Maximum rate of change

Problem 10.336 · hard

1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} x e^{- y}\\\frac{\partial}{\partial y} x e^{- y}\end{matrix}\right] = \left[\begin{matrix}e^{- y}\\- x e^{- y}\end{matrix}\right] \]
   ∇f.✓ Proved
2. \[ \left[\begin{matrix}e^{-1}\\- \frac{2}{e}\end{matrix}\right] \]
   ∇f(2, 1).✓ Proved
3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]
   The unit vector from P toward Q.✓ Proved
4. \[ - \frac{3 \sqrt{2}}{2 e} \]
   D_u f = ∇f · u.✓ Proved