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The multivariable chain rule

Problem 10.397 · easy

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

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

Answer \( 2 \)

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 dx/dt and dy/dt and fails to substitute the values into the partial derivatives. It asserts the final answer '2' without showing the intermediate steps required by the chain rule, making the derivation incomplete and logically disconnected.

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 dx/dt and dy/dt and fails to substitute the values into the partial derivatives. It asserts the final answer '2' without showing the intermediate steps required by the chain rule, making the derivation incomplete and logically disconnected.
gpt-oss:20b: fail (misleading) 2026-10-06 — The solution skips the crucial step of evaluating the partial derivatives, the derivatives of x and y, and plugging them into the chain‑rule formula. The final line "2 = 2" is not justified by any preceding computation, so a student could be misled into thinking the result is obvious without the intermediate work.
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly calculate or state the derivatives dx/dt and dy/dt, which are required to apply the chain rule formula stated in step 2. It jumps directly to the final numerical answer without showing the intermediate differentiation 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.