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Implicit differentiation

Problem 2.809 · medium

The curve \( \displaystyle 4 x^{2} - 2 x y + 3 y^{2} = 3 \) passes through \( \displaystyle (0, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.

\[ 3 \]
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
\[ \frac{d}{d x} \left(4 x^{2} - 2 x Y{\left(x \right)} + 3 Y^{2}{\left(x \right)}\right) = 8 x + \left(- 2 x + 6 Y{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} - 2 Y{\left(x \right)} \]
Every y term picks up a factor dy/dx.✓ Proved
\[ \frac{- 8 x + 2 y}{- 2 x + 6 y} = \frac{4 x - y}{x - 3 y} \]
Solve for dy/dx: minus F_x over F_y.✓ Proved
\[ 1 \cdot \frac{1}{3} = \frac{1}{3} \]
At the point.

Answer \( \frac{dy}{dx} = \frac{4 x - y}{x - 3 y}, \quad \left.\frac{dy}{dx}\right|_{(0,1)} = \frac{1}{3} \)

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: This is a plan sentence, not an equation. Differentiating both sides in x with y = y(x) and then solving for dy/dx is the correct method, and the later lines carry it out correctly.
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 implicit differentiation step gives y'=(2y-8x)/(-2x+6y). At (0,1) this evaluates to -1/3, not +1/3. The solution’s sign is incorrect.
qwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation, handles the algebraic manipulation to isolate dy/dx, and accurately evaluates the derivative at the given point.

Senior review claude-sonnet-5-5, 2026-10-13: pass — F_x = 8x-2y and F_y = -2x+6y, so dy/dx = (-8x+2y)/(-2x+6y) = (4x-y)/(x-3y), which is 1/3 at (0,1). The line 3 label mentions only the chain-rule factor, but the product rule on the xy term is part of the same differentiation step, so this is not a defect.

gpt-oss:20b: dismiss — The objection's own expression (2y-8x)/(-2x+6y) gives 2/6 = +1/3 at (0,1), not -1/3, and it equals (4x-y)/(x-3y) after multiplying top and bottom by -1.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — F_x = 8x-2y and F_y = -2x+6y, so dy/dx = (-8x+2y)/(-2x+6y) = (4x-y)/(x-3y), which is 1/3 at (0,1). The line 3 label mentions only the chain-rule factor, but the product rule on the xy term is part of the same differentiation step, so this is not a defect.
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies implicit differentiation, handles the algebraic manipulation to isolate dy/dx, and accurately evaluates the derivative at the given point.
gpt-oss:20b: fail (error) 2026-09-28 — The implicit differentiation step gives y'=(2y-8x)/(-2x+6y). At (0,1) this evaluates to -1/3, not +1/3. The solution’s sign is incorrect.
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The intermediate algebraic steps are verified as correct.
gpt-oss:20b: pass 2026-09-27
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 logic is sound and the result is correct.
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 2026-10-13 with SymPy 1.14.0.