The usual derivative is a local derivative in the sense that the value of the derivative at one point only depends on the value of the function in a neighborhood of that point. But the fractional derivative is not a local operator and cannot be due to some general theoretical result .

So the definition depends on the domain of definition of the functions under scrutiny. This is not the same definition if we are looking at functions defined on RR or on [0,1][0,1] or on [0,∞)[0,∞) and of course the derivative of say sinsin is not the same in these three cases. Same for the derivative of the constant function.

Please, you nay check

https://mathoverflow.net/questions/285186/why-are-there-so-many-fractional-derivatives

for more discussion

Thank you @Maged Gumaan Bin-Saad

There are also definitions of fractional derivative with a local character, there are not only global ones, think of it this way, the objective was to create a definition of derivative of arbitrary order and that would solve problems. They all have advantages and disadvantages, the trick is to know what problem you want to solve to choose which one suits you. I apologize for this unprofessional answer, in my opinion your question carries a rather long explanation. Not to mention that opinions are divided.

@Héctor José Thank you very much!

the fractional derivative has several definitions each definition depends on the theme for example our definition the motivation is to solve the singularity in the algebra the colombeau

@A. Charities thank you!