  The Chebyshev points are commonly used for spectral differentiation in
non-periodic domains. The rounding error in the Chebyshev approximation to the
$n$-the derivative increases at a rate greater than $n^{2m}$ for the $m$-th
derivative. The mapping technique of Kosloff and Tal-Ezer (\emph{J. Comp.
Physics}, vol. 104 (1993), p. 457-469) ameliorates this increase in rounding
error. We show that the argument used to justify the choice of the mapping
parameter is substantially incomplete. We analyze rounding error as well as
discretization error and give a more complete argument for the choice of the
mapping parameter. If the discrete cosine transform is used to compute
derivatives, we show that a different choice of the mapping parameter yields
greater accuracy.
