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Implicit differentiation

Problem 2.1278 · medium

The curve \( \displaystyle 2 x^{2} + x y + y^{2} = 11 \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.

\[ 11 \]
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
\[ \frac{d}{d x} \left(2 x^{2} + x Y{\left(x \right)} + Y^{2}{\left(x \right)}\right) = 4 x + \left(x + 2 Y{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} + Y{\left(x \right)} \]
Every y term picks up a factor dy/dx.✓ Proved
\[ \frac{- 4 x - y}{x + 2 y} \]
Solve for dy/dx: minus F_x over F_y.Reviewed
\[ - \frac{9}{4} \]
At the point.Reviewed

Answer \( \frac{dy}{dx} = \frac{- 4 x - y}{x + 2 y}, \quad \left.\frac{dy}{dx}\right|_{(2,1)} = - \frac{9}{4} \)

Lines: 1 proved, 4 reviewed. 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	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 and evaluates the derivative at the given point. The intermediate steps are algebraically sound and the final result matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies implicit differentiation and evaluates the derivative at the given point. The intermediate steps are algebraically sound and the final result matches the stated answer.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (misleading) 2026-09-29 — The sentence in step 3 claims that 'Every y term picks up a factor dy/dx', which is mathematically false for the term 'xy' (which requires the product rule, not just multiplying by dy/dx). While the final algebraic result is correct, the stated reasoning is incorrect and would teach a student a false rule for implicit differentiation.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The logic and calculations are sound.
gpt-oss:20b: pass 2026-09-29

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.