A New Fractional Derivative with Classical Properties

Abstract

We introduce a new fractional derivative which obeys classical properties including: linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, the Rolle's Theorem and the Mean Value Theorem. The definition, \[ D^\alpha (f)(t) = \lim_{\epsilon \rightarrow 0} \frac{f(te^{\epsilon t^{-\alpha}}) - f(t)}{\epsilon}, \] is the most natural generalization that uses the limit approach. For $0\leq \alpha < 1$, it generalizes the classical calculus properties of polynomials. Furthermore, if $\alpha = 1$, the definition is equivalent to the classical definition of the first order derivative of the function $f$. Furthermore, it is noted that there are $\alpha-$differentiable functions which are not differentiable.