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Derivative (')

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 n1 times with respect to the first argument, n2 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.
  • Derivative[-n][f] represents the n^(th) indefinite integral of f.
  • Derivative[{n1, n2, ...}][f] represents the derivative of f[{x1, x2, ...}] taken ni times with respect to xi. 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.
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