Question

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

• A. $e^x$
• B. $e^x+x$
• C. $e ^{\left(x+e^x\right)}$
• D. $e ^{( x +\ln x )}$

Answer 1

Answer: (C)

$\text { Let } f(x)=y \Rightarrow x=f^{-1}(y)=g(y) \Rightarrow x=e^{e^y} $
$\Rightarrow \frac{d x}{d y}=e^{e^y} \cdot e^y=e^{e^y+y}=g^{\prime}(y) $
$\text { hence } g^{\prime}(x)=e^{e^x+x}$