Implicit differentiation
Problem 2.1051 · medium
The curve \( \displaystyle x^{2} y - 2 x + y^{3} = 1 \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ 1 \]The point is on the curve.Reviewed
- Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.Reviewed
- \[ \frac{d}{d x} \left(x^{2} Y{\left(x \right)} - 2 x + Y^{3}{\left(x \right)}\right) = 2 x Y{\left(x \right)} + \left(x^{2} + 3 Y^{2}{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} - 2 \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- 2 x y + 2}{x^{2} + 3 y^{2}} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ - \frac{2}{7} \]At the point.Reviewed
Answer \( \frac{dy}{dx} = \frac{2 \left(- x y + 1\right)}{x^{2} + 3 y^{2}}, \quad \left.\frac{dy}{dx}\right|_{(2,1)} = - \frac{2}{7} \)
Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The steps are logically sound and the final answer is correct.
Every verdict on record (6)
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 steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (error) 2026-09-27 — The sign in the expression for dy/dx is incorrect. The correct derivative is (2xy-2)/(x^2+3y^2), giving 2/7 at (2,1), not -2/7.qwen3.6:27b-mlx: fail (misleading) 2026-09-27 — The sentence in step 3 claims 'Every y term picks up a factor dy/dx', which is mathematically false for terms involving products with x (like x^2 y) or constants. This incorrect heuristic obscures the necessary application of the product rule.gpt-oss:20b: fail (error) 2026-09-27 — Step 1 incorrectly states the point as (1,1) instead of (2,1), which misrepresents the given data.
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.