Derivative of \( \displaystyle 3 x \left(x^{2} + 3 x + 3\right) \)
Problem 2.202 · hard Beautiful Video
Differentiate \( \displaystyle f(x) = 3 x \left(x^{2} + 3 x + 3\right) \).
- \[ \frac{d}{d x} 3 x \left(x^{2} + 3 x + 3\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 3 \frac{d}{d x} x \left(x^{2} + 3 x + 3\right) \]constant-multiplePull out the constant factor 3.✓ Proved
- \[ = 3 x \frac{d}{d x} \left(x^{2} + 3 x + 3\right) + 3 \left(x^{2} + 3 x + 3\right) \frac{d}{d x} x \]productApply the product rule to the remaining expression.✓ Proved
- \[ = 3 x^{2} + 3 x \left(\frac{d}{d x} 3 + \frac{d}{d x} 3 x + \frac{d}{d x} x^{2}\right) + 9 x + 9 \]derivativeDifferentiate each term inside the parentheses.✓ Proved
- \[ = 3 x^{2} + 3 x \left(2 x + 3\right) + 9 x + 9 \]derivative algebraEvaluate the derivatives of the individual terms. Simplify the expression inside the parentheses.✓ Proved
- \[ = 9 x^{2} + 18 x + 9 \]algebra algebra simplifyDistribute x into the parentheses. Combine like terms. Distribute the 3 to get the final simplified form.✓ Proved
Answer \( 9 \left(x + 1\right)^{2} \)
✓ 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 |
| 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple, product, and sum rules in separate steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Senior review claude-sonnet-5-5, 2026-10-06: pass — The derivation is correct and every step is justified. Line 4 splits the sum and evaluates d/dx of x in one step, which is a negligible bundling of derivative rules.
unnamed panel model (passed and objected under the same prompt): dismiss — The objection names no specific line or defect, and SymPy proved every line equivalent; the derivative 9x^2+18x+9 = 9(x+1)^2 is correct and the rule labels are legitimate.agent:pipeline: uphold — Beautiful nomination: Since x(x^2+3x+3) = (x+1)^3 - 1, f = 3((x+1)^3 - 1). The messy cubic therefore has the derivative 9(x+1)^2, a perfect square.
Every verdict on record (16)
claude-sonnet-5-5: pass 2026-10-06 — The derivation is correct and every step is justified. Line 4 splits the sum and evaluates d/dx of x in one step, which is a negligible bundling of derivative rules.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple, product, and sum rules in separate steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 4 applies two rules at once: it differentiates the sum and then differentiates each term. Each step must change only one thing, so this is a defect.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final result is algebraically equivalent to the stated answer.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies the constant multiple, product, and sum rules in separate steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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 2026-09-26 with SymPy 1.14.0.