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Implicit differentiation with partial derivatives

Problem 10.510 · medium

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (1, 2) \) for the curve \( \displaystyle x y + \sin{\left(x \right)} + \cos{\left(y \right)} = \cos{\left(2 \right)} + \sin{\left(1 \right)} + 2 \).

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

Answer \( \frac{\cos{\left(1 \right)} + 2}{-1 + \sin{\left(2 \right)}} \)

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 solution fails to explicitly substitute the point (1, 2) into the partial derivatives before applying the formula. While the final numerical result is correct, the intermediate step labeled 'dy/dx = −F_x/F_y at the point' is mathematically incorrect because it equates a symbolic expression (or an expression evaluated at an unspecified point) to a specific number without showing the substitution F_x(1,2) and F_y(1,2). A student might incorrectly believe that dy/dx is simply -F_x/F_y without evaluating the partials at the specific point.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly substitute the point (1, 2) into the partial derivatives before applying the formula. While the final numerical result is correct, the intermediate step labeled 'dy/dx = −F_x/F_y at the point' is mathematically incorrect because it equates a symbolic expression (or an expression evaluated at an unspecified point) to a specific number without showing the substitution F_x(1,2) and F_y(1,2). A student might incorrectly believe that dy/dx is simply -F_x/F_y without evaluating the partials at the specific point.
gpt-oss:20b: pass 2026-10-12
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12

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.