Differentials
Problem 3.386 · easy
Find the differential \( \displaystyle dy \) of \( \displaystyle y = \frac{7}{x} \), and evaluate it at \( \displaystyle x = -3 \), \( \displaystyle dx = - \frac{1}{10} \).
- \[ \frac{d}{d x} \frac{7}{x} = - \frac{7}{x^{2}} \]dy = f′(x) dx.✓ Proved
- \[ - \frac{\left. - \frac{7}{x^{2}} \right|_{\substack{ x=-3 }}}{10} = \frac{7}{90} \]Substitute x and dx.✓ Proved
- The exact change is 7/93 ≈ 0.0752688; the differential is the tangent-line estimate of it.Reviewed
Answer \( dy = - \frac{7}{x^{2}}\,dx = \frac{7}{90} \)
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 accurate context regarding the relationship between the differential and the exact change.
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 accurate context regarding the relationship between the differential and the exact change.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution calculates the differential correctly but then incorrectly claims the 'exact change' is 7/93. The exact change is f(-3 - 1/10) - f(-3) = 7/(-3.1) - 7/(-3) = -70/31 + 7/3 = 7/93 is incorrect; the actual value is 7/93 only if the denominator was 3.1*3 which is 9.3, so 7/9.3 = 70/93. Wait, let's recompute. f(x) = 7/x. x = -3, dx = -0.1. New x = -3.1. f(-3.1) = 7/-3.1 = -70/31. f(-3) = -7/3. Delta y = -70/31 - (-7/3) = -70/31 + 7/3 = (-210 + 217) / 93 = 7/93. Okay, the arithmetic for exact change is actually correct. However, the problem only asked for the differential dy. The sentence adds unsolicited and potentially confusing information about 'exact change' which was not requested. More importantly, is the statement 'The exact change is 7/93' factually correct? Yes. Is it misleading? No. Is it an error? No. But wait, the prompt asks to review the solution. The solution provides the correct answer for dy. The extra sentence is just commentary. Is there an error in the commentary? No. Is there an error in the setup? No. Why would this fail? Let me re-read the prompt constraints. 'Find the differential dy... and evaluate it'. The solution does this. The extra sentence is just extra. Is it wrong? No. Is it misleading? No. Maybe it's just style? Or maybe I missed something. Let's look at the signs. x=-3, dx=-1/10. dy = -7/(-3)^2 * (-1/10) = -7/9 * -1/10 = 7/90. Correct. Exact change: 7/93. Correct. The statement is true. The verdict should be pass? Or is the inclusion of 'exact change' considered 'misleading' because it might confuse a student into thinking they need to calculate exact change? The prompt says 'nothing would teach a student something false'. It doesn't teach anything false. It teaches that dy is an approximation of delta y. That is true. So it should pass. Wait, looking at similar examples, often extra unsolicited info that is correct is just style or pass. But if the question is 'Find dy', and the solution finds dy, it's correct. Let me check if there's a subtle error. 'dy = f'(x) dx'. Correct. Substitution correct. Result correct. Commentary correct. I will mark it pass.gpt-oss:20b: fail (misleading) 2026-10-04 — The sentence claims the exact change is 7/93, which is incorrect; the correct differential is 7/90. This misleads a student about the relationship between the differential and the 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.