Gradient and directional derivatives
Problem 10.181 · medium
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2}}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2}}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2}}}\\\frac{y}{\sqrt{x^{2} + y^{2}}}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}\frac{3}{5}\\\frac{4}{5}\end{matrix}\right] \]∇f at the point.✓ Proved
- \[ \left[\begin{matrix}\frac{3}{5}\\\frac{4}{5}\end{matrix}\right] \]The unit direction u.✓ Proved
- \[ 1 \]D_u f = ∇f · u.✓ Proved
Answer \( 1 \)
✓ 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 | f differenced along the unit direction agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly computes the gradient, normalizes the direction vector, and calculates the dot product. The steps are logically sound and the result is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the gradient, normalizes the direction vector, and calculates the dot product. The steps are logically sound and the result is correct.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the gradient, normalizes the direction vector, and calculates the dot product. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-09-27
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.