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f(x)=x^x , x in (0,oo) and let g(x) be...

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

A

`1/(x(1+lnx))`

B

`1/(x(1+ln(g(x))))`

C

`1/(g(x).(1+ln(g(x)))`

D

non existent

Text Solution

Verified by Experts

The correct Answer is:
B
We have `f(g(x))=g(x)^(g(x))=x`
also `g(f(x))=x`
`impliesg^(')(f(x)).f^(')(x)=1impliesg^(')(f(x))=1/(f^(')(x))`
`impliesg^(')(f(x))=1/(x^(x).(1+lnx))`
`impliesg^(')(f(g(x)))=1/((g(x))^(g(x)).(1+ln(g(x))))`
`implies g^(')(x)=1/(x(1+lng(x)))`
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