Answer
$f'(z)=-\dfrac{1}{(z-1)^{2}}e^{z/(z-1)}$
Work Step by Step
$f(z)=e^{z/(z-1)}$
Differentiate using the chain rule:
$f'(z)=[e^{z/(z-1)}](\dfrac{z}{z-1})'=...$
Use the quotient rule to find $(\dfrac{z}{z-1})'$:
$...=[e^{z/(z-1)}][\dfrac{(z-1)(z)'-(z)(z-1)'}{(z-1)^{2}}]=...$
$...=[e^{z/(z-1)}][\dfrac{(z-1)(1)-(z)(1)}{(z-1)^{2}}]=...$
Simplify:
$...=\dfrac{z-1-z}{(z-1)^{2}}e^{z/(z-1)}=-\dfrac{1}{(z-1)^{2}}e^{z/(z-1)}$
