Gradient and directional derivatives
Problem 10.185 · medium
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2}}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2}}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2}}}\\\frac{y}{\sqrt{x^{2} + y^{2}}}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}\frac{3}{5}\\\frac{4}{5}\end{matrix}\right] \]∇f at the point.✓ Proved
- \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]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: fail (misleading) — The final sentence claims the directional derivative is "sqrt(5)/5, sqrt(5)/5", implying two values, whereas the directional derivative is a single scalar. This misstatement could confuse a student.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (misleading) 2026-09-27 — The final sentence claims the directional derivative is "sqrt(5)/5, sqrt(5)/5", implying two values, whereas the directional derivative is a single scalar. This misstatement could confuse a student.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (misleading) 2026-09-27 — The final line incorrectly lists two identical values for the directional derivative instead of a single scalar. The correct directional derivative is √5/5, not two numbers.
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 2026-09-27 with SymPy 1.14.0.