Implicit differentiation
Problem 2.1164 · medium
The curve \( \displaystyle x^{2} + 2 x y + 4 y^{2} = 1 \) passes through \( \displaystyle (1, 0) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ 1 \]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.
- \[ \frac{d}{d x} \left(x^{2} + 2 x Y{\left(x \right)} + 4 Y^{2}{\left(x \right)}\right) = 2 x + \left(2 x + 8 Y{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} + 2 Y{\left(x \right)} \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- 2 x - 2 y}{2 x + 8 y} = \frac{- x - y}{x + 4 y} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ -1 \]At the point.
Answer \( \frac{dy}{dx} = \frac{- x - y}{x + 4 y}, \quad \left.\frac{dy}{dx}\right|_{(1,0)} = -1 \)
Lines: 2 proved, 3 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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: passqwen3.6:27b-mlx: fail (error) — The solution claims 'Every y term picks up a factor dy/dx' in step 3, which is mathematically incorrect. The term 2xy requires the product rule, yielding 2y + 2x(dy/dx), not just a factor of dy/dx applied to the original term. While the final algebraic result is correct, the stated reasoning is false and would teach a student incorrect differentiation rules.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution claims 'Every y term picks up a factor dy/dx' in step 3, which is mathematically incorrect. The term 2xy requires the product rule, yielding 2y + 2x(dy/dx), not just a factor of dy/dx applied to the original term. While the final algebraic result is correct, the stated reasoning is false and would teach a student incorrect differentiation rules.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies implicit differentiation and evaluates the derivative at the specified point. The intermediate steps are algebraically sound and the final answer is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies implicit differentiation, handles the product rule for the xy term, and evaluates the derivative at the given point.gpt-oss:20b: pass 2026-09-28
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.