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ND

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

ND[expr, x, x0]
    gives a numerical approximation to the derivative of expr with respect to x at the point x0.
ND[expr, {x, n}, x0]
    gives a numerical approximation to the n-th derivative.
ND[{e1, e2, ...}, x, x0]
    threads element-wise over the first argument.

Default Method -> EulerSum uses Richardson extrapolation of forward, direction-Scale finite differences (works for non-analytic expr; needs integer n >= 0). Method -> NIntegrate uses Cauchy's integral formula via NResidue (needs expr analytic near x0; allows fractional/complex order). ND cannot recognize small numbers that should be zero -- Chop if needed.

Options: Method (EulerSum | NIntegrate), Scale (step size / contour radius / complex direction, default 1), Terms (EulerSum extrapolation terms, default 7), WorkingPrecision, PrecisionGoal, MaxRecursion.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= ND[Exp[x], x, 1]
Out[1]= 2.71828

In[2]:= ND[Cos[x]^3, {x, 2}, 0]
Out[2]= -3.0

In[3]:= ND[Sin[x], x, Pi I]
Out[3]= 11.592 + 1.32527e-10*I

In[4]:= ND[{Exp[x], Sin[x]}, x, 1]
Out[4]= {2.71828, 0.540302}

In[5]:= ND[Re[Cos[I y]], y, 1]          (* non-analytic: use EulerSum *)
Out[5]= 1.1752

In[6]:= ND[Abs[x], {x, 1}, 0, Scale -> 1 + I]
Out[6]= 0.707107 - 0.707107*I

In[7]:= ND[Sin[100 x], x, 0, Scale -> 1/100]
Out[7]= 100.0

In[8]:= ND[Exp[x^2], {x, 4}, 0, Method -> NIntegrate]
Out[8]= 12.0 - 3.3723e-15*I

Implementation notes

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= ND[Sin[x], x, 1]
Out[1]= 0.540302

In[1]:= ND[Gamma[x], x, 1]
Out[1]= -0.577216

In[1]:= ND[BesselJ[0, x], x, 2]
Out[1]= -0.576725

In[1]:= ND[Tan[x], {x, 2}, 1]
Out[1]= 11.4484

Notes

ND[expr, x, x0] numerically differentiates expr at x = x0. The first case recovers Cos[1] = 0.540302. The Gamma example gives Gamma'[1] = -EulerGamma, since PolyGamma[0, 1] = -EulerGamma. The Bessel example uses the identity BesselJ[0, x]' = -BesselJ[1, x], so the value is -BesselJ[1, 2]. The {x, 2} form takes the second derivative. The default Method -> EulerSum applies Richardson extrapolation to finite differences; Method -> NIntegrate uses Cauchy's integral formula and allows fractional or complex orders. ND cannot recognise small numbers that should be zero — Chop if needed.