1 answer

Find the derivative of #8^{3^{x^2}}# ?

Question:

I got the first step but I'm not sure what to do next

#ln(8)*8^{3^{x^2}}*\frac{d}{dx}(3x^2)#


Answers

#(dy)/(dx)=ln3*ln8*2x(3^(x^2)*8^(3^(x^2)))#

Explanation:

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)))#

.

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