' represents the derivative of a function f of one argument.
, ... ][
] 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.
' is equivalent to Derivative[
'' evaluates to Derivative[
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[
] and so on to pure functions. Whenever Derivative[
] is generated, Mathematica rewrites it as D[
]. If Mathematica finds an explicit value for this derivative, it returns this value. Otherwise, it returns the original Derivative form.
] 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.
]] will give a numerical approximation to a derivative.
See the Mathematica book: Section 2.2.8, Section 3.5.4.
See also: D, Dt.
Here is the first derivative of the sine function given as a pure function.
This gives the more familiar functional form.
Here is a function of two variables.
This gives the partial derivative with respect to the first variable once and with respect to the second variable three times.
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