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}#
