Gradient and directional derivatives
Problem 10.280 · medium
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2}}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2}}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2}}}\\\frac{y}{\sqrt{x^{2} + y^{2}}}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}\frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]∇f at the point.✓ Proved
- \[ \left[\begin{matrix}1\\0\end{matrix}\right] \]The unit direction u.✓ Proved
- \[ \frac{\sqrt{5}}{5} \]D_u f = ∇f · u.✓ Proved
Answer \( \frac{\sqrt{5}}{5} \)
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 | f differenced along the unit direction agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (misleading) — The solution claims the direction vector <1, 0> is a unit vector without explicitly stating or verifying its magnitude, which is a critical step in the definition of the directional derivative. While the vector happens to be unit length, omitting the normalization step teaches students that they can skip checking the magnitude of the direction vector.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (misleading) 2026-10-03 — The solution claims the direction vector <1, 0> is a unit vector without explicitly stating or verifying its magnitude, which is a critical step in the definition of the directional derivative. While the vector happens to be unit length, omitting the normalization step teaches students that they can skip checking the magnitude of the direction vector.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03
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/gradient_directional, checked 2026-10-03 with SymPy 1.14.0.