Derivative of \( \displaystyle x \left(3 x - 2\right) \)
Problem 2.1333 · medium
Differentiate \( \displaystyle f(x) = x \left(3 x - 2\right) \).
- \[ \frac{d}{d x} x \left(3 x - 2\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(3 x^{2} - 2 x\right) \]algebraExpand the expression using the distributive property.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 3 x^{2} \]sumApply the difference rule for derivatives.✓ Proved
- \[ = - 2 \frac{d}{d x} x + 3 \frac{d}{d x} x^{2} \]constant-multiplePull out the constant coefficients.✓ Proved
- \[ = 6 x - 2 \]power algebraApply the power rule to each term. Multiply the constants.✓ Proved
Answer \( 6 x - 2 \)
✓ 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.
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 |
| 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 (style) — Step 1 lacks a rule label. The step should be labeled with an appropriate rule (e.g., "product" or simply "derivative"), but the label is missing, violating the contract’s requirement that each step names its rule.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (style) 2026-10-03 — Step 1 lacks a rule label. The step should be labeled with an appropriate rule (e.g., "product" or simply "derivative"), but the label is missing, violating the contract’s requirement that each step names its rule.qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly expands the polynomial before differentiating, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary.gpt-oss:20b: fail (style) 2026-09-30 — Step 1 is missing a rule label; it should be "derivative". Step 3 incorrectly describes the rule as a difference rule, but it is actually the sum rule. Both issues violate the single‑rule‑per‑step and correct labeling requirements.
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.