To find the derivative we will use implicit differentiation.
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 of #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)#.
This way differentiating #x^6+y^6=1#, we get
#6x^5+6y^5(dy)/(dx)=0#, which gives us first derivative as
#(dy)/(dx)=-x^5/y^5#
Writing #6x^5+6y^5(dy)/(dx)=0# as #6x^5+6y^5y'=0# and differentiating it to get second derivative, we get
#30x^4+30y^4xxy'xxy'+6y^5xxy''=0#
or #30x^4+30y^4(y')^2+6y^5y''=0#
Now putting #y'=-x^5/y^5#
#30x^4+30y^4(-x^5/y^5)^2+6y^5y''=0#
or #30x^4+(30x^10)/y^6=-6y^5y''#
or #(30x^4(y^6+x^6))/y^6=-6y^5y''#
i.e. #y''=-(30x^4(y^6+x^6))/(6y^11)=-(5x^4(y^6+x^6))/(y^11)=-(5 x^4)/y^11#
