Let $z = f(x,y)$ be the differentiable function given implicitly by $x^3 +y^3 +z^3 + xyz =9$ and such that $f(0,1)=2$.
- Find $\nabla f$ at the point $P_0 = (0, 1)$.
- Find the rate and direction of the steepest increase of $f$ at $P_0$.
- Find $(D_u f)(P_0)$, where $u = (\mathbf{i}+\mathbf{j})/\sqrt{2}$.