Differentials
Problem 3.743 · easy
Find the differential \( \displaystyle dy \) of \( \displaystyle y = - x^{3} - 2 x - 1 \), and evaluate it at \( \displaystyle x = 2 \), \( \displaystyle dx = - \frac{1}{10} \).
- \[ \frac{d}{d x} \left(- x^{3} - 2 x - 1\right) = - 3 x^{2} - 2 \]dy = f′(x) dx.✓ Proved
- \[ \frac{\left(-1\right) \left. - 3 x^{2} - 2 \right|_{\substack{ x=2 }}}{10} = \frac{7}{5} \]Substitute x and dx.✓ Proved
- The exact change is 1341/1000 ≈ 1.341; the differential is the tangent-line estimate of it.
Answer \( dy = - 3 x^{2} - 2\,dx = \frac{7}{5} \)
Lines: 2 proved, 1 not checked. 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 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | 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: fail (misleading) — The solution claims the exact change is 1341/1000, but the actual change is Δy = f(1.9) - f(2) = 1.341, whereas the differential dy = 1.4. The text implies dy is an estimate of Δy, which is true, but the specific numbers provided are confusingly presented without clarifying that dy (1.4) is the approximation and Δy (1.341) is the exact value, potentially leading a student to think the calculation of dy resulted in 1.341 or that the 'exact change' label applies to the differential.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-26 — The solution claims the exact change is 1341/1000, but the actual change is Δy = f(1.9) - f(2) = 1.341, whereas the differential dy = 1.4. The text implies dy is an estimate of Δy, which is true, but the specific numbers provided are confusingly presented without clarifying that dy (1.4) is the approximation and Δy (1.341) is the exact value, potentially leading a student to think the calculation of dy resulted in 1.341 or that the 'exact change' label applies to the differential.gpt-oss:20b: pass 2026-10-26qwen3.6:27b-mlx: fail (misleading) 2026-10-25 — The solution claims the exact change is 1341/1000, but the actual change in y is -1341/1000. The sign error in the exact change makes the comparison with the differential (which is positive) misleading.gpt-oss:20b: pass 2026-10-25qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly computes the differential and its value. The final sentence provides correct context regarding the relationship between the differential and the exact change, without introducing errors.gpt-oss:20b: pass 2026-10-25
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 2026-10-26 with SymPy 1.14.0.