Related rates from an equation
Problem 3.572 · easy
If \( \displaystyle y = 3 x^{2} + x \) and \( \displaystyle \frac{dx}{dt} = -1 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = -2 \).
- Differentiate both sides with respect to t; by the chain rule dy/dt = (dy/dx)(dx/dt).Reviewed
- \[ \frac{d}{d x} \left(3 x^{2} + x\right) = 6 x + 1 \]dy/dx.✓ Proved
- \[ - \left. 6 x + 1 \right|_{\substack{ x=-2 }} = 11 \]Multiply by dx/dt at the given x.✓ Proved
Answer \( \frac{dy}{dt} = 11 \)
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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | moved x along at dx/dt, solved for y numerically, differenced in t |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and substitutes the given values accurately.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies the chain rule and substitutes the given values accurately.qwen3.6:27b-mlx: inconclusive 2026-10-10 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly computes the value of dy/dt. At x = -2, dy/dx = 6(-2) + 1 = -11. Multiplying by dx/dt = -1 gives dy/dt = 11. However, the egpt-oss:20b: pass 2026-10-10
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/related_rates_equation, checked 2026-10-10 with SymPy 1.14.0.