If `f(x)=log_(x)(log_(e)x)`, then `f'(x)`at `x=e` is equal to

Options

A. 1
B. 2
C. 0
D. `1/e`

Answer

D

Explanation

Given that `f(x)=log_(x)(log_(e)x)`
`=(log_(e)log_(e)x)/(log_(e)x)`
On differenting w.r.t x we get
`(log_(e)x . 1/(log_(e)x) . 1/x-log_(e)log_(e)x .1/x)/((log_(e)x)^(2))`
`impliesf'(x)=(1-log_(e)log_(e)x)/(x(log_(e)x)^(2))`
`implies f'(e)=(1-log_(e)log_(e)e)/(e(log_(e)e)^(2))`
`=(1-log_(e)1)/e=1/e`