Implicit differentiation
Problem 2.1274 · hard
The curve \( \displaystyle - x + 2 y + \sin{\left(x y \right)} = \sin{\left(1 \right)} + 1 \) passes through \( \displaystyle (1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ \sin{\left(1 \right)} + 1 \]The point is on the curve.✓ Proved
- Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.Reviewed
- \[ \frac{d}{d x} \left(- x + 2 Y{\left(x \right)} + \sin{\left(x Y{\left(x \right)} \right)}\right) = \left(x \cos{\left(x Y{\left(x \right)} \right)} + 2\right) \frac{d}{d x} Y{\left(x \right)} + Y{\left(x \right)} \cos{\left(x Y{\left(x \right)} \right)} - 1 \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- y \cos{\left(x y \right)} + 1}{x \cos{\left(x y \right)} + 2} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ \frac{1 - \cos{\left(1 \right)}}{\cos{\left(1 \right)} + 2} \]At the point.✓ Proved
Answer \( \frac{dy}{dx} = \frac{- y \cos{\left(x y \right)} + 1}{x \cos{\left(x y \right)} + 2}, \quad \left.\frac{dy}{dx}\right|_{(1,1)} = \frac{1 - \cos{\left(1 \right)}}{\cos{\left(1 \right)} + 2} \)
Lines: 4 proved, 1 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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: passqwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation, handles the chain rule for the mixed term sin(xy), and evaluates the result at the specified point. The algebraic steps are verified and the final answer matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies implicit differentiation, handles the chain rule for the mixed term sin(xy), and evaluates the result at the specified point. The algebraic steps are verified and the final answer matches the stated answer.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — The derivative of the term sin(xy) is incorrect in line 3. The product rule yields y*cos(xy) + x*y'*cos(xy), but the solution lists Y(x)*cos(x*Y(x)) - 1 as the non-derivative part, implying a sign error or incorrect application of the chain rule that leads to the wrong numerator in the final answer.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 2026-09-29 with SymPy 1.14.0.