  This work investigates finite differences and the use of interpolation models
to obtain approximations to the first and second derivatives of a function.
Here, it is shown that if a particular set of points is used in the
interpolation model, then the solution to the associated linear system (i.e.,
approximations to the gradient and diagonal of the Hessian) can be obtained in
$\mathcal{O}(n)$ computations, which is the same cost as finite differences,
and is a saving over the $\mathcal{O}(n^3)$ cost when solving a general
unstructured linear system. Moreover, if the interpolation points are formed
using a `regular minimal positive basis', then the error bound for the gradient
approximation is the same as for a finite differences approximation. Numerical
experiments are presented that show how the derivative estimates can be
employed within an existing derivative free optimization algorithm, thus
demonstrating one of the potential practical uses of these derivative
approximations.
