Question

How do you find `(df)/dy` and `(df)/dx` of `f(x,y)=(4x-2y)/(4x+2y)`, using the quotient rule?

Answer 1

`f_x (x,y) = (4y)/((2x + y)^2)`

`f_y (x,y) = (-4x)/((2x+y)^2)`

Explanation:

`f(x, y) = (2x - y)/(2x + y) = u/v`

`(partial f)/(partial x) = f_x = frac{u_x v - u v_x}{v^2} = frac{2(2x+y) - (2x-y)*2}{(2x + y)^2}`

`(partial f)/(partial y) = f_y = frac{u_y v - u v_y}{v^2} = frac{-1(2x+y) - (2x-y)*1}{(2x + y)^2}`