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 differentiating #xlny+ylnx=2#

#1xxlny+x xx1/yxx(dy)/(dx)+(dy)/(dx)lnx+yxx 1/x=0#

or #lny+x/y(dy)/(dx)+lnx(dy)/(dx)+y/x=0#

or #(x/y+lnx)(dy)/(dx)=-(y/x+lny)#

or #(dy)/(dx)=-(y/x+lny)/(x/y+lnx)=-(y^2+xylny)/(x^2+xylnx)#