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The multivariable chain rule

Problem 10.492 · easy

Let \( \displaystyle z = x e^{3 y} \) with \( \displaystyle x = s^{2} - t^{2} \), \( \displaystyle y = 2 s t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 1 \), \( \displaystyle t = 1 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} x e^{3 y}\\\frac{\partial}{\partial y} x e^{3 y}\end{matrix}\right] = \left[\begin{matrix}e^{3 y}\\3 x e^{3 y}\end{matrix}\right] \]
∂z/∂x and ∂z/∂y.✓ Proved
∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).
Reviewed
\[ 2 e^{6} \]
Substitute.✓ Proved

Answer \( 2 e^{6} \)

Lines: 2 proved, 1 reviewed. 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	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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the multivariable chain rule and the final substitution yields the correct result.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies the multivariable chain rule and the final substitution yields the correct result.
gpt-oss:20b: pass 2026-10-10
gpt-oss:20b: fail (misleading) 2026-10-10 — The solution skips the step showing that the ∂y/∂s term vanishes because x=0 at (s,t)=(1,1). A student could incorrectly assume the second term contributes, so the reasoning is incomplete.
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, which are necessary to apply the chain rule. It jumps directly to the final numerical answer without showing the intermediate values or the substitution process.

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.