Related rates from an equation
Problem 3.421 · easy
If \( \displaystyle y = x^{3} - 2 x \) 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
- \[ \frac{d}{d x} \left(x^{3} - 2 x\right) = 3 x^{2} - 2 \]dy/dx.✓ Proved
- \[ - 3 \left. 3 x^{2} - 2 \right|_{\substack{ x=3 }} = -75 \]Multiply by dx/dt at the given x.✓ Proved
Answer \( \frac{dy}{dt} = -75 \)
✓ 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 | 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, computes the derivative, and substitutes the given values accurately.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the chain rule, computes the derivative, and substitutes the given values accurately.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the chain rule and substitutes the given values to arrive at the correct result.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.