Implicit differentiation
Problem 2.1277 · medium
The curve \( \displaystyle x^{2} y - 4 x + y^{3} = -3 \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ -3 \]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.
- \[ \frac{d}{d x} \left(x^{2} Y{\left(x \right)} - 4 x + Y^{3}{\left(x \right)}\right) = 2 x Y{\left(x \right)} + \left(x^{2} + 3 Y^{2}{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} - 4 \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- 2 x y + 4}{x^{2} + 3 y^{2}} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ 0 \]At the point.
Answer \( \frac{dy}{dx} = \frac{2 \left(- x y + 2\right)}{x^{2} + 3 y^{2}}, \quad \left.\frac{dy}{dx}\right|_{(2,1)} = 0 \)
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 (misleading) — Step 3 claims 'Every y term picks up a factor dy/dx', which is false for the term x^2*y (which requires the product rule, yielding a term without dy/dx). This incorrect heuristic could lead a student to make errors in other implicit differentiation problems.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-03 — Step 3 claims 'Every y term picks up a factor dy/dx', which is false for the term x^2*y (which requires the product rule, yielding a term without dy/dx). This incorrect heuristic could lead a student to make errors in other implicit differentiation problems.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The logic is sound and the result is correct.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The algebraic steps are verified by the CAS checks provided.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 2026-10-13 with SymPy 1.14.0.