In the Apostol's Calculus (at first) integrals are treated without limits but derivatives are postponed to come after limits, their conceptual foundation.
I want to know that could the author have defined derivatives similar to the he defined integrals? If yes can you write how? If not, what is the conceptual difference between derivatives and integral that makes this happen?
edit: (My question is different from the general question of defining derivatives in terms of other things as it demands a similar way the author defines integrals.)
EDIT2:The author uses the darboux integral,Which is equivalent to the riemann integral;Now it now remains to be seen if there is a way to define a darboux-esque derivative too?