Implicit differentiation
Problem 2.802 · medium
The curve \( \displaystyle x^{3} - 2 x y + y^{3} = 2 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ 2 \]The point is on the curve.
- Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.Reviewed
- \[ \frac{d}{d x} \left(x^{3} - 2 x Y{\left(x \right)} + Y^{3}{\left(x \right)}\right) = 3 x^{2} + \left(- 2 x + 3 Y^{2}{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} - 2 Y{\left(x \right)} \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- 3 x^{2} + 2 y}{- 2 x + 3 y^{2}} = \frac{3 x^{2} - 2 y}{2 x - 3 y^{2}} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ -5 \]At the point.
Answer \( \frac{dy}{dx} = \frac{3 x^{2} - 2 y}{2 x - 3 y^{2}}, \quad \left.\frac{dy}{dx}\right|_{(1,-1)} = -5 \)
Lines: 2 proved, 1 reviewed, 2 not checked. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 2 | Reviewed | claude-sonnet-5-5 | claude-sonnet-5-5: This is a plan sentence, not an equation. Differentiating both sides with respect to x and then solving for dy/dx is the correct implicit-differentiation method, and the later lines carry it out correctly. |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | sympy.idiff differentiates the curve on its own and agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (misleading) — The sentence in step 3 claims that 'every y term picks up a factor dy/dx', which is mathematically false for terms involving both x and y (like -2xy) or pure x terms. This incorrect heuristic obscures the necessary application of the product rule.
Senior review claude-sonnet-5-5, 2026-10-13: pass — All equations are proved, and the derivative, the implicit-differentiation formula, and the value at (1,-1) are correct. The step 3 phrasing is slightly informal, but it is not a defect.
qwen3.6:27b-mlx: dismiss — The phrase is a loose but standard description: each differentiated occurrence of y yields a dy/dx factor (e.g. the y in -2xy gives -2x·y'), and the product rule on -2xy is correctly applied in the SymPy-proved line, so no false statement or wrong rule name results.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-13 — All equations are proved, and the derivative, the implicit-differentiation formula, and the value at (1,-1) are correct. The step 3 phrasing is slightly informal, but it is not a defect.qwen3.6:27b-mlx: fail (misleading) 2026-09-28 — The sentence in step 3 claims that 'every y term picks up a factor dy/dx', which is mathematically false for terms involving both x and y (like -2xy) or pure x terms. This incorrect heuristic obscures the necessary application of the product rule.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies implicit differentiation, handles the algebraic rearrangement properly, and evaluates the derivative at the given point.gpt-oss:20b: inconclusive 2026-09-27 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The derivative in step 3 is incorrect: the product rule for \(-2xy\) gives \(-2(y+x\,y')\), not \((-2x+3y^2)y'-2y\). This leads to a wrong intermediate exqwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-09-26
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/implicit_differentiation, checked 2026-10-13 with SymPy 1.14.0.