What's the derivative? I've seen some meanings of the derivative but I'm wondering if there is a more general context other than: (Quoting from "On proof and progress in Mathematics")


*

*Infinitesimal (dy/dx is not a ratio?).

*Symbolic, Ex. the derivative of $x^n$ is $nx^{n-1}$, the derivative of $\sin(x)$ is $\cos(x)$.

*Logical: using the formal definition of limit.

*Geometric: the derivative is the slope of a line tangent to the graph of the function.

*Rate: Ex. The instantaneous speed of $f(t)$, when t is the time.

*Approximation: The derivative of a function is the best linear approximation to the function near a point (if someone can deepen into this point, it would be great).


Also the derivative can be seen as a linear transformation and therefore it has a matrix representation. I've heard that the derivative is the inverse operation of the indefinite integral but a professor told me that this only holds for functions that fulfil the fundamental theorem of calculus.
Are these definitions always true, what's the more general definition of a derivative and why?
 A: *

*"Are these definitions always true" - I'm not sure what it means for a definition to be true or false. Yes, there are several definitions of "derivative" that apply in different contexts, and the nice thing is that they usually agree when the contexts overlap; for example the formal derivative of a polynomial $p$ is a polynomial $p'$ whose value at any (real) point is the slope of $p$ at that point. None of these definitions applies in all contexts.

*"What's the more general definition" - there isn't one that merits being called "the" definition, I'm afraid. The wiki article reviews many generalizations, and there are others.

*"why?" - why what? Maybe that'll help: Ehud de Shalit once told me that mathematicians can only handle linearity, so they linearize everything. Derivatives are a linearization, cohomology is linearization, etc. This emphasizes one aspect of the derivative, which is that it's the linear transformation which most closely resembles a given non-linear one. We take derivatives so we can apply linear algebra when it otherwise seems inapplicable.
A: I think you are missing the point of this part of Thurston's essay.  He exhibits a portion of a putatively infinite sequence of definitions of the derivative.  The implication here is that there is no one most general or best definition: mathematicians will continue to find new ways to view the derivative and new contexts in which derivatives can be defined.  
A: The notion of best linear approximation is the idea that's typically used to generalize the idea of differentiability to higher dimensions.
Another concept you might want to think of is that differentiability is usually a condition that's stronger than continuity.
A: I think in general, a derivative of a function is essentially a limit. (And a limit is precisely defined.)
