The multivariable chain rule
Problem 10.396 · easy
Let \( \displaystyle z = \sqrt{x^{2} + y^{2} + 4} \) with \( \displaystyle x = \cos{\left(t \right)} \), \( \displaystyle y = \sin{\left(t \right)} \). Use the chain rule to find \( \displaystyle \frac{dz}{dt} \) at \( \displaystyle t = \frac{\pi}{2} \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2} + 4}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2} + 4}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2} + 4}}\\\frac{y}{\sqrt{x^{2} + y^{2} + 4}}\end{matrix}\right] \]∂z/∂x and ∂z/∂y.✓ Proved
- dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt).
- \[ 0 \]At t = pi/2, where (x, y) = (0, 1).✓ 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: passqwen3.6:27b-mlx: fail (error) — The solution fails to compute the derivatives dx/dt and dy/dt or substitute the values of x and y into the partial derivatives. It jumps from the chain rule formula to the final answer without showing the necessary intermediate calculation steps.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to compute the derivatives dx/dt and dy/dt or substitute the values of x and y into the partial derivatives. It jumps from the chain rule formula to the final answer without showing the necessary intermediate calculation steps.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the chain rule and evaluates the derivatives at the specified point, yielding the correct result of 0.
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.