Gradient and directional derivatives

Problem 10.149 · medium

1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + 3 x y + y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} + 3 x y + y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x + 3 y\\3 x + 2 y\end{matrix}\right] \]
   ∇f.✓ Proved
2. \[ \left[\begin{matrix}8\\7\end{matrix}\right] \]
   ∇f at the point.✓ Proved
3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
   The unit direction u.✓ Proved
4. \[ \frac{6 \sqrt{5}}{5} \]
   D_u f = ∇f · u.✓ Proved