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If y=x^((log x)^(log(log,x)))," then "(d...

If `y=x^((log x)^(log(log,x)))," then "(dy)/(dx) is`

A

`(y)/(x)(lnx^(logx-1))+2lnxln(lnx)`

B

`(y)/(x)(logx)^(log(logx))(2log(logx)+1)`

C

`[(lnx)^(2)+2ln(lnx)]`

D

`(y)/(x)(logy)/(logx)(2log(logx)+1)`

Text Solution

Verified by Experts

The correct Answer is:
B, D
`y=x^((logx)^(log(logx)))`
`rArr" "logy=(logx)(logx)^(log(logx))" (1)"`
Taking log on both sides, we get
`log(logy)=log(logx)+log(logx)log(logx)`
Differentiating w.r.t., we get
`(1)/(logy).(1)/(y)(dy)/(dx)=(1)/(xlogx)+(2log(logx))/(logx)(1)/(x)`
`=(2log(logx)+1)/(xlogx)`
`rArr" "(dy)/(dx)=(y)/(x).(logy)/(logx)(2log(logx)+1)`
Substituting the value of log y from (1), we get
`(dy)/(dx)=(y)/(x)(logx)^(log(logx))(2log(logx)+1)`
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