Maximum rate of change

Problem 10.537 · hard

1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sin{\left(x y \right)}\\\frac{\partial}{\partial y} \sin{\left(x y \right)}\end{matrix}\right] = \left[\begin{matrix}y \cos{\left(x y \right)}\\x \cos{\left(x y \right)}\end{matrix}\right] \]
   ∇f.✓ Proved
2. \[ \left[\begin{matrix}4 \cos{\left(12 \right)}\\3 \cos{\left(12 \right)}\end{matrix}\right] \]
   ∇f(3, 4).✓ Proved
3. \[ \left[\begin{matrix}\frac{2 \sqrt{5}}{5}\\\frac{\sqrt{5}}{5}\end{matrix}\right] \]
   The unit vector from P toward Q.✓ Proved
4. \[ \frac{11 \sqrt{5} \cos{\left(12 \right)}}{5} \]
   D_u f = ∇f · u.✓ Proved