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, \[ D^α(f)(t) = \lim_{ε\rightarrow 0} \frac{f(te^{εt^{-α}}) - f(t)}ε, \] is the most natural generalization that uses the limit approach. For $0\le \alpha <1$, 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.

8 pages, 1 figure, submitted to Journal of American Mathematical Society

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