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The multivariable chain rule

Problem 10.440 · easy

Let \( \displaystyle z = \ln{\left(x^{2} + y^{2} + 1 \right)} \) with \( \displaystyle x = s \cos{\left(t \right)} \), \( \displaystyle y = s \sin{\left(t \right)} \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 2 \), \( \displaystyle t = 0 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \ln{\left(x^{2} + y^{2} + 1 \right)}\\\frac{\partial}{\partial y} \ln{\left(x^{2} + y^{2} + 1 \right)}\end{matrix}\right] = \left[\begin{matrix}\frac{2 x}{x^{2} + y^{2} + 1}\\\frac{2 y}{x^{2} + y^{2} + 1}\end{matrix}\right] \]
∂z/∂x and ∂z/∂y.✓ Proved
∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).
Reviewed
\[ \frac{4}{5} \]
Substitute.✓ Proved

Answer \( \frac{4}{5} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly applies the chain rule and the final substitution yields the correct value.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the chain rule and the final substitution yields the correct value.
gpt-oss:20b: pass 2026-10-08
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the chain rule and the final substitution yields the correct result. Although the intermediate step of substituting the specific values into the partial derivatives is omitted, the logic is sound and the final answer is correct.

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.