The multivariable chain rule
Problem 10.392 · easy
Let \( \displaystyle z = \sin{\left(x y \right)} \) with \( \displaystyle x = s^{2} - t^{2} \), \( \displaystyle y = 2 s t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 1 \), \( \displaystyle t = 1 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sin{\left(x y \right)}\\\frac{\partial}{\partial y} \sin{\left(x y \right)}\end{matrix}\right] = \left[\begin{matrix}y \cos{\left(x y \right)}\\x \cos{\left(x y \right)}\end{matrix}\right] \]∂z/∂x and ∂z/∂y.✓ Proved
- ∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).
- \[ 4 \]Substitute.✓ Proved
Answer \( 4 \)
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: passqwen3.6:27b-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 values or the substitution process.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — 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 values or the substitution process.gpt-oss:20b: fail (error) 2026-10-06 — The solution omits the actual computation of ∂x/∂s, ∂y/∂s, and the evaluation at s=1, t=1. It merely states “4 = 4” without showing how the value 4 is obtained, so the setup and conclusion are incomplete and potentially misleading.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to compute or state 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 calculation of ∂x/∂s and ∂y/∂s, making the derivation incomplete and unverifiable.
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 2026-10-06 with SymPy 1.14.0.