Let α be a given angle. Find the directional derivative of the function 2x + 3y in the direction u = cos αi + sin αj, at any point in the Plane.

Answer 1

The formula for directional derivative of a function f(x,y) in the direction u =〈a,b〉 is

D_uf(x,y) = ∇ f • 〈a, b〉, or

D_uf(x,y) = f_x(x,y)a + f_y(x,y)b

(where f_x and f_y are partial derivatives w/ respect to x and y, respectively).

Here,

f(x,y) = 2x + 3y.

f_x = 2 (disregarding the y term);

f_y = 3 (disregarding the x term).

u =〈a,b〉= cos(a)i + sin(a)j.

Thus,

D_uf(x,y) = 2cos(a) + 3sin(a).

Note that if f_x and f_y had retained any x and y terms, we would have left them as x and y because the question asks for the general directional derivative at any given x or y.