The geometrical meaning of ordinary derivative is simple and intuitive: For smooth function f which is differentiable at x, the local behavior of f around point x can be approximated by using the ordinary derivative: f(x + h) ≈ f(x) + hf′(x). The ordinary derivative gives the linear approximation of smooth function. Here we expect the fractional derivative to have the similar geometrical meaning. We hope for non-differentiable functions, the fractional derivative could give some kind approximation of its local behavior. A SIMPLE DEFINITION DIRECTLY FROM GEOMETRICAL MEANING: We expect that the fractional derivative could give nonlinear (power law) approximation of the local behavior of non-differentiable functions f(x + h) ≈ f(x) + h^α f^(α)(x)/Γ(α + 1) in which the function f is not differentiable because df ≈ (dx)^α so the classical derivative df/dx will diverge. Note that the purpose of adding the coefficient Γ(α + 1) is just to make the formal consistency with the Taylor series.

Thanks for the information.

I think you can construct a geometric interpretation of the local fractional derivatives in a natural way.

The attachment can illustrate this.

regards

Thanks.