Maximum rate of change
Problem 10.548 · medium
Find the maximum rate of change of \( \displaystyle f(x, y) = \sin{\left(x y \right)} \) at \( \displaystyle (1, 1) \), and the direction in which it occurs.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sin{\left(x y \right)}\\\frac{\partial}{\partial y} \sin{\left(x y \right)}\end{matrix}\right] = \left[\begin{matrix}y \cos{\left(x y \right)}\\x \cos{\left(x y \right)}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}\cos{\left(1 \right)}\\\cos{\left(1 \right)}\end{matrix}\right] \]∇f(1, 1).✓ Proved
- \[ \sqrt{2} \cos{\left(1 \right)} \]The maximum rate of change is ‖∇f‖, in the direction of ∇f.✓ Proved
Answer \( \|\nabla f\| = \sqrt{2} \cos{\left(1 \right)}\ \text{in the direction of}\ \left\langle \cos{\left(1 \right)}, \cos{\left(1 \right)} \right\rangle \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | the largest directional difference quotient over 3600 directions |
Reviewers
gpt-oss:20b: fail (misleading) — The solution states the direction of maximum increase as ⟨cos(1),cos(1)⟩, which is not a unit vector. The correct direction is the unit vector (1/√2,1/√2), obtained by normalizing the gradient.qwen3.6:27b-mlx: pass — The solution correctly identifies the gradient, evaluates it at the given point, and applies the standard theorem that the maximum rate of change is the magnitude of the gradient in the direction of the gradient. The final answer matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-13 — The solution states the direction of maximum increase as ⟨cos(1),cos(1)⟩, which is not a unit vector. The correct direction is the unit vector (1/√2,1/√2), obtained by normalizing the gradient.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the gradient, evaluates it at the given point, and applies the standard theorem that the maximum rate of change is the magnitude of the gradient in the direction of the gradient. The final answer matches the stated answer.gpt-oss:20b: fail (misleading) 2026-10-13 — The direction given should be the unit vector in the direction of the gradient, i.e. <1/√2, 1/√2>, not <cos 1, cos 1>.qwen3.6:27b-mlx: fail (misleading) 2026-10-13 — The solution claims the direction is \langle \cos(1), \cos(1) \rangle, which is not a unit vector. The direction of maximum rate of change must be specified as a unit vector (\frac{1}{\sqrt{2}}\langle \cos(1), \cos(1) \rangle) to be mathematically precise, otherwise the student may confuse the gradient vector with the direction vector.
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/max_rate_of_change, checked 2026-10-13 with SymPy 1.14.0.