Home›Calculus 3›Implicit differentiation with partial derivatives›Problem 10.362

Implicit differentiation with partial derivatives

Problem 10.362 · medium

The equation \( \displaystyle x^{2} z + y z^{2} = 1 \) defines \( \displaystyle z \) as a function of \( \displaystyle x \) and \( \displaystyle y \) near \( \displaystyle (-1, 2, -1) \). Find \( \displaystyle \frac{\partial z}{\partial x} \) there.

\[ 1 \]
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
\[ \frac{2}{3} \]
∂z/∂x = −F_x/F_z.✓ Proved

Answer \( \frac{2}{3} \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.

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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the Implicit Function Theorem formula ∂z/∂x = -F_x/F_z and verifies the point lies on the surface. The algebraic evaluation is correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the Implicit Function Theorem formula ∂z/∂x = -F_x/F_z and verifies the point lies on the surface. The algebraic evaluation is correct.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the Implicit Function Theorem. The partial derivatives F_x and F_z are calculated correctly, and the final evaluation at the point (-1, 2, -1) yields the correct result.

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 2026-10-06 with SymPy 1.14.0.