Derivative

f'
represents the derivative of a function f of one argument.

Derivative[n1, n2, ...][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.

DetailsDetails

  • 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.
  • Derivative[-n][f] represents the ^(th) indefinite integral of f.
  • Derivative[{n1, n2, ...}][f] represents the derivative of 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.

ExamplesExamplesopen allclose all

Basic Examples (1)Basic Examples (1)

Derivative of a defined function:

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This is equivalent to :

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Derivative at a particular value:

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This is equivalent to :

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The second derivative:

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