Dicrectional derivative?

Question

f(xy)=(x^3,y^(2.5)), find the directional derivative at (4, 5) in the direction theta = (2*pi)/3

Answer 1

I assume that the function should be written:
f(x,y)=(x^3 y^(2.5))
In words, "f is a function of x and y (not a function of xy, which is how you wrote it), and is equal to the product of x^3 and y^(2.5)."

Solution:
1. Find the partial derivative with respect to x:
3 x^2 y^2.5
1a. Evaluate it at (4,5):
3 (16) (25 sqrt(5)) = 1200 sqrt(5)

2. Find the partial derivative with respect to y:
2.5 x^3 y^1.5
2a. Evaluate it at (4,5):
2.5 (64) (5 sqrt(5)) = 800 sqrt(5)

3. Determine what proportions of the above 2 values are represented by the direction (2 pi / 3):
This angle is equivalent to 120 degrees. A unit "step" in this direction is equal to 0.5 units in the NEGATIVE x direction, and sqrt(3)/2 units in the positive y direction.

So we calculate (-0.5) times 1200 sqrt(5), plus (sqrt(3)/2) times 800 sqrt(5) to find the derivative in the direction (2 pi / 3).

Result: -600 sqrt(5) + 400 sqrt(15)