If f(x) `=|log_(e)|x||,` then f'(x) equals

Options

A. `(1)/(|x|)` , where `x ne 0`
B. `(1)/(x)` for `|x| gt 1` and `-(1)/(x)` for `|x| lt 1`
C. `-(1)/(x)` for `|x| gt 1` and `(1)/(x)` for `|x| lt 1`
D. `(1)/(x)` for `x gt 0` and `- (1)/(x)` for `x lt 0`

Explanation

For `xgt1," we have "f(x)=|log|x||=log x`
`"or "f'(x)=(1)/(x)`
For `xlt-1," we have "f(x)=|log|x||=log(-x)`
`"or "f'(X)=(1)/(x)`
For `0ltxlt1," we have "f(X)=|log|x||=-log x`
`"or "f'(x)=(-1)/(x)`
For `-1ltxlt0," we have "f(x)=-log (-x)`
`"or "f'(x)=(-1)/(x).`
`"Hence,"f(x)={{:((1)/(x)",",|x|gt1),(-(1)/(x)",",|x|lt1):}`