Implicit differentiation is a special case of the chain rule for derivatives. Generally differentiation problems involve functions i.e. #y=f(x)# - written explicitly as functions of #x#. However, some functions y are written implicitly as functions of #x#. So what we do is to treat #y# as #y=y(x)# and use chain rule. This means differentiating #y# w.r.t. #y#, but as we have to derive w.r.t. #x#, as per chain rule, we multiply it by #(dy)/(dx)#.
Hence implicit differential of #-y=x^4y-3x^2y^2+xy^4# is
#-(dy)/(dx)=(4x^3y+x^4(dy)/(dx))-3(2xy^2+x^2xx2y(dy)/(dx))+(1xxy^4+x xx4y^3(dy)/(dx))#
or #-(dy)/(dx)=4x^3y+x^4(dy)/(dx)-6xy^2-6x^2y(dy)/(dx)+y^4+4xy^3(dy)/(dx)#
or #6x^2y(dy)/(dx)-4xy^3(dy)/(dx)-x^4(dy)/(dx)-(dy)/(dx)=4x^3y-6xy^2+y^4#
or #(6x^2y-4xy^3-x^4-1)(dy)/(dx)=4x^3y-6xy^2+y^4#
or #(dy)/(dx)=(4x^3y-6xy^2+y^4)/(6x^2y-4xy^3-x^4-1)#
