Can you differentiate it ?

Question

If #y=x^(x^(x..........oo# then #x.dy/dx # is ?

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Answer 1 · 2018-06-18T16:03:19

#(dy)/(dx)=(y^2)/(x(1-ylnx))#

I think your Question is :

#color(red)(y=x^(x^(x...........oo)# , then , # x*(dy)/(dx)= ?#

Here,

#y=x^color(red)(x^(x^(x......oo##=x^color(red)(y)# .

#i.e. y=x^y#

Taking natural log, we get

#lny=ln(x^y)#

#=>lny=y*lnx#

Diff.w.r.t. #x#,

#1/y*(dy)/(dx)=y*1/x+lnx*(dy)/(dx)#

#=>1/y*(dy)/(dx)-lnx*(dy)/(dx)=y/x#

#=>(dy)/(dx){1/y-lnx}=y/x#

#=>(dy)/(dx){(1-ylnx)/y}=y/x#

#(dy)/(dx)=y/x*(y)/(1-ylnx)#

#(dy)/(dx)=(y^2)/(x(1-ylnx))#