Implicit differentiation
Problem 2.1045 · medium
The curve \( \displaystyle - x + 2 y + \sin{\left(x y \right)} = 2 \) passes through \( \displaystyle (0, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ 2 \]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
- \[ 0 \]At the point.✓ Proved
Answer \( \frac{dy}{dx} = \frac{- y \cos{\left(x y \right)} + 1}{x \cos{\left(x y \right)} + 2}, \quad \left.\frac{dy}{dx}\right|_{(0,1)} = 0 \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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, solves for dy/dx, and evaluates it at the given point. The steps are logically sound and the final answer is correct.
Every verdict on record (4)
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 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies implicit differentiation, including the product rule for the sin(xy) term, and accurately evaluates the derivative at the given point.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.