Related rates from an equation
Problem 3.615 · easy
\( \displaystyle x \) and \( \displaystyle y \) are functions of \( \displaystyle t \) related by \( \displaystyle x y = 8 \). If \( \displaystyle \frac{dx}{dt} = -2 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 4 \) and \( \displaystyle y = 2 \).
- \[ 8 \]The point satisfies the equation.✓ Proved
- Differentiate with respect to t: (y)·dx/dt + (x)·dy/dt = 0.
- \[ 1 \]Solve for dy/dt and substitute.✓ Proved
Answer \( \frac{dy}{dt} = 1 \)
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 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: fail (misleading) — The solution skips the critical step of substituting the specific values (x=4, y=2, dx/dt=-2) into the differentiated equation. It jumps from the general derivative to the final answer without showing the calculation, which prevents the student from verifying the arithmetic or understanding how the values are used.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (misleading) 2026-10-13 — The solution skips the critical step of substituting the specific values (x=4, y=2, dx/dt=-2) into the differentiated equation. It jumps from the general derivative to the final answer without showing the calculation, which prevents the student from verifying the arithmetic or understanding how the values are used.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies implicit differentiation to the constraint xy=8 and substitutes the given values to find the correct rate of 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/related_rates_equation, checked 2026-10-13 with SymPy 1.14.0.