`f(x)=x^x , x in (0,oo)` and let ` g(x)` be inverse of f(x) , then `g(x)'` must be

Options

A. `x(1+log x)`
B. `x(1+ log (x))`
C. `(1)/(x(1+log g(x))`
D. non-existent

Explanation

`f(x)=x^(x),g(x)" is inverse of "f(x)`
`therefore" "f(g(x))=x`
`rArr" "f'(g(x))cdotg'(x)=1`
`rArr" "g'(x)=(1)/(f'(g(x)))`
`"Now "log_(e)f(x)=xlog_(e)x`
`therefore" "(f'(x))/(f(x))=x(1)/(x)+log_(e)x=1+log_(e)x`
`therefore" "f'(x)=x^(x)(1+log_(e)x)`
`therefore" "f'(g(x))=(g(x))^(g(x))(1+log g(x))`
`"Now "f(g(x))=g(x)^(g(x))=x`
`therefore" "f'(g(x))=x(1+log g(x))`
`therefore" "g'(x)=(1)/(x(1+log g(x)))`