Find the directional derivative of f at the given point in the direction indicated by the angle θ. f(x, y) = 2ye−x, (0, 6), θ = 2π/3

Answer-

The directional derivative of f(x,y) at (0,6) in the direction θ = 2π/3 is, 7.73

Solution-

The unit vector in the direction of the given angle is

Given that,

Gradient of f(x,y)

Gradient of f(x,y) at (0,6)

= <(-2)(6)(1) , (2)(1)> = <-12 , 2>

The directional derivative of f(x,y) at (0,6) is

simplifying

5(x + -6) = 2(x + 3)

reorder the terms:

5(-6 + x) = 2(x + 3)

(-6 * 5 + x * 5) = 2(x + 3)

(-30 + 5x) = 2(x + 3)

reorder the terms:

-30 + 5x = 2(3 + x)

-30 + 5x = (3 * 2 + x * 2)

-30 + 5x = (6 + 2x)

solving

-30 + 5x = 6 + 2x

solving for variable 'x'.

move all terms containing x to the left, all other terms to the right.

add '-2x' to each side of the equation.

-30 + 5x + -2x = 6 + 2x + -2x

combine like terms: 5x + -2x = 3x

-30 + 3x = 6 + 2x + -2x

combine like terms: 2x + -2x = 0

-30 + 3x = 6 + 0

-30 + 3x = 6

add '30' to each side of the equation.

-30 + 30 + 3x = 6 + 30

combine like terms: -30 + 30 = 0

0 + 3x = 6 + 30

3x = 6 + 30

combine like terms: 6 + 30 = 36

3x = 36

divide each side by '3'.

x = 12

simplifying

x = 12

this should .

17

step-by-step explanation:

51/3=17

thus 17 is your answer.