Home›Calculus 3›The multivariable chain rule›Problem 10.348

The multivariable chain rule

Problem 10.348 · easy

Let \( \displaystyle z = x^{2} y - 3 y^{3} \) with \( \displaystyle x = s \cos{\left(t \right)} \), \( \displaystyle y = s \sin{\left(t \right)} \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 1 \), \( \displaystyle t = 0 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} y - 3 y^{3}\right)\\\frac{\partial}{\partial y} \left(x^{2} y - 3 y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}2 x y\\x^{2} - 9 y^{2}\end{matrix}\right] \]
∂z/∂x and ∂z/∂y.✓ Proved
∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).
\[ 0 \]
Substitute.✓ 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 partial derivatives of x and y with respect to s, nor does it substitute the values s=1, t=0 into the chain rule formula. Line 3 is a tautology that provides no calculation.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute the partial derivatives of x and y with respect to s, nor does it substitute the values s=1, t=0 into the chain rule formula. Line 3 is a tautology that provides no calculation.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute the partial derivatives of x and y with respect to s, nor does it substitute the specific values s=1, t=0 into the chain rule formula. Line 3 is an empty placeholder that does not constitute a calculation.
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.