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Implicit differentiation

Problem 2.755 · medium

The curve \( \displaystyle x^{3} + 2 x y + y^{3} = -2 \) 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.Reviewed
Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.
Reviewed
\[ \frac{d}{d x} \left(x^{3} + 2 x Y{\left(x \right)} + Y^{3}{\left(x \right)}\right) = 3 x^{2} + \left(2 x + 3 Y^{2}{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} + 2 Y{\left(x \right)} \]
Every y term picks up a factor dy/dx.✓ Proved
\[ \frac{- 3 x^{2} - 2 y}{2 x + 3 y^{2}} \]
Solve for dy/dx: minus F_x over F_y.Reviewed
\[ - \frac{1}{5} \]
At the point.Reviewed

Answer \( \frac{dy}{dx} = \frac{- 3 x^{2} - 2 y}{2 x + 3 y^{2}}, \quad \left.\frac{dy}{dx}\right|_{(1,-1)} = - \frac{1}{5} \)

✓ Nihil obstat Lines: 1 proved, 4 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	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
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 logic is sound and the steps are clearly justified.

Senior review claude-sonnet-5-5, 2026-10-12: pass — The math is correct. The point (1,-1) satisfies 1-2-1=-2. Differentiating gives 3x^2+2y+(2x+3y^2)y'=0, so y'=-(3x^2+2y)/(2x+3y^2), which is -1/5 at (1,-1). The line 3 label is loose, since the 2xy term uses the product rule, but this is not a defect.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — The math is correct. The point (1,-1) satisfies 1-2-1=-2. Differentiating gives 3x^2+2y+(2x+3y^2)y'=0, so y'=-(3x^2+2y)/(2x+3y^2), which is -1/5 at (1,-1). The line 3 label is loose, since the 2xy term uses the product rule, but this is not a defect.
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 logic is sound and the steps are clearly justified.
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 steps are logically sound and the final answer is 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 steps are logically sound and the final answer is correct.
gpt-oss:20b: fail (error) 2026-09-26 — The implicit differentiation omitted the constant term –2, leading to an incorrect derivative. The correct derivative is dy/dx = -(3x^2+2y)/(2x+3y^2), which at (1,−1) evaluates to −1, not −1/5.

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.