gives a numerical approximation to the derivative of expr with respect to x at the point x0.


gives a numerical approximation to the n^(th) derivative of expr.


  • To use ND, you first need to load the Numerical Calculus Package using Needs["NumericalCalculus`"].
  • The expression expr must be numeric when its argument x is numeric.
  • ND[expr,x,x0] is equivalent to ND[expr,{x,1},x0].
  • ND is unable to recognize small numbers that should in fact be zero. Chop may be needed to eliminate these spurious residuals.
  • The following options can be given:
  • MethodEulerSummethod to use
    Scale1size at which variations are expected
    Terms7number of terms to be used
    WorkingPrecisionMachinePrecisionprecision to use in internal computations
  • Possible settings for Method include:
  • EulerSumuse Richardson's extrapolation to the limit
    NIntegrateuse Cauchy's integral formula
  • With Method->EulerSum, ND needs to evaluate expr at x0.
  • If expr is not analytic in the neighborhood of x0, then the default method EulerSum must be used.
  • The option Scale->s is used to capture the scale of variation when using Method->EulerSum.
  • When the value of the derivative depends on the direction, the default is to the right. Other directions can be chosen with the option Scale->s, where the direction is s.
  • The option Terms->n gives the number of terms to use for extrapolation when using Method->EulerSum.
  • With Method->NIntegrate, the expression expr must be analytic in a neighborhood of the point x0.
  • The option Scale->r specifies the radius of the contour of integration to use with Method->NIntegrate.


open allclose all

Basic Examples  (1)

Click for copyable input
Click for copyable input
Click for copyable input

Scope  (1)

Generalizations & Extensions  (1)

Options  (7)

Applications  (1)

Properties & Relations  (3)

Neat Examples  (1)

See Also

NLimit  NSeries  D  NResidue