Differentials
Problem 3.435 · easy
Find the differential \( \displaystyle dy \) of \( \displaystyle y = \frac{6}{x} \), and evaluate it at \( \displaystyle x = 3 \), \( \displaystyle dx = \frac{1}{100} \).
- \[ \frac{d}{d x} \frac{6}{x} = - \frac{6}{x^{2}} \]dy = f′(x) dx.✓ Proved
- \[ \frac{\left. - \frac{6}{x^{2}} \right|_{\substack{ x=3 }}}{100} = - \frac{1}{150} \]Substitute x and dx.✓ Proved
- The exact change is -2/301 ≈ -0.00664452; the differential is the tangent-line estimate of it.
Answer \( dy = - \frac{6}{x^{2}}\,dx = - \frac{1}{150} \)
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.
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: fail (misleading) — The sentence claims the exact change is -2/301, which is incorrect; the true exact change from x=3 to x=3.01 is approximately -0.00731, not -0.00664. The differential value -1/150 is correct, but the accompanying statement misleads the reader.qwen3.6:27b-mlx: pass — The solution correctly computes the differential and provides accurate context regarding its relationship to the exact change.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-06 — The sentence claims the exact change is -2/301, which is incorrect; the true exact change from x=3 to x=3.01 is approximately -0.00731, not -0.00664. The differential value -1/150 is correct, but the accompanying statement misleads the reader.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly computes the differential and provides accurate context regarding its relationship to the exact change.gpt-oss:20b: fail (misleading) 2026-10-06 — The sentence claims the exact change is –2/301, but the true change from y(3)=2 to y(3+0.01)=6/3.01≈1.994 is about –0.0059, not –0.00664. Thus the statement misleads the reader about the relationship between the differential and the actual change.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly computes the differential and provides accurate context regarding its relationship to the exact 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 2026-10-06 with SymPy 1.14.0.