Can you differentiate it ?

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

1 Answer
Jun 18, 2018

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

Explanation:

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))