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Derivative
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[1][
f
].
f
'' evaluates to Derivative[2][
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[
,
, ... 
][
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, Section 3.5.4.
See also: D, Dt.
Further Examples
Here is the first derivative of the sine function given as a pure function.
In[1]:= 
Out[1]= 
This gives the more familiar functional form.
In[2]:= 
Out[2]= 
Here is a function of two variables.
In[3]:= 
This gives the partial derivative with respect to the first variable once and with respect to the second variable three times.
In[4]:= 
Out[4]= 
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