Calculus 3 Directional

Question

Suppose that f(x,y)=xy. The directional derivative of f(x,y) in the directional <2,−1> and at the point (x,y)=(3,−6) is?

Answer 1

Note that ∂f/∂x = y and ∂f/∂y = x and hence directional derivative of in the directional <2,-1> (with unit vector <2/√5,-1/√5>) is given by:

(2/√5)*∂f/∂x (3,-6) -(1/√5)*∂f/∂y(3,-6) = (2/√5)*(-6) - (1/√5)*(3)= -15/√5.

Answer 2

Since your vector is not a unit vector, it must be converted such that the new vector is v = <2/√5, -1/√5>. Then, the directional derivative is simply the scalar product of the unit vector and the gradient of your function ∇f(x,y).

∇f(x,y) = <y, x>

Thus, the directional derivative is

D(x, y) = <y, x> • <2/√5, -1/√5> = 2y/√5 - x/√5

Evaluating the directional derivative at the point (3, -6) yields

D(3, -6) = 2(-6)/√5 - 3/√5

D(3, -6) = -12/√5 - 3/√5

D(3, -6) = -15/√5