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