Home›Calculus 3›Implicit differentiation with partial derivatives›Problem 10.463

Implicit differentiation with partial derivatives

Problem 10.463 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(y^{2} + e^{x y}\right)\\\frac{\partial}{\partial y} \left(y^{2} + e^{x y}\right)\end{matrix}\right] = \left[\begin{matrix}y e^{x y}\\x e^{x y} + 2 y\end{matrix}\right] \]
F_x and F_y.✓ Proved
\[ \frac{1}{e \left(-2 + e^{-1}\right)} = \frac{1}{1 - 2 e} \]
dy/dx = −F_x/F_y at the point.✓ Proved

Answer \( \frac{1}{1 - 2 e} \)

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: fail (error) — The partial derivative with respect to x is calculated incorrectly; it should be y*exp(x*y) + 0, but the solution lists it as just y*exp(x*y) which is correct, wait. Let's re-read carefully. F(x,y) = y^2 + e^{xy} - (e^{-1} + 1) = 0. F_x = y * e^{xy}. F_y = 2y + x * e^{xy}. At (1, -1): F_x = (-1) * e^{1*(-1)} = -e^{-1}. F_y = 2(-1) + 1 * e^{-1} = -2 + e^{-1}. dy/dx = -F_x / F_y = -(-e^{-1}) / (-2 + e^{-1}) = e^{-1} / (e^{-1} - 2). The solution states: exp(-1)/(-2 + exp(-1)) = 1/(1 - 2*E). Let's check the algebra: e^{-1} / (e^{-1} - 2) = (1/e) / (1/e - 2) = (1/e) / ((1 - 2e)/e) = 1 / (1 - 2e). The algebraic simplification in line 2 is correct. The evaluation of the derivatives in line 1 is correct. Wait, look at line 1 again. Matrix([ [Derivative(y**2 + exp(x*y), x)], [Derivative(y**2 + exp(x*y), y)] ]) = Matrix([ [y*exp(x*y)], [x*exp(x*y) + 2*y] ]) Derivative of y^2 w.r.t x is 0. Derivative of e^{xy} w.r.t x is y*e^{xy}. So F_x = y*e^{xy}. Correct. Derivative of y^2 w.r.t y is 2y. Derivative of e^{xy} w.r.t y is x*e^{xy}. So F_y = 2y + x*e^{xy}. Correct. The setup is correct. The calculation is correct. The final answer is correct. Why did I think it might be wrong? Let me double check the sign. dy/dx = - F_x / F_y. F_x(1, -1) = -1 * e^{-1} = -1/e. F_y(1, -1) = 2(-1) + 1 * e^{-1} = -2 + 1/e. -dy/dx = - (-1/e) / (-2 + 1/e) = (1/e) / (1/e - 2). Multiply numerator and denominator by e: 1 / (1 - 2e). The solution is correct. I will mark it as pass.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The partial derivative with respect to x is calculated incorrectly; it should be y*exp(x*y) + 0, but the solution lists it as just y*exp(x*y) which is correct, wait. Let's re-read carefully. F(x,y) = y^2 + e^{xy} - (e^{-1} + 1) = 0. F_x = y * e^{xy}. F_y = 2y + x * e^{xy}. At (1, -1): F_x = (-1) * e^{1*(-1)} = -e^{-1}. F_y = 2(-1) + 1 * e^{-1} = -2 + e^{-1}. dy/dx = -F_x / F_y = -(-e^{-1}) / (-2 + e^{-1}) = e^{-1} / (e^{-1} - 2). The solution states: exp(-1)/(-2 + exp(-1)) = 1/(1 - 2*E). Let's check the algebra: e^{-1} / (e^{-1} - 2) = (1/e) / (1/e - 2) = (1/e) / ((1 - 2e)/e) = 1 / (1 - 2e). The algebraic simplification in line 2 is correct. The evaluation of the derivatives in line 1 is correct. Wait, look at line 1 again. Matrix([ [Derivative(y**2 + exp(x*y), x)], [Derivative(y**2 + exp(x*y), y)] ]) = Matrix([ [y*exp(x*y)], [x*exp(x*y) + 2*y] ]) Derivative of y^2 w.r.t x is 0. Derivative of e^{xy} w.r.t x is y*e^{xy}. So F_x = y*e^{xy}. Correct. Derivative of y^2 w.r.t y is 2y. Derivative of e^{xy} w.r.t y is x*e^{xy}. So F_y = 2y + x*e^{xy}. Correct. The setup is correct. The calculation is correct. The final answer is correct. Why did I think it might be wrong? Let me double check the sign. dy/dx = - F_x / F_y. F_x(1, -1) = -1 * e^{-1} = -1/e. F_y(1, -1) = 2(-1) + 1 * e^{-1} = -2 + 1/e. -dy/dx = - (-1/e) / (-2 + 1/e) = (1/e) / (1/e - 2). Multiply numerator and denominator by e: 1 / (1 - 2e). The solution is correct. I will mark it as pass.
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The partial derivative with respect to x is incorrect; the product rule yields exp(xy) + xy*exp(xy), not just y*exp(xy). Consequently, the final answer is wrong.

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 2026-10-09 with SymPy 1.14.0.