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Gradient and directional derivatives

Problem 10.79 · medium Video

Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} x e^{y}\\\frac{\partial}{\partial y} x e^{y}\end{matrix}\right] = \left[\begin{matrix}e^{y}\\x e^{y}\end{matrix}\right] \]
∇f.✓ Proved
\[ \left[\begin{matrix}1\\1\end{matrix}\right] \]
∇f at the point.
\[ \left[\begin{matrix}\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]
The unit direction u.
\[ \sqrt{2} \]
D_u f = ∇f · u.

Answer \( \sqrt{2} \)

A contour map of f(x, y) = x·e^(y): each curve joins points where f has the same value. At (1, 0) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

Lines: 1 proved, 3 not checked. 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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 solution claims the directional derivative equals "sqrt(2), sqrt(2)" which is incorrect; the dot product of the gradient and the unit vector yields a single scalar value sqrt(2).
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28
gpt-oss:20b: fail (misleading) 2026-09-28 — The solution claims the directional derivative equals "sqrt(2), sqrt(2)" which is incorrect; the dot product of the gradient and the unit vector yields a single scalar value sqrt(2).
qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: fail (error) 2026-09-27 — The final line incorrectly lists two values "sqrt(2), sqrt(2)" for the directional derivative. The dot product of ∇f=(1,1) and u=(1/√2,1/√2) yields a single scalar √2, not two separate numbers.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26

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 2026-10-13 with SymPy 1.14.0.