Home›Calculus 3›Gradient and directional derivatives›Problem 10.19

Gradient and directional derivatives

Problem 10.19 · medium Video

Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} x e^{y}\\\frac{\partial}{\partial y} x e^{y}\end{matrix}\right] = \left[\begin{matrix}e^{y}\\x e^{y}\end{matrix}\right] \]
∇f.✓ Proved
\[ \left[\begin{matrix}1\\1\end{matrix}\right] \]
∇f at the point.
\[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
The unit direction u.
\[ \frac{\sqrt{5}}{5} \]
D_u f = ∇f · u.

Answer \( \frac{\sqrt{5}}{5} \)

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 incorrectly presents the directional derivative as two numbers (sqrt(5)/5, sqrt(5)/5) instead of the single scalar value sqrt(5)/5. This could mislead a student into thinking the derivative has two components.
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 incorrectly presents the directional derivative as two numbers (sqrt(5)/5, sqrt(5)/5) instead of the single scalar value sqrt(5)/5. This could mislead a student into thinking the derivative has two components.
qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: pass 2026-09-27
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.