Implicit differentiation
Problem 2.801 · 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
- \[ - \frac{1}{5} \]At the point.
Answer \( \frac{dy}{dx} = \frac{3 x^{2} - 2 y}{2 x - 3 y^{2}}, \quad \left.\frac{dy}{dx}\right|_{(-1,1)} = - \frac{1}{5} \)
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: Line 2 is a sentence describing the method (differentiate implicitly, then solve for dy/dx), not an algebraic equivalence. It is accurate and matches what lines 3-4 do. |
| 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: fail (error) — The derivative in step 3 is incorrect: the term multiplying dy/dx should be –2x, not (–2x+3y²). The subsequent algebra still yields the correct value, but the justification is flawed.qwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The algebraic steps are verified and the final answer is correct.
Senior review claude-sonnet-5-5, 2026-10-13: pass — Line 3's derivative is correct, and solving for dy/dx gives -F_x/F_y = (3x^2-2y)/(2x-3y^2). At (-1,1) this is (3-2)/(-2-3) = -1/5, matching the stated answer. The point check -1+2+1 = 2 also holds.
gpt-oss:20b: dismiss — The objection is false. d/dx of -2xy is -2y - 2x·y' and d/dx of y^3 is 3y^2·y', so the y' coefficient is (-2x + 3y^2) and the remaining term is 3x^2 - 2y, exactly as line 3 states.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-13 — Line 3's derivative is correct, and solving for dy/dx gives -F_x/F_y = (3x^2-2y)/(2x-3y^2). At (-1,1) this is (3-2)/(-2-3) = -1/5, matching the stated answer. The point check -1+2+1 = 2 also holds.qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The algebraic steps are verified and the final answer is correct.gpt-oss:20b: fail (error) 2026-09-28 — The derivative in step 3 is incorrect: the term multiplying dy/dx should be –2x, not (–2x+3y²). The subsequent algebra still yields the correct value, but the justification is flawed.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates the result at the given point. The algebraic steps are verified and the final answer is correct.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The algebraic steps are verified and the final answer is correct.gpt-oss:20b: inconclusive 2026-09-26 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The derivative in step 3 is incorrect: the product rule for \(-2xy\) gives \(-2y-2x\,dy/dx\), not \(-2x+3y^2\) times \(dy/dx\). Consequently the expressio
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.