Home›Calculus 3›Implicit differentiation with partial derivatives›Problem 10.513

Implicit differentiation with partial derivatives

Problem 10.513 · easy

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (0, 1) \) for the curve \( \displaystyle x^{3} - 3 x y + y^{3} = 1 \).

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

Answer \( 1 \)

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 (0, 1) before computing the ratio, resulting in a tautology (1=1) rather than the correct value of dy/dx. The correct calculation requires substituting x=0 and y=1 into F_x and F_y.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to evaluate the partial derivatives at the specific point (0, 1) before computing the ratio, resulting in a tautology (1=1) rather than the correct value of dy/dx. The correct calculation requires substituting x=0 and y=1 into F_x and F_y.
gpt-oss:20b: pass 2026-10-12
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to evaluate the partial derivatives at the specific point (0, 1) and does not perform the division to calculate the final value. It merely states the formula and an identity, leaving the actual computation of dy/dx incomplete.

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 2026-10-12 with SymPy 1.14.0.