Answer
$-4$
Work Step by Step
In order to find the partial derivative, we will differentiate
with respect to $x$, by keeping $y$ as a constant to find the x-coordinate of the gradient vector, and vice versa:
$f_x=4x \\ f_y= 2y $
Write the gradient equation.
$\nabla f (-1,1)= \lt f_x,f_y \gt = \lt 4(-1), 2(1) \gt =\lt -4,2 \gt$
Thus, $v=\dfrac{u}{|u|} = \dfrac{\lt 3, -4 \gt }{\sqrt {3^2+(-4)^2}} =\lt \dfrac{3}{5}, \dfrac{-4}{5} \gt$
The directional derivative at that direction is given as:
$D_v f=\nabla f \cdot v=\lt -4,2 \gt \times \lt \dfrac{3}{5}, \dfrac{-4}{5} \gt =-4$
