Implicit differentiation
Problem 2.1472 · medium
The curve \( \displaystyle 2 x^{2} + x y + 3 y^{2} = 13 \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ 13 \]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
- \[ \frac{d}{d x} \left(2 x^{2} + x Y{\left(x \right)} + 3 Y^{2}{\left(x \right)}\right) = 4 x + \left(x + 6 Y{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} + Y{\left(x \right)} \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- 4 x - y}{x + 6 y} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ - \frac{9}{8} \]At the point.✓ Proved
Answer \( \frac{dy}{dx} = \frac{- 4 x - y}{x + 6 y}, \quad \left.\frac{dy}{dx}\right|_{(2,1)} = - \frac{9}{8} \)
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 and evaluates the derivative at the specified point. The intermediate steps are algebraically sound and the final result is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies implicit differentiation and evaluates the derivative at the specified point. The intermediate steps are algebraically sound and the final result is correct.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (misleading) 2026-10-03 — Step 3 claims that 'Every y term picks up a factor dy/dx', which is mathematically false (e.g., the derivative of 3y^2 is 6y dy/dx, not 3y^2 dy/dx). This incorrect reasoning leads to the wrong intermediate expression, even though the final answer happens to be correct due to a cancellation of errors or coincidence in the specific algebraic manipulation shown in Step 4.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.