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A New Fractional Derivative with Classical Properties

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 $0\leq α< 1$, it generalizes the classical calculus properties of polynomials. Furthermore, if $α= 1$, the definition is equivalent to the classical definition of the first order derivative of the function $f$. Furthermore, it is noted that there are $α-$differentiable functions which are not differentiable.