Research Article A Novel Theoretical Investigation of the Abu-Shady–Kaabar Fractional Derivative as a Modeling Tool for Science and Engineering Francisco Martínez 1 and Mohammed K. A. Kaabar 2,3 1 Department of Applied Mathematics and Statistics, Technological University of Cartagena, Cartagena 30203, Spain 2 Gofa Camp, Near Gofa Industrial College and German Adebabay, Nifas Silk-Lafto, Addis Ababa 26649, Ethiopia 3 Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia Correspondence should be addressed to Mohammed K. A. Kaabar; mohammed.kaabar@wsu.edu Received 27 June 2022; Accepted 15 September 2022; Published 26 September 2022 Academic Editor: Sania Qureshi Copyright © 2022 Francisco Martínez and Mohammed K. A. Kaabar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A newly proposed generalized formulation of the fractional derivative, known as Abu-Shady–Kaabar fractional derivative, is investigated for solving fractional differential equations in a simple way. Novel results on this generalized definition is proposed and verified, which complete the theory introduced so far. In particular, the chain rule, some important properties derived from the mean value theorem, and the derivation of the inverse function are established in this context. Finally, we apply the results obtained to the derivation of the implicitly defined and parametrically defined functions. Likewise, we study a version of the fixed point theorem for α-differentiable functions. We include some examples that illustrate these applications. The obtained results of our proposed definition can provide a suitable modeling guide to study many problems in mathematical physics, soliton theory, nonlinear science, and engineering. 1. Introduction Fractional calculus is theoretically considered as a natural extension of classical differential calculus, which has attracted many researchers, both from a more theoretical point of view and for its diverse applications in sciences and engineering. Thus, from a more theoretical perspective, various definitions of fractional derivatives have been initi- ated. Fractional definitions try to satisfy the usual properties of the classical derivative; however, the only property inher- ent in these definitions is the property of linearity. On the contrary, some of the drawbacks that these derivatives pres- ent can be located in the following: (i) The Riemann-Liouville derivative does not satisfy D α a ð1Þ =0, if α is not a natural number (ii) Fractional derivative statements do not possess some of the fundamental properties of classical derivatives, such as the product rule, the quotient rule, or the chain rule (iii) These derived proposals, in general, do not satisfy D α D β f = D α+β f (iv) The definition of the Caputo derivative implies that the function f must be differentiable in the ordinary sense More information on this definition of fractional deriva- tive can be found in [1, 2]. The locally formulated fractional derivative is established through certain quotients of increments. In this sense, Khalil et al. [3] introduced a locally defined derivative, called con- formable derivative. Some of the inconveniences that the previous fractional derivatives presented have been success- fully solved via this definition. Thus, for example, the afore- mentioned rules for the derivation of products and quotients of two functions or the chain rule are properties that have been satisfied by the conformable derivative. The physical Hindawi Computational and Mathematical Methods in Medicine Volume 2022, Article ID 4119082, 8 pages https://doi.org/10.1155/2022/4119082