Finding ∇f of given implicit fuction.

Question

Let $z = f(x,y)$ be the differentiable function given implicitly by $x^3 +y^3 +z^3 + xyz =9$ and such that $f(0,1)=2$.

1. Find $\nabla f$ at the point $P_0 = (0, 1)$.
2. Find the rate and direction of the steepest increase of $f$ at $P_0$.
3. Find $(D_u f)(P_0)$, where $u = (\mathbf{i}+\mathbf{j})/\sqrt{2}$.

Answer 1

Let me give you a few hints: $$\nabla f = \left\langle {\frac{{\partial f}}{{\partial x}},\frac{{\partial f}}{{\partial y}},\frac{{\partial f}}{{\partial z}}} \right\rangle$$ and $${D_u}f({P_0}) = \nabla f({P_0}) \cdot \hat u.$$ Also, we know that $\nabla f$ points in the direction of max increase, therefore $|\nabla f|$ gives the max rate. It does not matter if the function is defined implicitly, the procedure is the same. Even if $z = f(x,y)$, we can still write $g(x,y,z) = f(x,y) - z$.