Home›Calculus 3›Implicit differentiation with partial derivatives›Problem 10.359

Implicit differentiation with partial derivatives

Problem 10.359 · easy

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (1, 2) \) for the curve \( \displaystyle x^{2} y + y^{3} = 10 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} y + y^{3}\right)\\\frac{\partial}{\partial y} \left(x^{2} y + y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}2 x y\\x^{2} + 3 y^{2}\end{matrix}\right] \]
F_x and F_y.✓ Proved
\[ - \frac{4}{13} \]
dy/dx = −F_x/F_y at the point.✓ Proved

Answer \( - \frac{4}{13} \)

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 (misleading) — The solution skips the crucial step of substituting the point (1, 2) into the partial derivatives to obtain the numerical values -4 and 13. It presents the final arithmetic result without showing how the partial derivatives were evaluated, which obscures the application of the implicit differentiation formula dy/dx = -F_x/F_y.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (misleading) 2026-10-06 — The solution skips the crucial step of substituting the point (1, 2) into the partial derivatives to obtain the numerical values -4 and 13. It presents the final arithmetic result without showing how the partial derivatives were evaluated, which obscures the application of the implicit differentiation formula dy/dx = -F_x/F_y.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly state the formula for implicit differentiation (dy/dx = -F_x/F_y) or show the substitution of the point (1, 2) into the partial derivatives. It jumps from defining the partials to the final numerical answer without demonstrating the required calculus steps, making it impossible to verify the logic or learn the method.

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 2026-10-06 with SymPy 1.14.0.