Answer
$-4$
Work Step by Step
Our aim is to take the first partial derivative of the given function $f(x,y,z)$ with respect to $x$, by treating $y$ and $z$ as a constant, and vice versa:
$f_x=4x $ and $f_y= 2y $
The gradient equation is: $\nabla f (-1,1)= \lt 4x, 2y \gt = \lt 4(-1), 2(1) \gt =\lt -4,2 \gt$
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
$D_v f=\nabla f \cdot v=\lt -4,2 \gt \times \lt \dfrac{3}{5}, \dfrac{-4}{5} \gt =-4$
