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ND

 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 derivative of expr.
• 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:
 Method EulerSum method to use Scale 1 size at which variations are expected Terms 7 number of terms to be used WorkingPrecision MachinePrecision precision to use in internal computations
• Possible settings for Method include:
 EulerSum use Richardson's extrapolation to the limit NIntegrate use 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 , 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 .
 Scope   (1)
The expression and evaluation point may be complex:
ND is threaded element-wise:
 Options   (7)
Use the default Method->EulerSum when expr is not analytic in the neighborhood of x0:
Check:
An incorrect answer is obtained with :
Here is a derivative where the default method works poorly:
The correct answer is:
In this case the expression is analytic, so will work well:
Use Scale->s to capture the region of variation:
The scale of variation is around .01:
A value of Scale->s which is too large can be compensated by increasing the number of terms:
Use Scale to specify directional derivatives. The left and right derivatives of the nonanalytic function x:
Check:
Complex directions may also be specified:
Check:
Use the option Scale to avoid regions of non-analyticity when the method used is NIntegrate:
Shrinking the radius avoids the essential singularity at x=1:
Check:
Increasing the number of terms may improve accuracy. Here is a somewhat inaccurate approximation:
Check:
Increasing the number of terms produces a more accurate answer:
Increasing the number of terms further can produce nonsense due to numerical instability:
Combining an increase in the number of terms with a higher working precision often will reduce the error:
High-order derivatives with Method->EulerSum experience significant subtractive cancellation:
Using a higher working precision and additional terms produces an accurate answer:
For this problem, with default options produces a correct answer:
Higher-order derivatives will again experience numerical instability:
Increasing WorkingPrecision will improve the accuracy:
An alternative is to increase the radius of the contour of integration:
 Applications   (1)
ND is useful for differentiating functions which are only defined numerically. Here is such a function:
Here is the derivative of f[a, b][t] with respect to b evaluated at {a, b, t}={1, 2, 1}:
ND can be used as an aid in developing and testing a more robust function for finding the derivative:
Check:
The option uses Cauchy's integral formula to compute derivatives:
The equivalent computation can be performed using NResidue:
Mathematica has built-in code to compute derivatives of numerical functions:
The built-in numerical derivative code can be used. However, it is unable to capture the rapid oscillations: