Derivative of \( \displaystyle 3 x \left(- 3 x^{2} - 3 x - 1\right) \)
Problem 2.1218 · hard Video
Differentiate \( \displaystyle f(x) = 3 x \left(- 3 x^{2} - 3 x - 1\right) \).
- \[ \frac{d}{d x} 3 x \left(- 3 x^{2} - 3 x - 1\right) \]Start with the derivative of the function.✓ Proved
- \[ = 3 x \frac{d}{d x} \left(- 3 x^{2} - 3 x - 1\right) + \left(- 3 x^{2} - 3 x - 1\right) \frac{d}{d x} 3 x \]productApply the product rule.✓ Proved
- \[ = - 9 x^{2} + 3 x \frac{d}{d x} \left(- 3 x^{2} - 3 x - 1\right) - 9 x - 3 \]constant-multipleApply the constant multiple rule to the first term.✓ Proved
- \[ = - 9 x^{2} - 3 x \frac{d}{d x} 1 - 3 x \frac{d}{d x} 3 x + 3 x \frac{d}{d x} \left(- 3 x^{2}\right) - 9 x - 3 \]sumDistribute the derivative across the sum.✓ Proved
- \[ = - 27 x^{2} - 18 x - 3 \]power algebra algebra simplifyDifferentiate the power terms. Simplify the terms. Distribute the 3. Combine like terms.✓ Proved
Answer \( - 3 \left(3 x + 1\right)^{2} \)
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 |
| 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: fail (error) — Step 4 incorrectly expands the derivative of the sum inside the product. It treats the term -3*x as 3*x*Derivative(3*x,x), which applies a product rule where none is needed, and thus the subsequent steps are based on a wrong expression.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (error) 2026-10-03 — Step 4 incorrectly expands the derivative of the sum inside the product. It treats the term -3*x as 3*x*Derivative(3*x,x), which applies a product rule where none is needed, and thus the subsequent steps are based on a wrong expression.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the product rule, linearity of the derivative, and power rule. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: fail (error) 2026-09-29 — Step 4 incorrectly applies the derivative to the term -3*x; it uses +3 instead of -3, leading to a sign error that propagates through the subsequent steps.
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 2026-10-03 with SymPy 1.14.0.