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=2y $ and $f_y= 2x-6y $
The gradient equation is: $\nabla f = \lt f_x,f_y \gt =\lt 24 , 2x-6y \gt =\lt 2(5), 2(5)-6(5)\gt =\lt 10 , -20 \gt$
Thus, $v=\dfrac{u}{|u|} = \dfrac{\lt 4, 3 \gt }{\sqrt {4^2+3^2}} =\lt \dfrac{4}{5}, \dfrac{3}{5} \gt $
Directional derivative at that direction is:
$D_v f=\nabla f \cdot v=\lt 10 , -20 \gt \times \lt \dfrac{4}{5}, \dfrac{3}{5} \gt =-4$
