Question

If 4x^(2)+5x+xy=4 and y(4)=-20, find y'(4) by implicit differentiation.

I got y'=(-8x-5-y)/(x)

Answer 1

Differentiate the first equation with respect to x, with y being treated as a function of x.
8x + 5 + y + x dy/dx = 0
dy/dx = -8 -5/x -y/x
(That agrees with your formula).
The want the actual value at x=4.
When x=4, 64 + 20 + 4y = 4, so
80 + 4y = 0, and
y (@x=4)= -20
dy/dx (@x=4) = -8 -5/4 +20/4 = -17/4