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If f(x)=(log)x(log x),t h e nf^(prime)(x...

If `f(x)=(log)_x(log x),t h e nf^(prime)(x)` at `x=e` is equal to (a) `1/e` (b) e (c) 1 (d) zero

Text Solution

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`"Given "f(x)=log_(x)(log x)=(log_(e)(log_(e)x))/((log_(e)x))`
`"or "f'(x)=((1)/(log_(e)x)(1)/(x)log_(e)x-(1)/(x)log_(e)(log_(e)x))/((log_(e)x)^(3))`
`=((1)/(x)[1-log_(e)(log_(e)e)])/((log_(e)x)^(2))`
`"At "x=e," we get"`
`f'(e)=((1)/(e)[1-log_(e)(log_(e)e)])/((log_(e)e)^(2))=((1)/(e)[1-log_(e)1])/((1)^(2))`
`=(1)/(e)(1-0)=(1)/(e)`
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