Gradient and directional derivatives
Problem 10.275 · medium
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} x e^{y}\\\frac{\partial}{\partial y} x e^{y}\end{matrix}\right] = \left[\begin{matrix}e^{y}\\x e^{y}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}e\\2 e\end{matrix}\right] \]∇f at the point.✓ Proved
- \[ \left[\begin{matrix}1\\0\end{matrix}\right] \]The unit direction u.✓ Proved
- \[ e \]D_u f = ∇f · u.✓ Proved
Answer \( e \)
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: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to explicitly normalize the direction vector, although it happens to be a unit vector. More critically, the final line presents the result as an identity 'E = E' rather than stating the calculated value, and the intermediate steps lack the explicit dot product calculation required to justify the result.gpt-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.