Maximum rate of change
Problem 10.553 · hard
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle P(1, 0) \) in the direction toward \( \displaystyle Q(-1, 1) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2}}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2}}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2}}}\\\frac{y}{\sqrt{x^{2} + y^{2}}}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}1\\0\end{matrix}\right] \]∇f(1, 0).✓ Proved
- \[ \left[\begin{matrix}- \frac{2 \sqrt{5}}{5}\\\frac{\sqrt{5}}{5}\end{matrix}\right] \]The unit vector from P toward Q.✓ Proved
- \[ - \frac{2 \sqrt{5}}{5} \]D_u f = ∇f · u.✓ Proved
Answer \( D_{\mathbf u} f = - \frac{2 \sqrt{5}}{5} \)
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 |
| 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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The unit vector calculation in line 3 is incorrect. The vector from P(1, 0) to Q(-1, 1) is <-2, 1>, which has magnitude sqrt(5). The unit vector sho
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: inconclusive 2026-10-13 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The unit vector calculation in line 3 is incorrect. The vector from P(1, 0) to Q(-1, 1) is <-2, 1>, which has magnitude sqrt(5). The unit vector shoqwen3.6:27b-mlx: inconclusive 2026-10-13 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The unit vector calculation in line 3 is incorrect. The vector from P(1,0) to Q(-1,1) is <-2, 1>, which has magnitude sqrt(5). The unit vector shouldgpt-oss:20b: pass 2026-10-13
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.