Implicit differentiation with partial derivatives
Problem 10.462 · medium
The equation \( \displaystyle x^{2} z + y z^{2} = 3 \) defines \( \displaystyle z \) as a function of \( \displaystyle x \) and \( \displaystyle y \) near \( \displaystyle (2, -1, 1) \). Find \( \displaystyle \frac{\partial z}{\partial x} \) there.
- \[ 3 \]The point is on the surface.✓ Proved
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} z + y z^{2}\right)\\\frac{\partial}{\partial z} \left(x^{2} z + y z^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x z\\x^{2} + 2 y z\end{matrix}\right] \]F_x and F_z.✓ Proved
- \[ -2 \]∂z/∂x = −F_x/F_z.✓ Proved
Answer \( -2 \)
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: pass — The solution correctly applies the Implicit Function Theorem formula ∂z/∂x = -F_x/F_z and evaluates it at the given point to obtain the correct result.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the Implicit Function Theorem formula ∂z/∂x = -F_x/F_z and evaluates it at the given point to obtain the correct result.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly substitute the point (2, -1, 1) into the partial derivatives F_x and F_z before applying the Implicit Function Theorem formula. While the final numerical answer is correct, the steps shown are algebraic identities that do not demonstrate the evaluation process required to find the derivative at a specific point.
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 2026-10-09 with SymPy 1.14.0.