ABSTRACT
From Eq. (5.18), x = x(s). Thus,
f did ds ( .r (x) e:' -(P,,(x)) = P,,(x) = -d (P,,(s))-d 5.19)
From Eq. (5.18), ds I dx h
Thus, Eq. (5.19) gives
Substituting Eq. (5.16) into Eq. (5.21) and differentiating gives I I I 2 I
P,,(x) =,y1./o + 2" [(s - I) + s]i1 fa + 6[(s - 1)(s - 2) + s(s - 2) + s(s - 1)]i13;Q + ... J
Simplifying Eq. (5.22) yields
P I ( ) _ I (Air 2s - 1 A2j( 3s 2
- 6s + 2 A3'j( ) X -- 00+--0 0+ 0 0+'"
" h' 2 6
The second derivative is obtained as foIlows:
Substituting Eq. (5.23) into Eq. (5.24), differentiating, and simplifying yields
/I I d nI I 2 3 P,,(x) = hds (r,,(s)) = h2 (i1 fa + (s - I) i1 '.fa + ...)
(5.20)
(5.21)
(5.23)
(5.24)
(5.25)
Higher-order derivatives can be obtained in a similar manner. Recall that i1"f becomes less and less accurate as n increases. Consequently, higher-order derivatives become increasingly less accurate.
