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, $$D^{\alpha }(f)(t)=\underset{ϵ\to 0}{\lim }\frac{f(t{e}^{ϵt^{-\alpha }})-f(t)}{ϵ},$$ 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.