*x*(x**2+6*x+12) without using the product rule, which is a multi‑rule application and incorrect. The correct first step would be to apply the product rule, not a single derivative rule.
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 2026-09-27 with SymPy 1.14.0.