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Related rates from an equation

Problem 3.415 · easy

\( \displaystyle x \) and \( \displaystyle y \) are functions of \( \displaystyle t \) related by \( \displaystyle x y = 2 \). If \( \displaystyle \frac{dx}{dt} = -4 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 1 \) and \( \displaystyle y = 2 \).

\[ 2 \]
The point satisfies the equation.✓ Proved
Differentiate with respect to t: (y)·dx/dt + (x)·dy/dt = 0.
Reviewed
\[ 8 \]
Solve for dy/dt and substitute.✓ Proved

Answer \( \frac{dy}{dt} = 8 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation to the given constraint and substitutes the known values to find the rate of change. The steps are logically sound and the final answer is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies implicit differentiation to the given constraint and substitutes the known values to find the rate of change. The steps are logically sound and the final answer is correct.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies implicit differentiation to the given constraint and substitutes the known values to find the correct rate of change.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.