>>11382745In 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 :)
