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)

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