Suppose $f \left( x, y \right) = \frac{x}{y}$, $P = \left( -3, 1 \right)$ and $\mathbf{v} = -3 \mathbf{i} + 3 \mathbf{j}$.

A. Find the gradient of $f$.
$(\nabla f)(x,y) =$ $\mathbf{i} +$ $\mathbf{j}$
Note: Your answers should be expressions of x and y; e.g. "3x - 4y"

B. Find the gradient of $f$ at the point $P$.
$\left( \nabla f \right) \left( P \right) =$ $\mathbf{i} +$ $\mathbf{j}$
Note: Your answers should be numbers

C. Find the directional derivative of $f$ at $P$ in the direction of $\mathbf{v}$.
$(D_{\mathbf{u}} f)(P) =$
Note: Your answer should be a number

D. Find the maximum rate of change of $f$ at $P$.

Note: Your answer should be a number

E. Find the (unit) direction vector $\mathbf{w}$ in which the maximum rate of change occurs at $P$.
$\mathbf{w} =$ $\mathbf{i} +$ $\mathbf{j}$
Note: Your answers should be numbers