Specifications

University of Pretoria etd – Combrinck, M (2006)
apply polynomial-based differentiation techniques to data in the domain where they can be
approximated best by polynomial functions. These very specific properties of being
logarithmically sampled and exhibiting power-law or exponential decays make TEM data
distinctly unsuitable for the most common numerical differentiation schemes such as the
Lagrange formulas (Burden and Faires, 1993) which are based on functions exhibiting
polynomial behaviour.
4.4.2.1 Analytical relationship between derivative transforms in the various domains
The objective is to find an analytical relationship between the derivative of a function in the
linear domain (V’(t)) and the derivatives calculated in the semi-logarithmic (g’(t)) and
logarithmic domains (h’(ln(t))) respectively. The strategy is simply to apply the chain rule
of differentiation to functions presenting the data in the different domains.
Semi-logarithmic domain
In the semi-logarithmic domain the natural logarithms of the V values are taken and treated
as a function of (linear) time. It is presented by the function g(t)=ln(V(t)).
)8.4()(')()('
)(ln)(
tgtVtV
tVtg
⋅=∴
=
Logarithmic domain
In the logarithmic domain the natural logarithms of both the V values and times are taken
and the function h(ln(t))=g(t)=ln(V(t)) is plotted against ln(
t). A substitution of variables
is made to simplify the derivation.
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)9.4(
1
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ln'
ln
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lnln
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ln
t
tVthtV
eV
e
eVxh
eeV
eV
xh
eVDxh
eVxh
x
ettx
eVth
tVth
x
x
x
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x
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t
⋅⋅=∴
=⋅⋅
⋅=
=
=
==
=
=
Then
andlet Now,
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