Abstract
In order to convince the sceptical reader, we herein give another proof of the fact that the Leibniz rule ) ( ) ( ) ( v D u v u D uv D α α α + =
Highlights
In the following, we shall refer to the definition of the fractional derivative f (α ) (x) ≡ Dα f (x), 0 < α < 1, of a R → R function f (x), given by the expression [1,2]( ) f (α ) (x) = lim ∆α f (x) / hα h↓ 0 with ∑ ∆α f (x) = ∞ k =0 (−1) k αk f (x + (α − k)h)
Proof: We need the expression of the fractional difference ∆α, to apply (1), and in order to get it, we shall refer to the formula (5) which involves ∆(uv)
The above result can be translated as follows: if we cannot use the standard Leibniz rule because there is no standard derivative, we can just try again by substituting fractional derivative for derivative in the standard formula. All this rationale comes from the fact that if a function is not differentiable at x = c, but has a fractional derivative of order α at this point, it is locally equivalent to the function f (x) = f (c) + (x − c)α f (α ) (c) + o(h2α ), (8) α!
Summary
We shall refer to the definition of the fractional derivative f (α ) (x) ≡ Dα f (x) , 0 < α < 1 , of a R → R function f (x) , given by the expression [1,2]. For further reading on fractional derivative, see for instance [3,4,5,6]. This being the case, it is well established and taken for granted that the Leibniz rule D(uv) = (Du)v + u(Dv) is to be generalized in the fractional form. We shall try to answer these questions in the following
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