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Implicit differentiation

Problem 2.1273 · hard

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

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

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

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: pass
qwen3.6:27b-mlx: fail (misleading) — The comment in line 3 claims 'Every y term picks up a factor dy/dx', which is false for the term 4y (which yields 4 dy/dx, not a product rule expansion) and misleading for sin(xy) (which yields cos(xy)(y + x y')). The comment in line 4 incorrectly states the formula as '-F_x / F_y'; the correct implicit differentiation formula is '-F_x / F_y' only if F(x,y)=c, but here the derivative of the LHS with respect to y is F_y, and the derivative with respect to x is F_x + F_y y', leading to y' = -F_x/F_y. However, the text says 'minus F_x over F_y' which is the standard formula, but the previous line's comment was the primary error. Actually, looking closer: Line 3's comment is the main issue. It implies a uniform rule that doesn't exist. Also, Line 4's comment 'minus F_x over F_y' is technically correct for implicit differentiation of F(x,y)=C, but the phrasing in Line 3 is definitely misleading/wrong regarding the mechanics.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-10-03 — The comment in line 3 claims 'Every y term picks up a factor dy/dx', which is false for the term 4y (which yields 4 dy/dx, not a product rule expansion) and misleading for sin(xy) (which yields cos(xy)(y + x y')). The comment in line 4 incorrectly states the formula as '-F_x / F_y'; the correct implicit differentiation formula is '-F_x / F_y' only if F(x,y)=c, but here the derivative of the LHS with respect to y is F_y, and the derivative with respect to x is F_x + F_y y', leading to y' = -F_x/F_y. However, the text says 'minus F_x over F_y' which is the standard formula, but the previous line's comment was the primary error. Actually, looking closer: Line 3's comment is the main issue. It implies a uniform rule that doesn't exist. Also, Line 4's comment 'minus F_x over F_y' is technically correct for implicit differentiation of F(x,y)=C, but the phrasing in Line 3 is definitely misleading/wrong regarding the mechanics.
gpt-oss:20b: pass 2026-10-03
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 algebraic steps are verified and the logic is sound.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: inconclusive 2026-09-29 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The implicit differentiation is incorrect. The derivative of sin(xy) with respect to x is cos(xy)*(y + x*y'), but the solution's intermediate step (
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.