If fx y 3x24y2 find the value of the directional derivative

If f(x, y) = -3*x^2-4*y^2, find the value of the directional derivatives at the point (-3, 2) in the direction given by the angle theta = 2*pi/5. More specifically, find the directional derivative at the point (-3, 2) in the direction given by the angle theta = 2*pi/5.

Solution

unit direction vector v^ =<cos2pi/5 ,sin2pi/5>

f(x, y) = -3x²-4y²

gradient f =<-6x ,-8y>

gradient at (-3,2)

f =<-6*-3 ,-8*2>

f =<18,-16>

directional derivative D_u =f .v^

directional derivative D_u =<18,-16>.<cos2pi/5 ,sin2pi/5>

directional derivative D_u =(18cos(2pi/5) -16sin(2pi/5))

directional derivative D_u =-9.65