Question

Let $e^{f\left(x\right)}=\ell nx.$ If $g\left(x\right)$ is the inverse function of $f\left(x\right)$, then $g^{\prime }\left(x\right)$ equals to:

• A. $e^{e^x+x}(e^x+1)$
  
• B. $e^x+x$
• C. $e^{x+e^x}$
• D. $e^{x+1}$

Answer 1

The correct option is A $e^{e^x+x}(e^x+1)$

$e^{f\left(x\right)}=\ln x$

Taking log on both sides.

$f\left(x\right)\ln e=\ln \left(\ln x\right)f\left(x\right)=\ln \left(\ln x\right)Let,f\left(x\right)=yy=\ln \left(\ln x\right)\ln \left(x\right)=e^{y}x=e^{e^y}Replacexbyy\&ybyxy=e^{e^x}g\left(x\right)=e^{e^x}{g}^{{}^{\prime }}\left(x\right)=e^{e^x}.e^x{g}^{{}^{\prime }}\left(x\right)=e^{(e^x+x)}(e^x+1)$