Precise interpretation of the conformable fractional derivative

Abstract

Let $\alpha \in ]0,1[$. We prove that the existence of the conformable fractional derivative $T_{\alpha }f$ of a function $f:[0,\infty [⟶\mathbb{R}$ introduced by Khalil et al. in [R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014) 65-70] is equivalent to classical differentiability. Precisely the fractional $\alpha$-derivative of $f$ is the pointwise product $T_{\alpha }f(x)=x^{1-\alpha }{f}^{\prime }(x)$, $x>0$. This simplifies the recent results concerning conformable fractional calculus.