Implicit differentiation
Problem 2.1157 · medium
The curve \( \displaystyle x^{2} - 2 x y + y^{2} = 1 \) passes through \( \displaystyle (1, 0) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ 1 \]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.
- \[ \frac{d}{d x} \left(x^{2} - 2 x Y{\left(x \right)} + Y^{2}{\left(x \right)}\right) = 2 x + \left(- 2 x + 2 Y{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} - 2 Y{\left(x \right)} \]Every y term picks up a factor dy/dx.✓ Proved
- \[ 1 \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ 1 \]At the point.✓ Proved
Answer \( \frac{dy}{dx} = 1, \quad \left.\frac{dy}{dx}\right|_{(1,0)} = 1 \)
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 computation is incorrect: the term from \u201c-2xy\u201d should be \u201c-2y-2x\,dy/dx\u201d, leading to \u201cdy/dx = -1\u201d at (1,0), nqwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation and evaluates the derivative at the specified point. The intermediate steps are algebraically sound and lead to the correct result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies implicit differentiation and evaluates the derivative at the specified point. The intermediate steps are algebraically sound and lead to the correct result.gpt-oss:20b: inconclusive 2026-09-28 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The derivative computation is incorrect: the term from \u201c-2xy\u201d should be \u201c-2y-2x\,dy/dx\u201d, leading to \u201cdy/dx = -1\u201d at (1,0), nqwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution claims the derivative is 1, but implicit differentiation of x^2 - 2xy + y^2 = 1 yields dy/dx = (y-x)/(x-y). At (1,0), this evaluates to -1, not 1. The algebraic steps shown are also incorrect (e.g., the derivative expression in line 3 is wrong).gpt-oss:20b: inconclusive 2026-09-28 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The derivative in step 3 is incorrect: the term from \(-2xy\) should be \(-2y-2x\,dy/dx\), not \((-2x+2y)\,dy/dx-2y\)."}
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 2026-09-28 with SymPy 1.14.0.