f' represents the derivative of a function f of one argument.
Derivative[, , ... ][f] is the general form, representing a function obtained from f by differentiating times with respect to the first argument, times with respect to the second argument, and so on.
f' is equivalent to Derivative[f].
f'' evaluates to Derivative[f].
You can think of Derivative as a functional operator which acts on functions to give derivative functions.
Derivative is generated when you apply D to functions whose derivatives Mathematica does not know.
Mathematica attempts to convert Derivative[n][f] and so on to pure functions. Whenever Derivative[n][f] is generated, Mathematica rewrites it as D[f[#]&, #, n]. If Mathematica finds an explicit value for this derivative, it returns this value. Otherwise, it returns the original Derivative form.
Example: Cos' .
Derivative[-n][f] represents the n indefinite integral of f.
Derivative[, , ... ][f] represents the derivative of f[, , ... ] taken times with respect to . In general, arguments given in lists in f can be handled by using a corresponding list structure in Derivative.
N[f'[x]] will give a numerical approximation to a derivative.
See The Mathematica Book: Section 2.2.8 and Section 3.5.4.
See also: D, Dt.