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)#.
#e^cosy=x^3arctany#
then taking differential
#e^cosyxxd/(dy)cosyxx(dy)/(dx)=3x^2arctany+x^3xx1/(1+y^2)xx(dy)/(dx)#
or #-sinye^cosy(dy)/(dx)=3x^2arctany+x^3/(1+y^2)(dy)/(dx)#
or #(sinye^cosy+x^3/(1+y^2))(dy)/(dx)=-3x^2arctany#
and #(dy)/(dx)=-(3x^2arctany)/(sinye^cosy+x^3/(1+y^2))#
