The multivariable chain rule
Problem 10.339 · easy
Let \( \displaystyle z = x^{2} - 3 x y - 3 y^{2} \) with \( \displaystyle x = s^{2} - t^{2} \), \( \displaystyle y = 2 s t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 2 \), \( \displaystyle t = 0 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} - 3 x y - 3 y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} - 3 x y - 3 y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x - 3 y\\- 3 x - 6 y\end{matrix}\right] \]∂z/∂x and ∂z/∂y.✓ Proved
- ∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).
- \[ 32 \]Substitute.✓ Proved
Answer \( 32 \)
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 (style) — The solution skips the explicit computation of ∂x/∂s and ∂y/∂s and the substitution into the chain‑rule formula, so a reader cannot verify the final result. The reasoning is correct in principle, but the missing steps make the explanation incomplete.qwen3.6:27b-mlx: fail (error) — The solution fails to compute or state the partial derivatives of x and y with respect to s, nor does it evaluate x and y at the given point (s=2, t=0). It jumps directly to the final numerical result without showing the necessary intermediate steps required by the chain rule.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute or state the partial derivatives of x and y with respect to s, nor does it evaluate x and y at the given point (s=2, t=0). It jumps directly to the final numerical result without showing the necessary intermediate steps required by the chain rule.gpt-oss:20b: fail (style) 2026-10-04 — The solution skips the explicit computation of ∂x/∂s and ∂y/∂s and the substitution into the chain‑rule formula, so a reader cannot verify the final result. The reasoning is correct in principle, but the missing steps make the explanation incomplete.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute the partial derivatives of x and y with respect to s, nor does it evaluate x and y at the given point (s=2, t=0). It jumps directly to the final numerical answer without showing the necessary intermediate steps or verifying the chain rule application.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.