A critical analysis of the conformable derivative

  • Ahmed A. AbdelhakimEmail author
  • José A. Tenreiro Machado
Original Paper


We prove that conformable “fractional” differentiability of a function \(f:[0,\infty [\,\longrightarrow \mathbb {R}\) is nothing else than the classical differentiability. More precisely, the conformable \(\alpha \)-derivative of f at some point \(x>0\), where \(0<\alpha <1\), is the pointwise product \(x^{1-\alpha }f^{\prime }(x)\). This proves the lack of significance of recent studies of the conformable derivatives. The results imply that interpreting fractional derivatives in the conformable sense alters fractional differential problems into differential problems with the usual integer-order derivatives that may no longer properly describe the original fractional physical phenomena. A general fractional viscoelasticity model is analysed to illustrate this state of affairs. We also test the modelling efficiency of the conformable derivative on various fractional models. We find that, compared with the classical fractional derivative, the conformable framework results in a substantially larger error.


Fractional derivative Fractional differential equations Fractional analysis Viscoelasticity 

Mathematics Subject Classification

26A33 34A08 74D05 



The authors are grateful for the valuable comments of the anonymous referees that improved the presentation of this manuscript.

Compliance with ethical standards

Conflict of interest

The authors declares that there is no conflict of interest.

Human participants or animals

The study does not involve any human participants or animals and does not require any form of a consent.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Ahmed A. Abdelhakim
    • 1
    Email author
  • José A. Tenreiro Machado
    • 2
  1. 1.Mathematics Department, Faculty of ScienceAssiut UniversityAssiutEgypt
  2. 2.Department of Electrical EngineeringInstitute of Engineering of Polytechnic of PortoPortoPortugal

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