Answer
$f'(z)=-ze^{z}-2e^{2z}+1$
Work Step by Step
$f(z)=(1-e^{z})(z+e^{z})$
Differentiate using the product rule
$f'(z)=(1-e^{z})(z+e^{z})'+(z+e^{z})(1-e^{z})'=...$
$...=(1-e^{z})(1+e^{z})+(z+e^{z})(-e^{z})=...$
Evaluate the products and simplify
$...=1-e^{2z}-ze^{z}-e^{2z}=-ze^{z}-2e^{2z}+1$