Refute the extension of "grünwald–letnikov derivative" to non-integers.

The claim that Grünwald–Letnikov derivative is valid for fractional calculus is disproved by this simple test:

Suppose that the first order differentiation is unknown or difficult to obtain as fractional calculus and also we can only calculate the differentiation at its even orders (the multiples of the second order differentiation)
The limit of the second order differentiation is:
$$\lim_{h \rightarrow 0}\frac {f(x+2h)-2f(x+h)+f(h)} {h^2}$$
By iterating this limit we can construct a formula looks like grünwald–letnikov derivative but it recognize the even order differentiation, so for example:
$$D^{2}e^{-x}=D^{4}e^{-x}=D^{6}e^{-x}=D^{8}e^{-x}=D^{10}e^{-x}=\cdots =D^{2n}e^{-x}=e^{-x}$$
This constructed formula will treat $e^{-x}$ as a neutral function for all its differentiation according to its mean value.
Then if we try to evaluate a non-even order differentiation of $e^{-x}$
We will get that $D^{1}e^{-x}=e^{-x}$ but this result is not true.

Hence the claim of extending grünwald–letnikov derivative to non-integer values is false.