Implicit differentiation with partial derivatives
Problem 10.412 · easy
Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (0, 2) \) for the curve \( \displaystyle x y + \sin{\left(x \right)} + \cos{\left(y \right)} = \cos{\left(2 \right)} \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x y + \sin{\left(x \right)} + \cos{\left(y \right)}\right)\\\frac{\partial}{\partial y} \left(x y + \sin{\left(x \right)} + \cos{\left(y \right)}\right)\end{matrix}\right] = \left[\begin{matrix}y + \cos{\left(x \right)}\\x - \sin{\left(y \right)}\end{matrix}\right] \]F_x and F_y.✓ Proved
- \[ \frac{3}{\sin{\left(2 \right)}} \]dy/dx = −F_x/F_y at the point.✓ Proved
Answer \( \frac{3}{\sin{\left(2 \right)}} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to evaluate the partial derivatives at the point (0, 2) before applying the implicit differentiation formula. Specifically, F_x(0,2) = 2 and F_y(0,2) = -sin(2), so dy/dx should be -2/(-sin(2)) = 2/sin(2), not 3/sin(2).
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to evaluate the partial derivatives at the point (0, 2) before applying the implicit differentiation formula. Specifically, F_x(0,2) = 2 and F_y(0,2) = -sin(2), so dy/dx should be -2/(-sin(2)) = 2/sin(2), not 3/sin(2).gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to verify that the point (0, 2) actually lies on the curve. Substituting x=0 and y=2 into the equation yields cos(2) = cos(2), which is true, so the point is on the curve. However, the calculation of dy/dx is presented as a trivial identity '3/sin(2) = 3/sin(2)' without showing the substitution of the point into the partial derivatives. Specifically, F_x(0,2) = 2 + cos(0) = 3 and F_y(0,2) = 0 - sin(2) = -sin(2), so dy/dx = -3/(-sin(2)) = 3/sin(2). While the final answer is correct, the solution line 2 is logically incomplete as it does not demonstrate the evaluation step, merely stating the result equals itself. More critically, the prompt asks to review if the setup models the problem and if conclusions follow. The jump from line 1 to line 2 skips the crucial step of evaluating the partials at the specific point and applying the implicit differentiation formula dy/dx = -F_x/F_y. It asserts the result without derivation. This is misleading because it hides the actual work.
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 2026-10-07 with SymPy 1.14.0.