Here we use concept of Implicit differentiation, which 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#. As either #y# as a function of #x# cannot be separated or it doing so makes things complicated.
Hence, 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)#.
Here in #sqrty+6xy=5#, differentiating w.r.t. to #x# on both sides, we get
#1/(2sqrty)*(dy)/(dx)+6(1*y+x*(dy)/(dx))=0#
Observe that differential of #sqrty# w.r.t. #y# is #1/(2sqrty)# and as we have to differentiate w.r.t. #x#, we multiply by #(dy)/(dx)#. Similarly in second term, we consider #xy# as product of #x# and #y# and use product rule. The term #5# on RHS is constant and its differential is ofcourse is #0#.
or #1/(2sqrty)(dy)/(dx)+6y+6x(dy)/(dx)=0# ....(A)
or #(dy)/(dx)=-(6y)/(1/(2sqrty)+6x)=-(12ysqrty)/(1+12xsqrty)#
differentiating (A) again, we get
#1/2*(-1/2)*y^(-3/2)(dy)/(dx)+1/(2sqrty)(d^2y)/(dx^2)+6(dy)/(dx)+6(dy)/(dx)+6x(d^2y)/(dx^2)=0#
or #-1/(4ysqrty)[-(12ysqrty)/(1+12xsqrty)]+1/(2sqrty)(d^2y)/(dx^2)-12[(12ysqrty)/(1+12xsqrty)]+6x(d^2y)/(dx^2)=0#
or #(d^2y)/(dx^2)(1/(2sqrty)+6x)=12[(12ysqrty)/(1+12xsqrty)]-1/(4ysqrty)[(12ysqrty)/(1+12xsqrty)]#
or #(d^2y)/(dx^2)[(1+12xsqrty)/(2sqrty)]=(144ysqrty)/(1+12xsqrty)-3/(1+12xsqrty)=(144ysqrty-3)/(1+12xsqrty)#
i.e. #(d^2y)/(dx^2)=(2sqrty(144ysqrty-3))/(1+12xsqrty)^2#
= #(288y^2-6sqrty)/(1+12xsqrty)^2#
