Implicit differentiation with partial derivatives
Problem 10.365 · easy
Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (0, 1) \) for the curve \( \displaystyle y^{2} + e^{x y} = 2 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(y^{2} + e^{x y}\right)\\\frac{\partial}{\partial y} \left(y^{2} + e^{x y}\right)\end{matrix}\right] = \left[\begin{matrix}y e^{x y}\\x e^{x y} + 2 y\end{matrix}\right] \]F_x and F_y.✓ Proved
- \[ - \frac{1}{2} \]dy/dx = −F_x/F_y at the point.✓ Proved
Answer \( - \frac{1}{2} \)
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 y*exp(x*y) to F_x instead of F_y, leading to a swapped numerator and denominator in the implicit differentiation formula.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — 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 y*exp(x*y) to F_x instead of F_y, leading to a swapped numerator and denominator in the implicit differentiation formula.qwen3.6:27b-mlx: fail (error) 2026-10-06 — 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 in the context of implicit differentiation, or rather, F_x should be y*exp(x*y) but the setup implies F_x is the derivative of the LHS w.r.t x treating y as constant? No, F_x is partial derivative. Let's re-evaluate. F(x,y) = y^2 + e^{xy} - 2 = 0. F_x = y e^{xy}. F_y = 2y + x e^{xy}. At (0,1), F_x = 1*e^0 = 1. F_y = 2(1) + 0 = 2. dy/dx = -F_x/F_y = -1/2. The equation in line 1 says F_x = y*exp(x*y). This is correct for partial derivative. F_y = x*exp(x*y) + 2*y. This is correct. The calculation in line 2 is correct. Why did I think it was wrong? Ah, the prompt says 'unchecked' for equations. I need to check the sentences. There are no sentences. Wait, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 1 is an equation. Line 2 is an equation. There are no sentences to review? Or is the 'Stated answer' part of the solution? The solution provided is just two equations. If there are no sentences, I cannot find a sentence error. However, usually these prompts have text. Let me look closer. '1. [equation, unchecked] ... 2. [equation, unchecked] ...'. There are no sentences. If there are no sentences, can I fail it? The instructions say 'The sentences are what you are here for.' If there are no sentences, perhaps it passes by default? Or is the lack of explanation a style issue? But the verdict options are pass/fail. If it's correct, it's pass. The math is correct. F_x = y e^{xy}. F_y = 2y + x e^{xy}. dy/dx = -F_x/F_y. At (0,1), -1/2. The solution is correct. But wait, is the first equation actually correct? Derivative(y**2 + exp(x*y), x) treating y as constant is y*exp(x*y). Correct. Derivative(y**2 + exp(x*y), y) treating x as constant is 2y + x*exp(x*y). Correct. The solution is mathematically sound. However, often these 'adversarial' reviews catch subtle errors. Is there one? No. The solution is correct. But the format is extremely sparse. Is 'style' a fail? Yes. 'style (correct, but a sentence could be clearer)'. But there are no sentences. This is a bit of a trick. If I must find something wrong, maybe the lack of context? But the instructions say 'Check that... each conclusion follows...'. It does. I will mark it pass because the math is correct and there are no sentences to critique for logical flow or false statements.gpt-oss:20b: pass 2026-10-06
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.