Related rates from an equation
Problem 3.618 · easy
If \( \displaystyle y = x^{2} + x \) and \( \displaystyle \frac{dx}{dt} = -3 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = -3 \).
- Differentiate both sides with respect to t; by the chain rule dy/dt = (dy/dx)(dx/dt).Reviewed
- \[ \frac{d}{d x} \left(x^{2} + x\right) = 2 x + 1 \]dy/dx.✓ Proved
- \[ - 3 \left. 2 x + 1 \right|_{\substack{ x=-3 }} = 15 \]Multiply by dx/dt at the given x.✓ Proved
Answer \( \frac{dy}{dt} = 15 \)
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 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies the chain rule and substitutes the given values accurately.qwen3.6:27b-mlx: fail (error) 2026-10-13 — The calculation in step 3 is incorrect. Substituting x = -3 into 2x + 1 yields -5, and multiplying by dx/dt = -3 gives 15, but the sign is wrong: (-5) * (-3) = 15 is actually correct arithmetic, wait. Let me re-evaluate. 2(-3) + 1 = -6 + 1 = -5. -5 * -3 = 15. The arithmetic is correct. Why did I think it was wrong? Let me re-read the prompt. 'find dy/dt when x = -3'. y = x^2 + x. dy/dx = 2x + 1. At x = -3, dy/dx = -5. dx/dt = -3. dy/dt = (-5)(-3) = 15. The solution is correct. Wait, is there a subtle issue? No. The solution is sound. I will mark it as pass.gpt-oss:20b: pass 2026-10-13
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 2026-10-13 with SymPy 1.14.0.