Gradient and directional derivatives
Problem 10.184 · medium
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} x e^{y}\\\frac{\partial}{\partial y} x e^{y}\end{matrix}\right] = \left[\begin{matrix}e^{y}\\x e^{y}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}e\\- e\end{matrix}\right] \]∇f at the point.✓ Proved
- \[ \left[\begin{matrix}\frac{3}{5}\\\frac{4}{5}\end{matrix}\right] \]The unit direction u.✓ Proved
- \[ \frac{\left(-1\right) e}{5} = - \frac{e}{5} \]D_u f = ∇f · u.✓ Proved
Answer \( - \frac{e}{5} \)
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 line incorrectly lists two values for the directional derivative; the dot product yields a single scalar, -E/5. The solution otherwise correctly sets up the gradient and unit direction.qwen3.6:27b-mlx: pass — The solution correctly computes the gradient, normalizes the direction vector, and calculates the dot product to find the directional derivative.
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 to find the directional derivative.gpt-oss:20b: fail (misleading) 2026-09-27 — The final line incorrectly lists two values for the directional derivative; the dot product yields a single scalar, -E/5. The solution otherwise correctly sets up the gradient and unit direction.qwen3.6:27b-mlx: pass 2026-09-27gpt-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.