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The multivariable chain rule

Problem 10.341 · easy

Let \( \displaystyle z = x^{2} y + 3 y^{3} \) with \( \displaystyle x = \cos{\left(t \right)} \), \( \displaystyle y = \sin{\left(t \right)} \). Use the chain rule to find \( \displaystyle \frac{dz}{dt} \) at \( \displaystyle t = 0 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} y + 3 y^{3}\right)\\\frac{\partial}{\partial y} \left(x^{2} y + 3 y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}2 x y\\x^{2} + 9 y^{2}\end{matrix}\right] \]
∂z/∂x and ∂z/∂y.✓ Proved
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt).
\[ 1 \]
At t = 0, where (x, y) = (1, 0).✓ Proved

Answer \( 1 \)

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 or substitute them into the chain rule formula. It jumps from the partial derivatives to the final answer without showing the necessary intermediate steps involving the time derivatives of x and y.

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 or substitute them into the chain rule formula. It jumps from the partial derivatives to the final answer without showing the necessary intermediate steps involving the time derivatives of x and y.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute or state the values of dx/dt and dy/dt at t=0, which are required by the chain rule formula stated in line 2. It jumps directly to the final answer without showing the necessary intermediate derivatives.
gpt-oss:20b: fail (misleading) 2026-10-04 — The solution skips the essential step of evaluating the partial derivatives and the time derivatives of x and y at t=0. It merely states "1 = 1" without showing that ∂z/∂x=0, ∂z/∂y=1, dx/dt=0, dy/dt=1, and hence dz/dt=1. This omission could mislead a student into thinking the result is obvious without the necessary calculations.

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.