Can derivatives be defined without a limit (as are integrals defined in Apostol's Calculus) 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?
 A: See the following two papers, which I found rather quickly (less than 30 seconds) by googling:
[1] Oved Shisha, Derivative without limit, Journal of Mathematical Analysis and Applications 113 #1 (January 1986), 280-287.
[2] Mordechai Falkowitz, Oved Shisha, and Nehad N. Morsi, Note on derivative without limit, Journal of Mathematical Analysis and Applications 127 #2 (1 November 1987), 595-597.
A: The problem with any other definition of the derivative of a function is that it is hiding it limit nature. One can says, as someone said in other answer, that the derivative is the slope of the "tangent" at a point or that it is "the best linear approximation" to the function. 
But all these notions rely, under the hood, in the concept of limit, so a derivative cannot be defined rigorously without the use of limits. The same for integrals.
So any definition of derivative that doesnt use limits (or some notion that include the concept of limit, by example supremum, nets and other set-theoretic constructs) cannot be said that are rigorous or formal definitions, more like informal or intuitive definitions.
In short: you can define in mathematics things in many ways, but not all of these definitions can be considered formal or rigorous.
EDIT: my point of view is similar to this one, relative to the difference between intuitive notion and formal definition.
A: Don't know about the book/author you're talking about. However you could define the derivative at one point as the best linear approximation of the function. This could certainly be done and will be equivalent to the typical definition of the derivative. Not sure however how much could be said about derivatives without ever using limits.
Maybe you would have a definition, but coultn't use it for much.
