If we accept the 'more symmetrical' notation of the derivative...
(1)
...do we also have to accept that for a function like is ?... or not? ...
Merry Christmas from Italy
implies that the derivative exists at , which excludes the posibility of something like . Also this can be done with the derivative from the right being at (as implied in my comment about an earlier version using an asymmetric form for the difference quotient).