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The multivariable chain rule

Problem 10.340 · easy

Let \( \displaystyle z = x^{2} - 2 x y - 3 y^{2} \) with \( \displaystyle x = 2 t + 1 \), \( \displaystyle y = t^{2} - 1 \). Use the chain rule to find \( \displaystyle \frac{dz}{dt} \) at \( \displaystyle t = 1 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} - 2 x y - 3 y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} - 2 x y - 3 y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x - 2 y\\- 2 x - 6 y\end{matrix}\right] \]
∂z/∂x and ∂z/∂y.✓ Proved
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt).
\[ 0 \]
At t = 1, where (x, y) = (3, 0).✓ Proved

Answer \( 0 \)

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 compute the derivatives dx/dt and dy/dt, nor does it evaluate the partial derivatives at the specific point (x,y) corresponding to t=1. It jumps directly to the final answer without performing the required chain rule calculation.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute the derivatives dx/dt and dy/dt, nor does it evaluate the partial derivatives at the specific point (x,y) corresponding to t=1. It jumps directly to the final answer without performing the required chain rule calculation.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute the derivatives dx/dt and dy/dt, which are required by the chain rule formula stated in line 2. Line 3 asserts the result is 0 without showing the calculation or substituting the values of the partial derivatives and the time derivatives.
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.