The multivariable chain rule
Problem 10.389 · medium
Let \( \displaystyle z = \sqrt{x^{2} + y^{2} + 4} \) with \( \displaystyle x = s \cos{\left(t \right)} \), \( \displaystyle y = s \sin{\left(t \right)} \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 1 \), \( \displaystyle t = 0 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2} + 4}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2} + 4}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2} + 4}}\\\frac{y}{\sqrt{x^{2} + y^{2} + 4}}\end{matrix}\right] \]∂z/∂x and ∂z/∂y.✓ Proved
- ∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).
- \[ \frac{\sqrt{5}}{5} \]Substitute.✓ Proved
Answer \( \frac{\sqrt{5}}{5} \)
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 (error) — The solution omits the explicit computation of ∂x/∂s and ∂y/∂s, does not evaluate the chain‑rule expression at s=1, t=0, and ends with a tautological equality instead of a derived value. The steps are incomplete and could mislead a student into thinking the answer was simply asserted rather than derived.qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and the final substitution yields the correct value. Although intermediate steps are omitted, the logic is sound and the result is correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The solution omits the explicit computation of ∂x/∂s and ∂y/∂s, does not evaluate the chain‑rule expression at s=1, t=0, and ends with a tautological equality instead of a derived value. The steps are incomplete and could mislead a student into thinking the answer was simply asserted rather than derived.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the chain rule and the final substitution yields the correct value. Although intermediate steps are omitted, the logic is sound and the result is correct.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution skips the calculation of the partial derivatives of x and y with respect to s, and fails to substitute the specific values of x and y (derived from s=1, t=0) into the expressions for ∂z/∂x and ∂z/∂y. It presents the final numerical equality without showing the intermediate steps required to verify the chain rule application.
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.