Implicit differentiation with partial derivatives
Problem 10.314 · medium
Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (1, -1) \) for the curve \( \displaystyle x y + \sin{\left(x \right)} + \cos{\left(y \right)} = -1 + \cos{\left(1 \right)} + \sin{\left(1 \right)} \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x y + \sin{\left(x \right)} + \cos{\left(y \right)}\right)\\\frac{\partial}{\partial y} \left(x y + \sin{\left(x \right)} + \cos{\left(y \right)}\right)\end{matrix}\right] = \left[\begin{matrix}y + \cos{\left(x \right)}\\x - \sin{\left(y \right)}\end{matrix}\right] \]F_x and F_y.✓ Proved
- \[ \frac{1 - \cos{\left(1 \right)}}{\sin{\left(1 \right)} + 1} \]dy/dx = −F_x/F_y at the point.✓ Proved
Answer \( \frac{1 - \cos{\left(1 \right)}}{\sin{\left(1 \right)} + 1} \)
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 (1, -1) before applying the formula. It incorrectly uses the symbolic expressions for F_x and F_y in the final ratio instead of their numerical values at the given point.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to evaluate the partial derivatives at the point (1, -1) before applying the formula. It incorrectly uses the symbolic expressions for F_x and F_y in the final ratio instead of their numerical values at the given point.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the implicit function theorem formula dy/dx = -F_x/F_y. The partial derivatives are computed correctly, and the evaluation at the point (1, -1) yields the stated answer.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.