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The multivariable chain rule

Problem 10.390 · medium

Let \( \displaystyle z = \sqrt{x^{2} + y^{2} + 4} \) with \( \displaystyle x = s^{2} - t^{2} \), \( \displaystyle y = 2 s t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 2 \), \( \displaystyle t = 0 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2} + 4}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2} + 4}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2} + 4}}\\\frac{y}{\sqrt{x^{2} + y^{2} + 4}}\end{matrix}\right] \]
∂z/∂x and ∂z/∂y.✓ Proved
∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).
\[ \frac{8 \sqrt{5}}{5} \]
Substitute.✓ Proved

Answer \( \frac{8 \sqrt{5}}{5} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution skips the calculation of the partial derivatives of x and y with respect to s, and fails to evaluate the intermediate terms at the specific point (s=2, t=0) before substitution. While the final answer is correct, the step labeled 'Substitute' is mathematically unjustified as presented because it does not show the derivation of the value 8*sqrt(5)/5 from the chain rule formula and the given functions.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-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 evaluate the intermediate terms at the specific point (s=2, t=0) before substitution. While the final answer is correct, the step labeled 'Substitute' is mathematically unjustified as presented because it does not show the derivation of the value 8*sqrt(5)/5 from the chain rule formula and the given functions.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-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 evaluate the intermediate terms at the given point (s=2, t=0) before substitution. It presents the final answer without showing the necessary chain rule application steps.

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.