Maximum rate of change
Problem 10.385 · medium
Find the directional derivative of \( \displaystyle f(x, y) = \sin{\left(x y \right)} \) at \( \displaystyle P(1, 1) \) in the direction toward \( \displaystyle Q(1, 2) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sin{\left(x y \right)}\\\frac{\partial}{\partial y} \sin{\left(x y \right)}\end{matrix}\right] = \left[\begin{matrix}y \cos{\left(x y \right)}\\x \cos{\left(x y \right)}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}\cos{\left(1 \right)}\\\cos{\left(1 \right)}\end{matrix}\right] \]∇f(1, 1).✓ Proved
- \[ \left[\begin{matrix}0\\1\end{matrix}\right] \]The unit vector from P toward Q.✓ Proved
- \[ \cos{\left(1 \right)} \]D_u f = ∇f · u.✓ Proved
Answer \( D_{\mathbf u} f = \cos{\left(1 \right)} \)
✓ Nihil obstat 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. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ 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 | difference quotient along the direction |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly computes the gradient, identifies the direction vector, normalizes it to a unit vector, and calculates the dot product. All steps are mathematically sound.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly computes the gradient, identifies the direction vector, normalizes it to a unit vector, and calculates the dot product. All steps are mathematically sound.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.