The multivariable chain rule
Problem 10.448 · easy
Let \( \displaystyle z = x y^{2} + 3 x \) with \( \displaystyle x = s + 2 t \), \( \displaystyle y = s - t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 1 \), \( \displaystyle t = 1 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x y^{2} + 3 x\right)\\\frac{\partial}{\partial y} \left(x y^{2} + 3 x\right)\end{matrix}\right] = \left[\begin{matrix}y^{2} + 3\\2 x y\end{matrix}\right] \]∂z/∂x and ∂z/∂y.✓ Proved
- ∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).
- \[ 3 \]Substitute.✓ Proved
Answer \( 3 \)
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: fail (misleading) — The solution skips the explicit substitution of the partial derivatives of x and y with respect to s and the evaluation at s=1,t=1. The final line “3 = 3” does not show how the value 3 was obtained, which could mislead a student into thinking the result is trivial or that the chain rule was applied automatically without computation.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 by the chain rule formula in step 2. It jumps directly to the final answer without showing the intermediate calculation.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-08 — The solution skips the explicit substitution of the partial derivatives of x and y with respect to s and the evaluation at s=1,t=1. The final line “3 = 3” does not show how the value 3 was obtained, which could mislead a student into thinking the result is trivial or that the chain rule was applied automatically without computation.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 by the chain rule formula in step 2. It jumps directly to the final answer without showing the intermediate calculation.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-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 intermediate derivatives at the specified point (s=1, t=1). It simply asserts the final answer without showing the necessary chain rule calculations.
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.