Home›Calculus 3›Implicit differentiation with partial derivatives›Problem 10.509

Implicit differentiation with partial derivatives

Problem 10.509 · medium

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

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

Answer \( - \frac{e}{2 + e} \)

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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The partial derivatives in step 1 are incorrect: F_x should be y*e^(xy) and F_y should be 2y + x*e^(xy), but the solution lists them as y*e^(xy) and

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The partial derivatives in step 1 are incorrect: F_x should be y*e^(xy) and F_y should be 2y + x*e^(xy), but the solution lists them as y*e^(xy) and
gpt-oss:20b: pass 2026-10-12
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The partial derivatives in line 1 are incorrect; the derivative of y^2 with respect to x is 0, not y*exp(x*y), and the derivative with respect to y is 2y, not x*exp(x*y) + 2y. The terms are swapped and mixed up.

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.