Partial derivative help please

let f( x , y) = 7x - y / x + 8y

∂f./ ∂.x = A(x+8y) - B(7x - y) / (x+8y)^2


A=
B=
C=

Thank you

You must mean f(x, y) = (7x –y )/ (x + 8y).

Taking partials wrt x just says to treat y like is was a constant.

So ∂f / ∂x = [(x+8y)•∂(7x–y) / ∂x – (7x–y)•∂(x+8y) / ∂x] / (x+8y)² =

[(x+8y)•7 – (7x–1)•1] / (x+8y)²

So A = 7, B = 1 and I see no C in the problem anywhere.

It's just the quotient rule for ordinary derivatives:

(u / v)' = (u' v - v' u) / v²

You can do that because, when evaluating ∂f/∂x, you treat y as a constant. So, you have
u = 7x - y
v = x + 8y
u' = ∂u/∂x = 7
v' = ∂v/∂x = 1

Match up the results and
A = u' = 7
B = v' = 1
C = ?? = There ain't no C