Problem: Derivative of the inverse function

Problem

$y=f\left( x \right)=e^{2x-2}$

1. Find $f'\left( x \right)$ and show that $f'\left( x \right)>0$ for all $x$ (Thus, $f$ is strictly increasing and it has an inverse $x=g(y)$ )
2. Find $g'\left( 1 \right)$ without finding $g$
3. Find $g\left( y \right)$ and then $g'\left( 1 \right)$ and confirm that the result is the same.