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The multivariable chain rule

Problem 10.496 · medium

Let \( \displaystyle z = x \cos{\left(y \right)} + y e^{x} \) with \( \displaystyle x = s + 2 t \), \( \displaystyle y = s - t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 2 \), \( \displaystyle t = 1 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x \cos{\left(y \right)} + y e^{x}\right)\\\frac{\partial}{\partial y} \left(x \cos{\left(y \right)} + y e^{x}\right)\end{matrix}\right] = \left[\begin{matrix}y e^{x} + \cos{\left(y \right)}\\- x \sin{\left(y \right)} + e^{x}\end{matrix}\right] \]
∂z/∂x and ∂z/∂y.✓ Proved
∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).
\[ - 4 \sin{\left(1 \right)} + \cos{\left(1 \right)} + 2 e^{4} \]
Substitute.✓ Proved

Answer \( - 4 \sin{\left(1 \right)} + \cos{\left(1 \right)} + 2 e^{4} \)

Lines: 2 proved, 1 not checked. 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	Not checked	—	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: pass
qwen3.6:27b-mlx: fail (error) — The solution skips the calculation of the partial derivatives of x and y with respect to s, and fails to evaluate the partial derivatives of z with respect to x and y at the specific point (x=3, y=1) corresponding to s=2, t=1. It jumps directly from the chain rule formula to the final numerical answer without showing the intermediate substitution steps.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution skips the calculation of the partial derivatives of x and y with respect to s, and fails to evaluate the partial derivatives of z with respect to x and y at the specific point (x=3, y=1) corresponding to s=2, t=1. It jumps directly from the chain rule formula to the final numerical answer without showing the intermediate substitution steps.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies the multivariable chain rule and the final numerical evaluation is correct.

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 2026-10-10 with SymPy 1.14.0.