# 2.11  Finite Difference Approximations of Derivatives

Suppose the derivative of a function f : →  is needed at a specific point x[0]. If an analytic expression for f ′ is unavailable, the derivative can be approximated based upon a finite difference:6

[2.131]

The scalar h should be small, but not so small that roundoff error distorts the result. Often, the value f{x[0]) is already known. If valuing f at two additional points will be computationally expensive, one valuation can be avoided by using the alternative approximation:

[2.132]

We call [2.131] a central approximation and [2.132] a forward approximation. These generalize to multiple dimensions. A partial derivative of a function f : n →  can be approximated at point x[0] with a central approximation:

[2.133]

where ei is a vector whose only nonzero component is its ith component, which equals 1. The corresponding forward approximation is

[2.134]

###### Exercises
2.18

Consider f : 2 → 2:

[2.135]

Value the Jacobian matrix of f at x = (1, 1) three different ways:

1. analytically,
2. with a central approximation,
3. with a forward approximation.

In items (b) and (c), use h = .00001.