We give interior estimates for first derivatives of solutions to a type of complex Monge-Ampère equations in convex domains. We also show global estimates for first derivatives of solutions in arbitrary domains. These global estimates are then used to show interior regularity of solutions to the complex Monge-Ampère equations in hyperconvex domains having a bounded exhaustion function which is globally Lipschitz. Finally we give examples of domains which have such an exhaustion function and domains which do not.