If f(x)=log(x)(log(e)x), then f'(x)at x=...

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

A

1

B

2

C

0

D

`(1)/(e)`

Text Solution

Verified by Experts

The correct Answer is:

D

Given , ` f(x) = log_(x) (log_(e) x) = (log_(e) log_(e) x)/(log_(e) x )`
` because log_(e) x= (log_(e)x)/(log_(e) a)`
on differentiating both sides w.r.t.x., we get
`f'(x)= (log_(e) x(d)/(dx) (log_(e) log_(e)x)-log_(e) log_(e) s(d)/(dx) (log_(e) x))/((log_(e) x)^(2))`
`rArr f'(x) = (1-log_(e) log_(e)x)/(x(log_(e)x)^(2))`
` rArr f'(x) = (1-log_(e) log_(e)e)/(x(log_(e)e)^(2))=(1-log_(e)1)/(e) = (1)/(e)` ltrgt `[because log_(e) 1 = 0 and log_(e) e = 1]`

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