Maximum rate of change

Problem 10.437 · medium

1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2}}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2}}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2}}}\\\frac{y}{\sqrt{x^{2} + y^{2}}}\end{matrix}\right] \]
   ∇f.✓ Proved
2. \[ \left[\begin{matrix}\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]
   ∇f(1, 1).✓ Proved
3. \[ \left[\begin{matrix}- \frac{2 \sqrt{13}}{13}\\\frac{3 \sqrt{13}}{13}\end{matrix}\right] \]
   The unit vector from P toward Q.✓ Proved
4. \[ \frac{\sqrt{26}}{26} \]
   D_u f = ∇f · u.✓ Proved