Implicit differentiation
Problem 2.1473 · medium
The curve \( \displaystyle 4 x^{2} - 2 x y + y^{2} = 4 \) passes through \( \displaystyle (1, 0) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ 4 \]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
- \[ \frac{d}{d x} \left(4 x^{2} - 2 x Y{\left(x \right)} + Y^{2}{\left(x \right)}\right) = 8 x + \left(- 2 x + 2 Y{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} - 2 Y{\left(x \right)} \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- 8 x + 2 y}{- 2 x + 2 y} = \frac{4 x - y}{x - y} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ 4 \]At the point.✓ Proved
Answer \( \frac{dy}{dx} = \frac{4 x - y}{x - y}, \quad \left.\frac{dy}{dx}\right|_{(1,0)} = 4 \)
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, solves for dy/dx, and evaluates it at the given point. The intermediate steps are algebraically sound and the logic is clear.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The intermediate steps are algebraically sound and the logic is clear.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — The derivative calculation in line 3 is incorrect; the product rule term for -2xy should yield -2y - 2x(dy/dx), but the solution incorrectly writes -2y + (-2x + 2y)(dy/dx). Consequently, the final formula for dy/dx is wrong.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.