Implicit differentiation with partial derivatives
Problem 10.464 · easy
Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (1, 1) \) for the curve \( \displaystyle x^{2} + x y^{2} = 2 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + x y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} + x y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x + y^{2}\\2 x y\end{matrix}\right] \]F_x and F_y.✓ Proved
- \[ - \frac{3}{2} \]dy/dx = −F_x/F_y at the point.✓ Proved
Answer \( - \frac{3}{2} \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ 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 | solved the equation numerically near the point and differenced |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (misleading) — The solution skips the crucial step of evaluating the partial derivatives at the point (1, 1) before applying the formula. It presents the final numerical equality without showing how the values -3 and 2 were derived from the expressions 2x + y^2 and 2xy, which obscures the method for a student.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (misleading) 2026-10-09 — The solution skips the crucial step of evaluating the partial derivatives at the point (1, 1) before applying the formula. It presents the final numerical equality without showing how the values -3 and 2 were derived from the expressions 2x + y^2 and 2xy, which obscures the method for a student.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the implicit function theorem formula dy/dx = -F_x/F_y. The partial derivatives are calculated correctly, and the evaluation at (1,1) yields the correct result.
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_partials, checked 2026-10-09 with SymPy 1.14.0.