Home›Calculus 3›Implicit differentiation with partial derivatives›Problem 10.418

Implicit differentiation with partial derivatives

Problem 10.418 · easy

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (-1, 2) \) for the curve \( \displaystyle x^{3} - 3 x y + y^{3} = 13 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{3} - 3 x y + y^{3}\right)\\\frac{\partial}{\partial y} \left(x^{3} - 3 x y + y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}3 x^{2} - 3 y\\- 3 x + 3 y^{2}\end{matrix}\right] \]
F_x and F_y.✓ Proved
\[ 1 \cdot \frac{1}{5} = \frac{1}{5} \]
dy/dx = −F_x/F_y at the point.✓ Proved

Answer \( \frac{1}{5} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to evaluate the partial derivatives at the specific point (-1, 2) before applying the formula. It presents the final numerical answer without showing the substitution step, making it impossible to verify that the calculation was performed correctly or that the Implicit Function Theorem conditions were checked.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to evaluate the partial derivatives at the specific point (-1, 2) before applying the formula. It presents the final numerical answer without showing the substitution step, making it impossible to verify that the calculation was performed correctly or that the Implicit Function Theorem conditions were checked.
gpt-oss:20b: fail (misleading) 2026-10-07 — The solution states the derivative formula and immediately gives 1/5 = 1/5 without evaluating F_x and F_y at (-1,2). It omits the necessary calculation, so a student could be misled into thinking the result is obvious without justification.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly evaluate the partial derivatives at the point (-1, 2) to justify the result. It jumps from the general symbolic expressions for F_x and F_y directly to the final numerical answer without showing the substitution step, which is the core of the problem.

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 2026-10-07 with SymPy 1.14.0.