Implicit differentiation with partial derivatives
Problem 10.416 · medium
The equation \( \displaystyle y z + e^{x z} = 1 + e^{2} \) 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.
- \[ 1 + e^{2} \]The point is on the surface.✓ Proved
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(y z + e^{x z}\right)\\\frac{\partial}{\partial z} \left(y z + e^{x z}\right)\end{matrix}\right] = \left[\begin{matrix}z e^{x z}\\x e^{x z} + y\end{matrix}\right] \]F_x and F_z.✓ Proved
- \[ \frac{\left(-1\right) e^{2}}{1 + 2 e^{2}} = - \frac{e^{2}}{1 + 2 e^{2}} \]∂z/∂x = −F_x/F_z.✓ Proved
Answer \( - \frac{e^{2}}{1 + 2 e^{2}} \)
✓ 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: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the implicit function theorem formula ∂z/∂x = -F_x/F_z. The partial derivatives F_x and F_z are computed correctly, and the evaluation at the point (2, 1, 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 2026-10-07 with SymPy 1.14.0.