Computational effort of a derivation

How great is the computational effort of a derivation ?

If I calculate e.g. f(x)=N*4N*2N , than the effort would scale with N^3 -but what is the effort for df(x)/dx ?

Multiplying 3 numbers does not scale as N^3.  It scales as ln(N)^3.

I don't think there's an answer for the general case, because there are too many functions.  How are you even characterizing function growth?