Appendix 2. The effect on the complete expectation of life of changing the ageing parameter in the Gompertz “law”
Consider the function
Z = - l n t p x
Under Gompertz mortality, this function becomes
Z = x eku du .
For given fixed x but varying senescence parameter k therefore
dZ/dk = x u eku du ,
which on integration and rearrangement yields
dZ/dk = (k t x+t - x+t
W e n o w n o t e t h a t s i n c e t p x = e - Z ,
d ( t p x ) / d k = - t p x d Z / d k .
Under the Gompertz “law”, therefore,
d ( t p x ) / d k = ( - k t t p x x + t + t p x x + t
- x t p x ) / k 2 .
Integration of (A2.6) with respect to t from 0 to yields
d(ex)/dk = [1 - (x + k) ex]/k2 .
Multiplication of (A2.7) by k2 and differentiation with respect to k (keeping x constant) leads to
d2(ex)/dk2 = -[(x+3k) d(ex)/dk + ex] / k2 ,
a process which may be repeated to obtain higher-order derivatives.