Differentials
Problem 3.390 · easy
Find the differential \( \displaystyle dy \) of \( \displaystyle y = \frac{3}{x} \), and evaluate it at \( \displaystyle x = 4 \), \( \displaystyle dx = \frac{1}{20} \).
- \[ \frac{d}{d x} \frac{3}{x} = - \frac{3}{x^{2}} \]dy = f′(x) dx.✓ Proved
- \[ \frac{\left. - \frac{3}{x^{2}} \right|_{\substack{ x=4 }}}{20} = - \frac{3}{320} \]Substitute x and dx.✓ Proved
- The exact change is -1/108 ≈ -0.00925926; the differential is the tangent-line estimate of it.Reviewed
Answer \( dy = - \frac{3}{x^{2}}\,dx = - \frac{3}{320} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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 | 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 provides accurate context regarding its relationship to the exact change.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the differential and provides accurate context regarding its relationship to the exact change.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 factually accurate and helpful.gpt-oss:20b: pass 2026-10-04
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.