**13.3****
****Differentiation** (.NET, C#, CSharp, VB, Visual Basic, F#)

The Differentiate() method on **OneVariableFunction**
(Section 13.1)
computes the derivative of a function at a given *x*-value.
For example, if f is **OneVariableFunction**,
this code estimates the derivative at 0:

Code Example – C# calculus

double d = f.Differentiate( 0 );

Code Example – VB calculus

Dim D As Double = F.Differentiate(0)

**NOTE—****Class
Polynomial provides a method for constructing the exact derivative of
a polynomial. See ****Section ****13.4****
for more information.**

To perform differentiation, every **OneVariableFunction**
has an **IDifferentiator** object associated
with it. **NMath** provides class
**RiddersDifferentiator**, which
computes the derivative of a given function at a given *x*-value
by Ridders' method of polynomial extrapolation, and implements the **IDifferentiator** interface.

Extrapolations of higher and higher order are produced.
Iteration stops when either the estimated error is less than a specified
error* tolerance*, the error estimate
is significantly worse than the previous order, or the *maximum
order* is reached.

The default **IDifferentiator** for a **OneVariableFunction** is an instance of
**RiddersDifferentiator**. To
achieve more control over how differentiation is performed, you can instantiate
your own **RiddersDifferentiator**. For
instance, this code uses the Tolerance property
to set the error tolerance to a non-default value, and the MaximumOrder property to set the maximum order,
then calls the Differentiate() method to
differentiate function f at π:

Code Example – C# calculus

var ridders = new RiddersDifferentiator();

ridders.Tolerance = 1e-6;

ridders.MaximumOrder = 20;

double d = ridders.Differentiate( f, Math.PI );

Code Example – VB calculus

Dim Ridders As New RiddersDifferentiator()

Ridders.Tolerance = "1e-6"

Ridders.MaximumOrder = 20

Dim D As Double = Ridders.Differentiate(F, Math.PI)

Setting the error tolerance to a value less than zero ensures that the Ridders differentiation is of the maximum order:

Code Example – C# calculus

var ridders = new RiddersDifferentiator();

ridders.Tolerance = -1;

double d = ridders.Differentiate( f, 1 );

Code Example – VB calculus

Dim Ridders As New RiddersDifferentiator()

Ridders.Tolerance = -1

Dim D As Double = Ridders.Differentiate(F, 1)

Read-only properties are provided for accessing information about a derivative approximation, once it has been computed:

● ErrorEstimate gets an estimate of the error of the derivative just computed.

● Order gets the order of the final polynomial extrapolation.

● ToleranceMet gets a boolean value indicating whether or not the error estimate for the derivative approximation is less than the tolerance.

● Tableau gets a matrix of successive approximations produced while computing the derivative. Successive columns in the matrix contain higher orders of extrapolation; successive rows decreasing step size.

For instance, this code checks whether the error tolerance was met before proceeding:

Code Example – C# calculus

var ridders = new RiddersDifferentiator();

double d = ridders.Differentiate( f, Math.PI );

if ( ridders.ToleranceMet ) {

// Do something here...

}

Code Example – VB calculus

Dim Ridders As New RiddersDifferentiator()

Dim D As Double = ridders.Differentiate(F, Math.PI)

If Ridders.ToleranceMet Then

' Do something here...

End If