No, really you don't need Taylor series at all; see for example Conal Elliott's "Beautiful Differentiation"[1], which shows how to handle arbitrary derivatives of multidimensional functions and never uses Taylor series for anything; all you need to know is that d/dx sin(x) = cos(x), etc. Just as a warning, Conal's paper is quite dense and is probably not a good introduction.

[1] http://conal.net/papers/beautiful-differentiation/