Implicit differentiation
Problem 2.930 · medium
The curve \( \displaystyle x^{2} - 2 x y + 3 y^{2} = 2 \) 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.✓ Proved
- Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.
- \[ \frac{d}{d x} \left(x^{2} - 2 x Y{\left(x \right)} + 3 Y^{2}{\left(x \right)}\right) = 2 x + \left(- 2 x + 6 Y{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} - 2 Y{\left(x \right)} \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- 2 x + 2 y}{- 2 x + 6 y} = \frac{x - y}{x - 3 y} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ 0 \]At the point.✓ Proved
Answer \( \frac{dy}{dx} = \frac{x - y}{x - 3 y}, \quad \left.\frac{dy}{dx}\right|_{(1,1)} = 0 \)
Lines: 4 proved, 1 not checked. 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | 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 | ✓ 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 | sympy.idiff differentiates the curve on its own and agrees |
Reviewers
gpt-oss:20b: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The derivative in step 3 is incorrect: the term \(-2xy\) differentiates to \(-2y-2x\,dy/dx\), not \((-2x+6y)dy/dx-2y\)."}qwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The intermediate steps and final result are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The intermediate steps and final result are 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 term \(-2xy\) differentiates to \(-2y-2x\,dy/dx\), not \((-2x+6y)dy/dx-2y\)."}qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies implicit differentiation, handles the product rule for the xy term, and evaluates the derivative at the given point to obtain the correct result.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 2026-09-26 with SymPy 1.14.0.