A New Fractional Derivative with Classical Properties

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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, Dα(f)(t) = ε→ ₀ f(teεᵗ⁻α) - f(t)ε, is the most natural generalization that uses the limit approach. For 0≤ 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.