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The multivariable chain rule

Problem 10.446 · easy

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} y - y^{3}\right)\\\frac{\partial}{\partial y} \left(x^{2} y - y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}2 x y\\x^{2} - 3 y^{2}\end{matrix}\right] \]
∂z/∂x and ∂z/∂y.✓ Proved
∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).
\[ 9 \]
Substitute.✓ Proved

Answer \( 9 \)

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 fails to compute or state the partial derivatives of x and y with respect to s (∂x/∂s and ∂y/∂s), which are required for the chain rule application in step 2. It jumps directly to the final numerical answer without showing the intermediate evaluation of the partials or the substitution process.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to compute or state the partial derivatives of x and y with respect to s (∂x/∂s and ∂y/∂s), which are required for the chain rule application in step 2. It jumps directly to the final numerical answer without showing the intermediate evaluation of the partials or the substitution process.
gpt-oss:20b: pass 2026-10-08
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to compute the partial derivatives of x and y with respect to s, nor does it evaluate the partials of z with respect to x and y at the specific point (s=1, t=1). It jumps directly to the final numerical answer without showing the necessary intermediate steps or verifying the values used in the chain rule.

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 2026-10-08 with SymPy 1.14.0.