`"If "y=a^(x^(a^(x...oo)))", prove that "(dy)/(dx)=(y^(2)(logy))/(x[1-y(logx)(logy)]).`

Explanation

`y=a^(x^(a^x..oo))`
`y-a^(x^(y)`
`rArr log y =log x^(x^(y))`
`x^(y).loga`
`y-a^(x^(y)`
`rArr log y =log x^(x^(y))`
`x^(y).loga`
`rArr log(log y)=log x^y+log(log a)`
`=ylog x+log +log (log a)`
Differentiate both sides with respect to x
`1/(y log y )dy/dx=y/x+log x. dy/dx +0`
`rArr (1/(y log y )-log x ) dy/dx = y/x `
`rArr ((1-y log x log y ) )/ (y log y ) .dy /dx = y/x`
`dy/dx = (y^2 log y )/(x(1-y log x log y)`