Use implicit differentiate to find y', where 5xy/x+y = (x^2 + 2)?

Answer 1

5xy=(x+y)(x^2+2)
5xy=x^3+2x+x^2y+2y
5xy'+5y=3x^2+2+x^2y'+2xy+2y'
5xy'-x^2y'-2y'=3x^2-5y+2xy
y'(5x-x^2-2)=3x^2-5y+2xy)
y'=(3x^2-5y+2xy)/(5x-x^2-2)

Answer 2

(5xy/x+y)' = 5*[(y+xy')*(x+y) - xy*(1+y')]/(x+y)^2

(x^2+2)' = 2*x

so we have

5*[(y+xy')*(x+y) - xy*(1+y')]/(x+y)^2 = 2*x
.
y' = [2*x*(x+y)^2/5 - y*(x+y) + xy] / [x*(x+y)-xy]

Answer 3

5 [(y + xy')*(x+y) - (xy)(1 + y')]/(x+y)^2 = 2x

From there, isolate y'.