Consider a function& f (x) = x^ x , x ∈ [1, ∞). If g (x) is the...

$f\left(x\right) = x^{x} ⇒ log f\left(x\right) = xlog x$
Differentiating with respect to $x$, we get,

$f^{′}$\left(x\right) = x^{x} $\left(1 + \log x\right)$
Now, by property $g^{′} \left(f \left(x\right)\right) = \dfrac{1}{f^{′} \left(x\right)} {∵ g\left(x\right)\text{ is }\text{ the }\text{ inverse }\text{ of } f \left(x\right)}$$<br />Putting$x = 2$<br />$g^{'}\left(f\left(2\right)\right) = \dfrac{1}{f^{'}\left(2\right)}⇒ g^{'} \left(4\right) =\dfrac{1}{{2}^{2} \left(1 + \log 2\r right)}=\dfrac{1}{4}\left(\log_{2e} e\right)$</p>$