In the case of real functions the viewpoint of the derivative as the best linear approximation can be overlooked just because the definition is so simple, but you can't avoid this viewpoint when you try to define derivatives in functions f: R^n -> R for n higher than 1. Technically speaking, in the case R->R the value of the derivative is the slope of the best linear approximation, not the linear map itself, but it could have just as well been defined as a linear map L:R->R such that
lim of (f(x+h)-f(x) - L(h))/h = 0
h->0
I'll leave it as an exercise to see that these two definitions are actually equivalent :)
The fundamental theorem of calculus is also remarkable in the following sense. When I was a child, I tried to think about what exactly the speed means. Sure you can take the ratio of distance with time between two distinct times, but that would be the average speed. What is the speed right now? Turns out the intuitive notion as taking the limit of the average speed as the future time point tends to the present time works out well. And then you might wonder, if you know the average time in the time interval, you know how far you traveled (by multiplying by the time), could you do a similar thing with instantaneous speeds? There are infinitely many speeds, because there are infinitely many time points at which you could have measured the speed, so what do you add and multiply? And what the FTC says is exactly that the intuitive thing to do: to consider instantaneous speeds as average speeds for small time intervals and calculating the distance in that way, while making the intervals smaller and smaller, the limit is remarkably the actual distance that you traveled! That seals the deal: derivatives are the right thing to look at and calculus is based!
