Implicit differentiation with partial derivatives
Problem 10.415 · medium
The equation \( \displaystyle x^{2} + y^{2} + z^{2} = 3 \) defines \( \displaystyle z \) as a function of \( \displaystyle x \) and \( \displaystyle y \) near \( \displaystyle (-1, -1, 1) \). Find \( \displaystyle \frac{\partial z}{\partial x} \) there.
- \[ 3 \]The point is on the surface.✓ Proved
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} + z^{2}\right)\\\frac{\partial}{\partial z} \left(x^{2} + y^{2} + z^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 z\end{matrix}\right] \]F_x and F_z.✓ Proved
- \[ 1 \]∂z/∂x = −F_x/F_z.✓ Proved
Answer \( 1 \)
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 |
| 3 | ✓ 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 substitute the point (-1, -1, 1) into the derived formula -F_x/F_z. It asserts the final answer is 1 without showing the calculation -2x/2z evaluated at the point, which yields -(-1)/1 = 1. While the final number is correct, the step is logically incomplete and skips the essential evaluation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to substitute the point (-1, -1, 1) into the derived formula -F_x/F_z. It asserts the final answer is 1 without showing the calculation -2x/2z evaluated at the point, which yields -(-1)/1 = 1. While the final number is correct, the step is logically incomplete and skips the essential evaluation.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 specific point (-1, -1, 1), resulting in a generic expression rather than the required numerical value. Additionally, the final line asserts the answer is 1 without showing the substitution of x=-1 and z=1 into -F_x/F_z.
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.