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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 2.2.8Section 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]=