Term 1:

#(d[x^3])/dx = 3x^2#

Term 2:

Use the quotient rule, #(d[(g/(h))])/dx = {g'h - h'g}/(h)^2#

let #g = y#, then #g' = dy/dx#, #h = x^2#, #h' = 2x # and #h^2 = x^4#

#(d[(y/(x^2))])/dx = { x^2dy/dx - 2xy}/x^4#

#(d[(y/(x^2))])/dx = 1/x^2dy/dx - 2y/x^3#

Term 3:

#(d[y^3])/dx = 3y^2(dy/dx)#

Term 4:

Use the product rule #(d[gh])/dx = g'h + gh'#

let #g = 8x#, then #h = y#, #g' = 8#, and #h' = dy/dx#

#(d[8xy])/dx = 8y + 8xdy/dx#

Put the derivatives back into their corresponding locations in the equation:

#3x^2 + 1/x^2dy/dx - 2y/x^3 + 3y^2(dy/dx) = 8y + 8xdy/dx#

Move the terms containing #dy/dx# to the left and the others to the right:

#1/x^2dy/dx + 3y^2(dy/dx) - 8xdy/dx = 8y + 2y/x^3 - 3x^2#

Factor #dy/dx# from the left:

#{1/x^2 + 3y^2 - 8x}dy/dx = 8y + 2y/x^3 - 3x^2#

Multiply both sides by #x^3#:

#{x + 3y^2x^3 - 8x^4}dy/dx = 8yx^3 + 2y - 3x^5#

Divide both sides by the coefficient of #dy/dx#:

#dy/dx = {8yx^3 + 2y - 3x^5}/{x + 3y^2x^3 - 8x^4}#