Differentials
Problem 3.393 · easy
Find the differential \( \displaystyle dy \) of \( \displaystyle y = - 2 x^{3} - x - 1 \), and evaluate it at \( \displaystyle x = -2 \), \( \displaystyle dx = - \frac{1}{50} \).
- \[ \frac{d}{d x} \left(- 2 x^{3} - x - 1\right) = - 6 x^{2} - 1 \]dy = f′(x) dx.✓ Proved
- \[ - \frac{\left. - 6 x^{2} - 1 \right|_{\substack{ x=-2 }}}{50} = \frac{1}{2} \]Substitute x and dx.✓ Proved
- The exact change is 31551/62500 ≈ 0.504816; the differential is the tangent-line estimate of it.Reviewed
Answer \( dy = - 6 x^{2} - 1\,dx = \frac{1}{2} \)
Lines: 2 proved, 1 reviewed. 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | central difference quotient of f at x times dx |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly computes the differential and evaluates it. The final sentence provides correct context regarding the relationship between the differential and the exact change, without introducing errors.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the differential and evaluates it. The final sentence provides correct context regarding the relationship between the differential and the exact change, without introducing errors.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the differential and evaluates it. The final sentence provides correct context regarding the relationship between the differential and the exact change, which is pedagogically sound.gpt-oss:20b: fail (misleading) 2026-10-04 — The sentence claiming the exact change is 31551/62500 ≈0.504816 is incorrect; the differential 1/2 is the linear approximation, not the exact change. This would mislead a student about the relationship between differential and actual change.
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
generator:structured/differentials, checked 2026-10-04 with SymPy 1.14.0.