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 ?

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I dunno lol.

Well if f=8N^3, then f'=24N^2, so the time it takes to calculate f goes as N^3 and the time it takes to calculate f' goes as N^2, right? 

Or do you mean long it takes to actually find the derivative?  I can't believe that's related at all to how long it takes to calculate the function itself.  It would take days for a computer to calculate N^2000! but I can find the derivative in a second. (Errr... about two seconds actually, in Maple)