Find the directional derivative for f xy x2 3xy 4y3 at th

Find the directional derivative for f (x,y) = x2 - 3xy + 4y3 at the point (-2,0)in the direction vector

Solution

directional derivative is given by--
d = grad(f).[unit vetor in the direction of <1,2>] (at the point (-2,0) [dot product]
So, d = [(2x - 3y)i + (-3x + 12y²)j] [(i+2j)/(1 + 4)]

= (2x-3y)/5 + 2(-3x + 12y²)/5

So at (-2,0);(substituting x =-2 and y=0 in the previous formula)

d = -4/5 + 12/5 = 8/5

So, directional derivative = 8/5