Question

Consider a function $f\left(x\right)=x^{x}, \, \forall x\in \left[1 , \in fty\right)$ . If $g\left(x\right)$ is the inverse function of $f\left(x\right)$ , then the value of $g^{'} \left(4\right)$ is equal to

• A. $log_{2} e$
• B. $\frac{1}{2}log_{2 e} e$
• C. $\frac{1}{4}log_{2 e} e$
• D. $\frac{1}{2}log_{e} 2 e$

Answer 1

Answer: (C)

$f\left(x\right)=x^{x}\Rightarrow log f \left(x\right)=xlog ⁡ x$
Differentiating with respect to $x$ , we get,
$f^{'} \left(x\right) = x^{x} \left(1 + log x\right)$
Now, by property $g^{\prime}(f(x))=\frac{1}{f^{\prime}(x)}$ { $\because g\left(x\right)$ is the inverse of $f\left(x\right)$ }
Putting $x=2$
$g^{'}\left(f \left(2\right)=\frac{1}{f^{'} \left(2\right)}\Rightarrow g^{'}\left(4\right)=\frac{1}{2^{2} \left(1 + log 2\right)}=\frac{1}{4}\left(log\right)_{2 e} e\right)$