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The multivariable chain rule

Problem 10.398 · easy

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

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

Answer \( 2 e^{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly applies the multivariable chain rule. The partial derivatives are correct, and the evaluation at t=0 (where x=1, y=-1) yields the correct result 2e^2.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the multivariable chain rule. The partial derivatives are correct, and the evaluation at t=0 (where x=1, y=-1) yields the correct result 2e^2.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the multivariable chain rule. The partial derivatives are correct, and the evaluation at t=0 (where x=1, y=-1) yields the correct result 2e^2.

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.