Solution 9.10.3.8 S G a = a G @G @a = a K(s + a) s(s + b) @ @a K(s + a) s(s + b) = as(s + b) K(s + a)  K s(s + b)  = a (s + a) ;; T c (s) = K(s + a) s(s + b) 1+ K(s + a) s(s + b) = K(s + a) s 2 +(b+ K)s + aK : S T c a = a T c @T c @a = a K(s + a) s 2 +(b+ K)s + aK @ @a K(s + a) s 2 +(b+ K)s + aK = a(s 2 +(b + K)s + aK) K(s+ a) " ;K 2 (s + a) [s 2 +(b+ K)s + aK] 2 + K s 2 +(b + K)s + aK # = a(s 2 +(b + K)s + aK) s + a " ;Ks;Ka+ s 2 +(b+ K)s + aK [s 2 +(b + K)s + aK] 2 # = as(s + b) s 2 +(b+ K)s + aK ;; S G b = b G @G @b = b K(s + a) s(s + b) @ @b K(s + a) s(s + b) = bs(s + b) K(s + a)  ;sK(s + a) (s 2 + bs) 2  = ;bs s(s + b) ;; S T c b = b T c @T c @b = b K(s + a) s 2 +(b+ K)s + aK @ @b K(s + a) s 2 +(b + K)s + aK = b(s 2 +(b+ K)s + aK) K(s + a)  ;sK(s + a) [s 2 +(b+ K)s + aK] 2  = ;bs s 2 +(b+ K)s + aK S G K = K G @G @a = K K(s + a) s(s + b) @ @K K(s + a) s(s + b) = Ks(s + b) K(s + a)  (s + a) s(s + b)  = 1 S T c K = K T c @T c @K = K K(s + a) s 2 +(b+ K)s + aK @ @K K(s + a) s 2 +(b+ K)s + aK = (s 2 +(b + K)s + aK) (s + a) " ;K(s + a) 2 [s 2 +(b+ K)s + aK] 2 + K s 2 +(b + K)s + aK # = K(s 2 +(b+ K)s + aK) s + a " ;(s 2 +2as + a 2 )+s 2 +(b + K)s + aK [s 2 +(b + K)s + aK] 2 # = K(s 2 +(b+ K)s + aK) s + a " (K + b;2a)s + aK ;a 2 [s 2 +(b+ K)s + aK] 2 # = K[(K + b;2a)s + aK ;a 2 ] (s + a)(s 2 +(b+ K)s + aK)