The multivariable chain rule
Problem 10.345 · medium
Let \( \displaystyle z = x \cos{\left(y \right)} + y e^{x} \) with \( \displaystyle x = s t \), \( \displaystyle y = s + t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 1 \), \( \displaystyle t = 1 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x \cos{\left(y \right)} + y e^{x}\right)\\\frac{\partial}{\partial y} \left(x \cos{\left(y \right)} + y e^{x}\right)\end{matrix}\right] = \left[\begin{matrix}y e^{x} + \cos{\left(y \right)}\\- x \sin{\left(y \right)} + e^{x}\end{matrix}\right] \]∂z/∂x and ∂z/∂y.✓ Proved
- ∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).Reviewed
- \[ - \sin{\left(2 \right)} + \cos{\left(2 \right)} + 3 e = \sqrt{2} \cos{\left(\frac{\pi}{4} + 2 \right)} + 3 e \]Substitute.✓ Proved
Answer \( \sqrt{2} \cos{\left(\frac{\pi}{4} + 2 \right)} + 3 e \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 | substituted first, then differenced numerically |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the multivariable chain rule and the algebraic simplification is verified. The intermediate step of evaluating partials at s=1, t=1 (implied by the substitution in line 3) is consistent with the problem statement.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the multivariable chain rule and the algebraic simplification is verified. The intermediate step of evaluating partials at s=1, t=1 (implied by the substitution in line 3) is consistent with the problem statement.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the chain rule and the algebraic simplification in line 3 is valid.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/multivariable_chain_rule, checked 2026-10-04 with SymPy 1.14.0.