Derivative of \( \displaystyle x \left(- 4 x^{2} + 6 x - 3\right) \)
Problem 2.438 · hard
Differentiate \( \displaystyle f(x) = x \left(- 4 x^{2} + 6 x - 3\right) \).
- \[ \frac{d}{d x} x \left(- 4 x^{2} + 6 x - 3\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(- 4 x^{2} + 6 x - 3\right) + \left(- 4 x^{2} + 6 x - 3\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = - 4 x^{2} + x \frac{d}{d x} \left(- 4 x^{2} + 6 x - 3\right) + 6 x - 3 \]derivative constant-multipleDifferentiate the first part of the product. Simplify the derivative of x.✓ Proved
- \[ = - 4 x^{2} + x \left(6 - 8 x\right) + 6 x - 3 \]derivativeDifferentiate the second part of the sum.✓ Proved
- \[ = - 12 x^{2} + 12 x - 3 \]algebra algebra simplifyDistribute x into the parentheses. Remove parentheses. Combine like terms.✓ Proved
Answer \( - 12 x^{2} + 12 x - 3 \)
✓ 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 |
| 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 (style) — Step 4 incorrectly labels the simplification of 1*(-4*x**2 + 6*x - 3) as "constant-multiple"; it is a plain algebraic simplification.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Step 4 incorrectly labels the simplification of 1*(-4*x**2 + 6*x - 3) as "constant-multiple"; it is a plain algebraic simplification.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule and subsequent differentiation steps. Each step isolates a single operation, and the labels accurately reflect the rules applied.gpt-oss:20b: fail (style) 2026-09-20 — Step 4 incorrectly labels the simplification of 1*(-4*x**2 + 6*x - 3) as a "constant-multiple" rule; it is merely an algebraic simplification (removing the factor 1).qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20
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.