<p>A new fractional derivative operator with a non-singular generalized exponential kernel and its extension, which includes a singular kernel, as well as the associated fractional integral operator, have recently been introduced. In this paper, we utilize the Laplace transform as an effective and useful tool to investigate and study the characterization, properties and relationships of generalized exponential kernel fractional derivatives. We discuss the unique existence of solutions to IVPs with differential equations incorporating the considered fractional derivatives and introduce a sufficient condition for existence. In addition, we provide solutions for some IVPs involving the studied fractional derivatives. Next, as a case study, we formulate a fractional extension of the logistic equation that includes the studied extended derivatives, and explore its dynamic behavior. This study focuses on the features of the generalized exponential kernel fractional derivative operator, especially the features of the extended version which provide useful suggestions regarding the modeling issue, that can later be used in fractional calculus.</p>