Maximum rate of change
Problem 10.429 · hard
Find the directional derivative of \( \displaystyle f(x, y) = e^{x} \sin{\left(y \right)} \) at \( \displaystyle P(2, 1) \) in the direction toward \( \displaystyle Q(5, 5) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} e^{x} \sin{\left(y \right)}\\\frac{\partial}{\partial y} e^{x} \sin{\left(y \right)}\end{matrix}\right] = \left[\begin{matrix}e^{x} \sin{\left(y \right)}\\e^{x} \cos{\left(y \right)}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}e^{2} \sin{\left(1 \right)}\\e^{2} \cos{\left(1 \right)}\end{matrix}\right] \]∇f(2, 1).✓ Proved
- \[ \left[\begin{matrix}\frac{3}{5}\\\frac{4}{5}\end{matrix}\right] \]The unit vector from P toward Q.✓ Proved
- \[ \frac{4 e^{2} \cos{\left(1 \right)}}{5} + \frac{3 e^{2} \sin{\left(1 \right)}}{5} = \frac{\left(4 \cos{\left(1 \right)} + 3 \sin{\left(1 \right)}\right) e^{2}}{5} \]D_u f = ∇f · u.✓ Proved
Answer \( D_{\mathbf u} f = \frac{\left(4 \cos{\left(1 \right)} + 3 \sin{\left(1 \right)}\right) e^{2}}{5} \)
✓ 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly computes the gradient, the unit direction vector, and their dot product. All steps are logically sound and algebraically 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/max_rate_of_change, checked 2026-10-08 with SymPy 1.14.0.