The multivariable chain rule
Problem 10.441 · easy
Let \( \displaystyle z = x^{2} - 2 x y + 3 y^{2} \) with \( \displaystyle x = e^{t} \), \( \displaystyle y = 2 t \). Use the chain rule to find \( \displaystyle \frac{dz}{dt} \) at \( \displaystyle t = 0 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} - 2 x y + 3 y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} - 2 x y + 3 y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x - 2 y\\- 2 x + 6 y\end{matrix}\right] \]∂z/∂x and ∂z/∂y.✓ Proved
- dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt).
- \[ -2 \]At t = 0, where (x, y) = (1, 0).✓ Proved
Answer \( -2 \)
Lines: 2 proved, 1 not checked. 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 | 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: fail (style) — The solution skips the explicit application of the chain rule at t=0; it states the final equality without showing the substitution of ∂z/∂x, ∂z/∂y, dx/dt, and dy/dt. While the result is correct, the missing step could confuse a student learning the method.qwen3.6:27b-mlx: pass — The solution correctly applies the multivariable chain rule and evaluates the derivatives at the specified point t=0.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the multivariable chain rule and evaluates the derivatives at the specified point t=0.gpt-oss:20b: fail (style) 2026-10-08 — The solution skips the explicit application of the chain rule at t=0; it states the final equality without showing the substitution of ∂z/∂x, ∂z/∂y, dx/dt, and dy/dt. While the result is correct, the missing step could confuse a student learning the method.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the multivariable chain rule and evaluates the partial derivatives and time derivatives at the specified point t=0. The arithmetic is correct.
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 2026-10-08 with SymPy 1.14.0.