Question

Can you differentiate it ?

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

Answer 1

`(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))`