First, the equation will be rewritten to get #yx^(-4)-3y^(-2)-y=8#
Now we differentiate with respect to #x#:
#d/(dx)[yx^(-4)-3y^(-2)-y]=d/(dx)[8]#
#d/(dx)[yx^(-4)]-d/(dx)[3y^(-2)]-d/(dx)[y]=d/(dx)[8]#
#d/(dx)[y]x^(-4)-4yx^(-5)-d/(dx)[3y^(-2)]-d/(dx)[y]=0#
Using the chain rule, we get #d/(dx)=d/(dy)*(dy)/(dx)#
#d/(dy)[y]* (dy)/(dx)x^(-4)-4yx^(-5)-d/(dy)[3y^(-2)]* (dy)/(dx)-d/(dy)[y]*(dy)/(dx)=0#
#(dy)/(dx)x^(-4)-4yx^(-5)+6y^(-3)* (dy)/(dx)-(dy)/(dx)=0#
#(dy)/(dx)[x^(-4)+6y^(-3)-1]-4yx^(-5)=0#
#(dy)/(dx)[x^(-4)+6y^(-3)-1]=4yx^(-5)#
#(dy)/(dx)=(4yx^(-5))/(x^(-4)+6y^(-3)-1)#