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.