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
