Implicit differentiation with partial derivatives
Problem 10.316 · medium
Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (1, 2) \) for the curve \( \displaystyle y^{2} + e^{x y} = 4 + e^{2} \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(y^{2} + e^{x y}\right)\\\frac{\partial}{\partial y} \left(y^{2} + e^{x y}\right)\end{matrix}\right] = \left[\begin{matrix}y e^{x y}\\x e^{x y} + 2 y\end{matrix}\right] \]F_x and F_y.✓ Proved
- \[ - \frac{2 e^{2}}{4 + e^{2}} \]dy/dx = −F_x/F_y at the point.✓ Proved
Answer \( - \frac{2 e^{2}}{4 + e^{2}} \)
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: passqwen3.6:27b-mlx: fail (error) — The partial derivative with respect to x is incorrect; the derivative of y^2 with respect to x is 0, not y*exp(x*y). The solution incorrectly attributes the term from the exponential to the y^2 term or conflates the derivatives.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The partial derivative with respect to x is incorrect; the derivative of y^2 with respect to x is 0, not y*exp(x*y). The solution incorrectly attributes the term from the exponential to the y^2 term or conflates the derivatives.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The partial derivative with respect to x is calculated incorrectly; the term 2y*y' is missing from the differentiation of y^2 with respect to x, and the product rule for e^(xy) yields y*e^(xy) + x*y'*e^(xy), not just y*e^(xy). The setup in line 1 is fundamentally wrong.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.