Related rates from an equation
Problem 3.469 · easy
\( \displaystyle x \) and \( \displaystyle y \) are functions of \( \displaystyle t \) related by \( \displaystyle x y = 5 \). If \( \displaystyle \frac{dx}{dt} = 3 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 1 \) and \( \displaystyle y = 5 \).
- \[ 5 \]The point satisfies the equation.✓ Proved
- Differentiate with respect to t: (y)·dx/dt + (x)·dy/dt = 0.Reviewed
- \[ -15 \]Solve for dy/dt and substitute.✓ Proved
Answer \( \frac{dy}{dt} = -15 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly applies implicit differentiation to the given constraint and substitutes the values accurately to find the rate of change.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies implicit differentiation to the given constraint and substitutes the values accurately to find the rate of change.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies implicit differentiation to the constraint equation and substitutes the given values to find the 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 2026-10-06 with SymPy 1.14.0.