find directional derivative given angle

Question

If f(x,y) = 1x^2 - 3y^2, find the value of the directional derivative at the point (-4, -1) in the direction given by the angle 𝜃 = 2pi / 3

Answer 1

Unit vector in direction 2π/3: u = <cos2π/3, sin2π/3> = <- 1/2, √3/2>

f'_x(x, y) = 2x; f'_x(- 4, - 1) = - 8; f'_y(x, y) = - 6y; f'_y(-4, - 1) = 6

So, D_u(- 4, - 1) = - 8·(-1/2) + 6(√3/2) = 4 + 3√3