Maximum rate of change
Problem 10.486 · medium
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y + y^{3} \) at \( \displaystyle P(-1, 2) \) in the direction toward \( \displaystyle Q(-3, 4) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} y + y^{3}\right)\\\frac{\partial}{\partial y} \left(x^{2} y + y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}2 x y\\x^{2} + 3 y^{2}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}-4\\13\end{matrix}\right] \]∇f(-1, 2).✓ Proved
- \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]The unit vector from P toward Q.✓ Proved
- \[ \frac{17 \sqrt{2}}{2} \]D_u f = ∇f · u.✓ Proved
Answer \( D_{\mathbf u} f = \frac{17 \sqrt{2}}{2} \)
✓ 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, determines the unit vector in the direction from P to Q, and calculates the dot product. All steps are mathematically sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly computes the gradient, determines the unit vector in the direction from P to Q, and calculates the dot product. All steps are mathematically sound.gpt-oss:20b: pass 2026-10-10gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10
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 2026-10-10 with SymPy 1.14.0.