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Implicit differentiation

Problem 2.814 · medium

The curve \( \displaystyle x^{2} - 2 x y + 3 y^{2} = 6 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.

\[ 6 \]
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
\[ \frac{d}{d x} \left(x^{2} - 2 x Y{\left(x \right)} + 3 Y^{2}{\left(x \right)}\right) = 2 x + \left(- 2 x + 6 Y{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} - 2 Y{\left(x \right)} \]
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
\[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]
At the point.Reviewed

Answer \( \frac{dy}{dx} = \frac{x - y}{x - 3 y}, \quad \left.\frac{dy}{dx}\right|_{(1,-1)} = \frac{1}{2} \)

✓ Nihil obstat Lines: 2 proved, 3 reviewed. 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	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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The algebraic steps are verified and the logic is sound.

Senior review claude-sonnet-5-5, 2026-10-13: pass — The math is correct. F_x = 2x - 2y and F_y = -2x + 6y, so dy/dx = -F_x/F_y = (x-y)/(x-3y), which equals 2/4 = 1/2 at (1,-1). Line 4 combines solving for dy/dx with cancelling a common factor of 2, and line 1 does not show the substitution. These are minor stylistic compressions, not errors.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — The math is correct. F_x = 2x - 2y and F_y = -2x + 6y, so dy/dx = -F_x/F_y = (x-y)/(x-3y), which equals 2/4 = 1/2 at (1,-1). Line 4 combines solving for dy/dx with cancelling a common factor of 2, and line 1 does not show the substitution. These are minor stylistic compressions, not errors.
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 algebraic steps are verified and the logic is sound.
gpt-oss:20b: pass 2026-09-28
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 algebraic steps are verified and the logic is sound.
gpt-oss:20b: inconclusive 2026-09-27 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The algebra in step 4 is incorrect: the derivative of \(-2xy\) should contribute \(-2y-2x\,y'\), leading to \(y'=(y-x)/(-x+3y)\). The solution writes the
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 algebraic steps are verified as correct.
gpt-oss:20b: inconclusive 2026-09-26 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The derivative of the term \u201c-2xy\u201d was computed incorrectly. It should be \u201c-2y - 2x\,dy/dx\u201d, not the expression shown in step 3. This m

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.