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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.
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.
\[ 1 \]
At the point.

Answer \( \frac{dy}{dx} = 1, \quad \left.\frac{dy}{dx}\right|_{(1,0)} = 1 \)

Lines: 1 proved, 4 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. 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	Not checked	—	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The derivative step is incorrect: the derivative of \(-2xy\) should be \(-2y-2x\,dy/dx\), not the expression shown. Consequently the subsequent conclusion
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the formula for dy/dx derived from the implicit differentiation, which is required by the problem statement ('Find dy/dx'). It jumps from the derivative expression to the final numerical answer without showing the algebraic isolation of dy/dx or the substitution step clearly.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to explicitly state the formula for dy/dx derived from the implicit differentiation, which is required by the problem statement ('Find dy/dx'). It jumps from the derivative expression to the final numerical answer without showing the algebraic isolation of dy/dx or the substitution step clearly.
gpt-oss:20b: inconclusive 2026-10-03 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The derivative step is incorrect: the derivative of \(-2xy\) should be \(-2y-2x\,dy/dx\), not the expression shown. Consequently the subsequent conclusion
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), n
qwen3.6:27b-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 2026-10-13 with SymPy 1.14.0.