Related rates from an equation
Problem 3.574 · easy
If \( \displaystyle y = x^{2} - 5 \) and \( \displaystyle \frac{dx}{dt} = -4 \), 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).
- \[ \frac{d}{d x} \left(x^{2} - 5\right) = 2 x \]dy/dx.✓ Proved
- \[ - 4 \left. 2 x \right|_{\substack{ x=-3 }} = 24 \]Multiply by dx/dt at the given x.✓ Proved
Answer \( \frac{dy}{dt} = 24 \)
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 | Not checked | — | 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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The calculation in step 3 is incorrect: substituting x = -3 into 2x gives -6, and multiplying by dx/dt = -4 yields 24, but the sign is wrong in the
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: inconclusive 2026-10-10 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The calculation in step 3 is incorrect: substituting x = -3 into 2x gives -6, and multiplying by dx/dt = -4 yields 24, but the sign is wrong in theqwen3.6:27b-mlx: fail (error) 2026-10-10 — The calculation in step 3 is incorrect: 2*(-3) = -6, and -6 * (-4) = 24 is false; the correct product is 24, wait, -6 * -4 is 24. Let me re-read carefully. 2*x at x=-3 is -6. dx/dt is -4. -6 * -4 = 24. The math is actually correct. Why did I think it was wrong? Ah, I see. The prompt says 'find dy/dt'. The solution calculates 24. Let me double check the sign. dy/dx = 2x. At x=-3, dy/dx = -6. dx/dt = -4. dy/dt = (-6)*(-4) = 24. The solution is correct. Wait, looking at the provided solution text: '-4*Subs(2*x, x, -3) = 24'. Subs(2*x, x, -3) is -6. -4 * -6 is 24. The equation is correct. Is there a subtle error? No. It seems correct. Let me look for 'misleading' or 'style'. The notation 'Subs' is non-standard but clear. The logic is sound. Verdict: pass.gpt-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.