Implicit differentiation with partial derivatives
Problem 10.315 · medium
The equation \( \displaystyle x^{2} + x z + y^{2} z = -2 \) defines \( \displaystyle z \) as a function of \( \displaystyle x \) and \( \displaystyle y \) near \( \displaystyle (2, 2, -1) \). Find \( \displaystyle \frac{\partial z}{\partial x} \) there.
- \[ -2 \]The point is on the surface.✓ Proved
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + x z + y^{2} z\right)\\\frac{\partial}{\partial z} \left(x^{2} + x z + y^{2} z\right)\end{matrix}\right] = \left[\begin{matrix}2 x + z\\x + y^{2}\end{matrix}\right] \]F_x and F_z.✓ Proved
- \[ - \frac{1}{2} \]∂z/∂x = −F_x/F_z.✓ Proved
Answer \( - \frac{1}{2} \)
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 the partial derivatives at the given point to obtain the correct result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Implicit Function Theorem formula ∂z/∂x = -F_x/F_z and evaluates the partial derivatives at the given point to obtain the correct result.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution applies the Implicit Function Theorem formula incorrectly. For F(x,y,z)=0, the correct formula is ∂z/∂x = -F_x/F_z. The solution computes F_x = 2x+z and F_z = x+y^2, but then claims the result is -1/2 without showing the substitution of the point (2,2,-1) into these expressions to get -F_x/F_z = -(3)/(6) = -1/2. While the final number is correct, the step labeled '∂z/∂x = −F_x/F_z' is presented as an equation '-1/2 = -1/2' which obscures the necessary evaluation step, making it misleading about how the result was derived from the partials. More critically, the notation in step 2 implies a matrix equality that doesn't directly yield the scalar derivative without the explicit division step, and step 3 skips the crucial evaluation of F_x and F_z at the point (2,2,-1). A student might think the ratio of the symbolic expressions is -1/2 identically, which is false.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.