We know that,
#color(red)((P)lnx^n=n*lnxto#[ Power /coefficient rule ]
#color(blue)((S)ln(X*Y)=lnX+lnY)to#[ product/sum rule ]
Let,
#y=8^(3^(x^2))...to(1)#
For simplicity we take, #N=3^(x^2)#
#:.y=8^N#
Taking natural log ,we get
#lny=ln8^N=Nln8 ...tocolor(red)(Apply(P)# ,
where , #N=3^(x^2)#
#=>lny=3^(x^2)*ln8...to(2)#
Again taking natural log ,
#ln(lny)=ln(3^(x^2)*ln8)...tocolor(blue)(Apply(S)#
#=>ln(lny)=ln(3^(x^2))+ln(ln8)#
#=>ln(lny)=x^2*ln3+ln(ln8)#
Diff.w.r.t. #x# ,we get
#1/lny*1/y(dy)/(dx)=2x*ln3+0#
#=>1/(ylny)*(dy)/(dx)=2xln3#
#=>(dy)/(dx)=ylny(2xln3)#
From #(1) and (2)# ,we have
#(dy)/(dx)=8^(3^(x^2))3^(x^2)ln8(2xln3)#
#=>(dy)/(dx)=ln3*ln8*2x(3^(x^2)*8^(3^(x^2)))#
In short :
#y=8^(3^(x^2))#
#=>(dy)/(dx)=8^(3^(x^2))*ln8d/(dx)(3^(x^2))#
#=>(dy)/(dx)=8^(3^(x^2))*ln8{3^(x^2)ln3d/(dx)(x^2)}#
#=>(dy)/(dx)=8^(3^(x^2))*ln8{3^(x^2)ln3(2x)}#
#=>(dy)/(dx)=ln3*ln8*2x(3^(x^2)*8^(3^(x^2)))#
