Home›Calculus 3›Implicit differentiation with partial derivatives›Problem 10.317

Implicit differentiation with partial derivatives

Problem 10.317 · 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
\[ \frac{\left(-1\right) e}{2 + e} = - \frac{e}{2 + e} \]
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: fail (error) — The partial derivatives in step 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 2*y, not x*exp(x*y) + 2*y. Consequently, the formula for dy/dx is applied to wrong expressions.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The partial derivatives in step 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 2*y, not x*exp(x*y) + 2*y. Consequently, the formula for dy/dx is applied to wrong expressions.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The partial derivative with respect to x is incorrect; the term y^2 has derivative 0 with respect to x, but the solution lists y*exp(x*y) as the total derivative or confuses the components. Specifically, F_x should be y*exp(x*y) and F_y should be 2*y + x*exp(x*y). The matrix shown in line 1 has the entries swapped or mislabeled relative to standard F_x, F_y ordering, and the text claims F_x is the first component which is y*exp(x*y), but the matrix shows the first component as y*exp(x*y) and second as x*exp(x*y) + 2*y. Wait, let's look closer. F = y^2 + e^{xy} - 1 - e = 0. F_x = y e^{xy}. F_y = 2y + x e^{xy}. The matrix in line 1 is [[y e^{xy}], [x e^{xy} + 2y]]. This matches [F_x, F_y]^T. So the matrix is correct. However, the label says "F_x and F_y". The second line uses -E/(2+E). At (1,1), F_x = 1*e^1 = e. F_y = 2(1) + 1*e^1 = 2+e. dy/dx = -F_x/F_y = -e/(2+e). The math is actually correct. Why did I think it was wrong? Let me re-read carefully. Line 1: Matrix([ [Derivative(..., x)], [Derivative(..., y)] ]) = Matrix([ [y*exp(x*y)], [x*exp(x*y) + 2*y] ]). This is correct. Line 2: -E/(2+E) = -E/(2+E). This is a tautology checking the final value. The solution is correct. Wait, is there a subtle error? The problem asks to use partial derivatives. The solution defines F_x and F_y correctly. It evaluates them correctly. The formula dy/dx = -F_x/F_y is correct. The evaluation at (1,1) gives -e/(2+e). The stated answer is -e/(2+e). Everything seems correct. Is there a style issue? The solution is very brief. But it is not wrong. Let me check the prompt again. "Find what is wrong". If nothing is wrong, pass. But usually these have a trick. Let's look at the matrix again. Derivative(y**2 + exp(x*y), x) is y*exp(x*y). Correct. Derivative(y**2 + exp(x*y), y) is 2*y + x*exp(x*y). Correct. The matrix is [[F_x], [F_y]]. The text says "F_x and F_y". This is consistent. The second line just asserts the final answer. It doesn't show the substitution step explicitly, but it's not incorrect. It's just a check. I will mark it as pass.
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.