Home›Calculus 3›Implicit differentiation with partial derivatives›Problem 10.310

Implicit differentiation with partial derivatives

Problem 10.310 · easy

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (2, 1) \) for the curve \( \displaystyle x^{2} y + y^{3} = 5 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} y + y^{3}\right)\\\frac{\partial}{\partial y} \left(x^{2} y + y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}2 x y\\x^{2} + 3 y^{2}\end{matrix}\right] \]
F_x and F_y.✓ Proved
\[ - \frac{4}{7} \]
dy/dx = −F_x/F_y at the point.✓ Proved

Answer \( - \frac{4}{7} \)

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 evaluating the partial derivatives at the point (2, 1) before applying the formula. It presents the final numerical equality without showing the substitution, which fails to demonstrate the method requested ('Use partial derivatives...').

Every verdict on record (4)

qwen3.6:27b-mlx: fail (misleading) 2026-10-04 — The solution skips the crucial step of evaluating the partial derivatives at the point (2, 1) before applying the formula. It presents the final numerical equality without showing the substitution, which fails to demonstrate the method requested ('Use partial derivatives...').
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly evaluate the partial derivatives at the point (2, 1) before applying the formula. While the final numerical answer is correct, the step jumping from the symbolic expressions to the final value without showing the substitution is logically incomplete and skips the core calculation required by the problem statement.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.