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The multivariable chain rule

Problem 10.491 · easy

Let \( \displaystyle z = \ln{\left(x^{2} + y^{2} + 1 \right)} \) with \( \displaystyle x = 2 t + 1 \), \( \displaystyle y = t^{2} - 1 \). Use the chain rule to find \( \displaystyle \frac{dz}{dt} \) at \( \displaystyle t = 0 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \ln{\left(x^{2} + y^{2} + 1 \right)}\\\frac{\partial}{\partial y} \ln{\left(x^{2} + y^{2} + 1 \right)}\end{matrix}\right] = \left[\begin{matrix}\frac{2 x}{x^{2} + y^{2} + 1}\\\frac{2 y}{x^{2} + y^{2} + 1}\end{matrix}\right] \]
∂z/∂x and ∂z/∂y.✓ Proved
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt).
\[ \frac{4}{3} \]
At t = 0, where (x, y) = (1, -1).✓ Proved

Answer \( \frac{4}{3} \)

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 fails to explicitly calculate or state the values of dx/dt and dy/dt, which are required components of the chain rule formula stated in line 2. Line 3 jumps directly to the final numerical answer without showing the substitution of these derivatives, making the logical connection between the formula and the result opaque and incomplete.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly calculate or state the values of dx/dt and dy/dt, which are required components of the chain rule formula stated in line 2. Line 3 jumps directly to the final numerical answer without showing the substitution of these derivatives, making the logical connection between the formula and the result opaque and incomplete.
gpt-oss:20b: pass 2026-10-10
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly calculate or state the derivatives dx/dt and dy/dt, which are required components of the chain rule formula stated in step 2. While the final numerical answer is correct, the omission of these intermediate steps makes the derivation incomplete and skips the core mechanical part of the problem.

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 2026-10-10 with SymPy 1.14.0.