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)))
