Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-682214120230101A new fractional derivative operator and applications12771282700310.22075/ijnaa.2022.26841.3423ENMouhssineZakariaDepartment of Mathematics, Faculty of Science Tetouan, Abdelmalek Essaad University, Tetouan, Morocco0000-0002-8939-9585AbdelazizMoujahidDepartment of Mathematics, Faculty of Science Tetouan, Abdelmalek Essaad University, Tetouan, MoroccoMahjoubIkhoubaDepartment of Mathematics, Faculty of Science Tetouan, Abdelmalek Essaad University, Tetouan, MoroccoJournal Article20220412We introduce a new fractional derivative which obeys classical properties including linearity, product rule, power rule, vanishing derivatives for constant functions, chain rule, quotient rule, Rolle's Theorem and the Mean Value Theorem:<br />$$<br />D^\alpha(f)(t)=\lim _{\epsilon \rightarrow 0} \frac{f\left(t e^{\frac{1}{\Gamma(1-\alpha)}} e^{-\alpha}\right)-f(t)}{\epsilon},<br />$$<br />this definition is comfortable with the classical definition of the Caputo Fractional Operator.https://ijnaa.semnan.ac.ir/article_7003_5c8f6ff4033fccd6eadc0fd001ee4ac8.pdf